Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations 9789813341982, 9789813341999

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Mathematical Physics Studies

Fumio Hiai

Quantum f-Divergences in von Neumann Algebras Reversibility of Quantum Operations

Mathematical Physics Studies Series Editors Giuseppe Dito, Dijon, France Edward Frenkel, Berkeley, CA, USA Sergei Gukov, Pasadena, CA, USA Yasuyuki Kawahigashi, Tokyo, Japan Maxim Kontsevich, Bures-sur-Yvette, France Nicolaas P. Landsman, Nijmegen, Gelderland, The Netherlands Bruno Nachtergaele, Davis, CA, USA

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Fumio Hiai

Quantum f -Divergences in von Neumann Algebras Reversibility of Quantum Operations

Fumio Hiai Abiko, Chiba, Japan

ISSN 0921-3767 ISSN 2352-3905 (electronic) Mathematical Physics Studies ISBN 978-981-33-4198-2 ISBN 978-981-33-4199-9 (eBook) https://doi.org/10.1007/978-981-33-4199-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To the memory of Dénes Petz (1953–2018)

Preface

After I wrote in 1981 the joint paper [61] with M. Tsukada and M. Ohya on sufficiency of von Neumann subalgebras and the relative entropy in a specialized situation, I wanted to extend the sufficiency notion to general von Neumann subalgebras. But I felt some difficulty in doing that because of the lack of the conditional expectation onto a general von Neumann subalgebra. In the meantime, I was shocked by D. Petz’ remarkable paper [105] in 1986, where he settled the problem mentioned above by making use of the generalized conditional expectation due to L. Accardi and C. Cecchini [1] in 1982. In his 1988 paper [106] Petz followed the same idea to introduce the notion of sufficient (or reversible) channels (more precisely, unital normal 2-positive maps) between von Neumann algebras, and he further collaborated with A. Jenˇcová on the subject in [68, 69]. After the joint work of the 2011 paper [63], I thought that it would be nice to extend the materials for [63] to the von Neumann algebra setting, including a renewal of Petz’ theory on sufficiency and reversibility in von Neumann algebras. But that was not undertaken due to my laziness before I looked at the papers [20, 66], where the sandwiched Rényi divergence, a new Rényi divergence extensively developed in recent years, was extended to the von Neumann algebra setting. Those papers strongly motivated me to develop a general theory of quantum f divergences in von Neumann algebras with applications to the sufficiency and reversibility problem. I planned to first write three papers on three different types of f -divergences (the standard f -divergences, the maximal f -divergences, and the measured f -divergences) in von Neumann algebras. The first two papers were published in [54, 55], which are surveyed in Chaps. 2 and 4 of this monograph. However, I wrote the third one as Chap. 5 instead of publishing a separate paper, so Chap. 5 of the monograph is new. In Chaps. 6–8, the main part of the monograph, I revisit the work [68, 105, 106, 108] with slight extensions and some modifications. Acknowledgments I would like to thank Milán Mosonyi, Anna Jenˇcová, and Hideki Kosaki. I greatly enjoyed joint work with Mosonyi in [58, 63]. Many results and proofs in this monograph are modeled on those in [58, 63] in the finitedimensional case. I thank Mosonyi also for inviting me to lectures for an intensive vii

viii

Preface

course [56] in April 2019 at the Department of Mathematical Analysis, Budapest University of Technology and Economics. Appendix A of this monograph is a short extract from those lecture notes. I learned from Jenˇcová important points in Sects. 3.2 and 3.3, in particular, I am grateful to her for permitting me to include Lemma 3.5, which is due to her. Discussions with Kosaki in recent joint work [57] on operator connections of unbounded positive operators and positive forms were helpful in writing Appendix D. I am grateful to Masayuki Nakamura (Springer Japan) for his constant support, without which it would not have been possible for me to write this monograph. This work was supported in part by JSPS KAKENHI Grant Number JP17K05266. Chiba, Japan October 2020

Fumio Hiai

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Standard f -Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Variational Expression of Standard f -Divergences.. . . . . . . . . . . . . . . . . . 2.3 Properties of Standard f -Divergences . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7 7 10 14

3 Rényi Divergences and Sandwiched Rényi Divergences .. . . . . . . . . . . . . . . . 3.1 Rényi Divergences.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Description of Rényi Divergences .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Sandwiched Rényi Divergences . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19 19 22 26

4 Maximal f -Divergences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Definition and Basic Properties .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Further Properties of Maximal f -Divergences . . .. . . . . . . . . . . . . . . . . . . . 4.3 Minimal Reverse Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41 41 45 47

5 Measured f -Divergences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Variational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Optimal Measurements.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 51 55 65

6 Reversibility and Quantum Divergences . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 6.1 Petz’ Recovery Map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 6.2 A Technical Lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 84 6.3 Preservation of Connes’ Cocycle Derivatives .. . . .. . . . . . . . . . . . . . . . . . . . 88 6.4 Reversibility via Standard f -Divergences . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 6.5 Reversibility via Sandwiched Rényi Divergences . . . . . . . . . . . . . . . . . . . . 100 7 Reversibility and Measurements . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 7.1 Approximation of Connes’ Cocycle Derivatives and Approximate Reversibility . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 7.2 Reversibility via Measurements . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 114

ix

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Contents

8 Preservation of Maximal f -Divergences . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 8.1 Maximal f -Divergences and Operator Connections . . . . . . . . . . . . . . . . . 119 8.2 Proof of the Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 A Preliminaries on von Neumann Algebras . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Introduction of von Neumann Algebras .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Tomita–Takesaki Modular Theory .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 Standard Forms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4 Relative Modular Operators.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.5 τ -Measurable Operators.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.6 Haagerup’s Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.7 Connes’ Cocycle Derivatives . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.8 Kosaki’s Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

131 131 134 137 141 146 148 154 159

B Preliminaries on Positive Self-Adjoint Operators . . . .. . . . . . . . . . . . . . . . . . . . 163 C Operator Convex Functions on (0, 1) .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171 D Operator Connections of Normal Positive Functionals .. . . . . . . . . . . . . . . . . 175 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 191

Chapter 1

Introduction

The notion of quantum divergences has played a significant role in quantum information, which defines important quantum quantities to discriminate between states of a quantum system. A quantum system is mathematically described, in most cases, by an operator algebra A on a Hilbert space, either finite-dimensional or infinite-dimensional, and a quantum divergence is generally given as a function S(ψϕ) of two states or more generally, two positive linear functionals ψ and ϕ on A. This monograph is aimed at presenting a comprehensive survey of quantum f -divergences and showing their significant role in the reversibility problem of quantum operations in the general von Neumann algebra setting. A general and rigorous framework of quantum information should be that of von Neumann algebras, while, in the present circumstances, quantum information literature is mostly presented in the finite-dimensional setting or the matrix setting. Indeed, the reader can consult the textbooks [50, 100, 109, 135] for broad and intensive developments of quantum information in the last quarter century. As widely believed, the von Neumann algebra framework is the most suitable in studying non-commutative (= quantum) probability and integration from the pure mathematical side [32, 99, 118] and also in developing quantum statistical mechanics from the mathematical physics side [22, 23]. The idea provides a good motivation to study quantum information in the von Neumann algebra setting. C ∗ algebras, a more abstract class of operator algebras, are useful as well in studying quantum physics, but discussions in C ∗ -algebras can be, in many cases, reduced to those in von Neumann algebras by taking the GNS (Gelfand–Naimark–Segal) construction associated with a relevant state on a C ∗ -algebra. Furthermore, several well-developed machines are at our disposal in theory of von Neumann algebras such as, starting from the structure theory based on the Tomita–Takesaki theory, the standard forms, the relative modular operators, the non-commutative Lp -spaces, and so on. Those basics of von Neumann algebras are briefly surveyed, for the convenience of the reader, in Appendix A of the monograph as preliminaries for quantum information in the von Neumann algebra setting. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_1

1

2

1 Introduction

In the first half of the monograph we present a comprehensive survey of quantum f -divergences, which are introduced for normal positive linear functionals ψ, ϕ on a von Neumann algebra M with a parametrization of convex functions (often operator convex functions) f on (0, ∞). In Chaps. 2–5 we deal with three different types of quantum f -divergences. First, the standard f -divergences Sf (ψϕ) discussed in Chap. 2 are defined by making use of the relative modular operators due to Araki [11], which are a special case of quasi-entropies introduced by Petz [103, 104]. The most notable example (when f (t) = t log t) is the relative entropy D(ψϕ) having a long history as the quantum version of the Kullback–Leibler divergence in classical theory, first introduced by Umegaki [134] in semifinite von Neumann algebras and extended by Araki [10, 11] to the general von Neumann algebra case. The content of Chap. 2 is a survey from [54], and a main point there is a variational expression (see Theorem 2.5) of Sf (ψϕ) extending Kosaki’s expression [79] of the relative entropy. See [54] for the proofs of the results given in Chap. 2, and more detailed discussions in the finite-dimensional case are found in [58, 63]. The maximal f -divergences  Sf (ψϕ) are discussed in Chap. 4. When specialized to the finite-dimensional case, those f -divergences are defined as the 1/2 −1/2 1/2 1/2 trace value of the operator perspective Dϕ f (Dϕ Dψ Dϕ )Dϕ of the density operators Dψ , Dϕ , whose detailed discussions in the finite-dimensional case are found in [58, 92]. A special example of the maximal f -divergence (when f (t) = t log t) is a variant of the relative entropy introduced by Belavkin and Staszewski [17], denoted by DBS (ψϕ). The content of Chap. 4 is a survey from [55], where, apart from the definition of  Sf (ψϕ), the main results are the integral expression (see Theorem 4.8) and the variational expression (see Theorem 4.17) in terms of reverse tests introduced following Matsumoto’s idea in [92]. Assume that f is an operator convex function on (0, ∞). Among many properties of the quantum f -divergence Sf (resp.,  Sf ), the most important is the monotonicity property (also called the DPI, the data-processing inequality) Sf (ψ ◦ γ ϕ ◦ γ ) ≤ Sf (ψϕ),

 Sf (ψ ◦ γ ϕ ◦ γ ) ≤  Sf (ψϕ)

(1.1)

for any unital normal Schwarz (resp., simply positive) map γ : N → M between von Neumann algebras N, M (in particular, normal completely positive maps or quantum channels). Here, a linear map γ : N → M is a Schwarz map if γ (x ∗ x) ≥ γ (x)∗ γ (x) for all x ∈ N. In Chap. 5 we introduce the measured f -divergences Sfmeas (ψϕ) by the supremum of Sf (M(ψ)M(ϕ)) := n j =1 ϕ(Aj )f (ψ(Aj )/ϕ(Aj )), the classical f -divergence of M(ψ) := (ψ(Aj )) and M(ϕ) = (ϕ(Aj )) over all measurements (or quantum-classical channels) pr M = (Aj )1≤j ≤n in M. The projectively measured f -divergence Sf (ψϕ) is also defined to be the supremum taken with restriction to projective measurements in M. pr The measured f -divergences Sfmeas and Sf are also defined by using  Sf in place of +  Sf because Sf (ψϕ) = Sf (ψϕ) for ψ, ϕ ∈ M∗ if M is an abelian von Neumann algebra. By construction we have the relation Sf . Sf ≤ Sfmeas ≤ Sf ≤  pr

1 Introduction

3

A more intrinsic explanation of the above relation is given at the end of Sect. 5.1. pr For the measured f -divergences Sfmeas and Sf , we discuss in Chap. 5 their basic properties, variational expressions, and the existence of optimal measurements. This part of the measured f -divergences in the von Neumann algebra setting is new, while discussions in the finite-dimensional case are found in [58, 91]. So the presentation of Chap. 5 is mostly self-contained including proofs. Together with the relative entropy, among the most used quantum divergences is the α-Rényi divergence (or the α-Rényi relative entropy) Dα (ψϕ) with parameter α ∈ (0, ∞) \ {1}, which is essentially the standard f -divergence for f (t) = t α (for α > 1) or −t α (for 0 < α < 1). In recent years it has also been widely α (ψϕ) with parameter known that another variant of the α-Rényi divergence D α ∈ [1/2, ∞) \ {1}, called the sandwiched α-Rényi divergence, is equally useful in quantum information, in particular, in quantum state discrimination, see [96, 97] for example. In the finite-dimensional case with the density operators Dψ , Dϕ , the α are given by divergences Dα and D Tr Dψα Dϕ1−α 1 log , α−1 ψ(1) 1−α   1−α α Tr Dϕ2α Dψ Dϕ2α 1 α (ψϕ) := D log . α−1 ψ(1) Dα (ψϕ) :=

The monograph [130] is a good source for Rényi-type quantum divergences and their mathematical backgrounds in the finite-dimensional setting. The sandwiched α has recently been extended to the von Neumann algebra Rényi divergence D setting by Berta, Scholz and Tomamichel [20] and Jenˇcová [66, 67], while the description of Dα in von Neumann algebras is found in [54]. Chapter 3 is a survey from those articles. In particular, Sect. 3.3 contains a proof of the fact that the two α in [20, 66] are equivalent, which is different from that given in definitions of D α was given in [67]. [66]. Moreover, yet another equivalent definition of D In Chaps. 6–8, the second half of the monograph, we develop the reversibility theory for quantum operations between von Neumann algebras via quantum f divergences. Let γ : N → M be a unital normal 2-positive map, and let ψ, ϕ ∈ M∗+ . We say that γ is reversible for {ψ, ϕ} if there exists a unital normal 2-positive map β : M → N such that ψ ◦γ ◦β =ψ

and ϕ ◦ γ ◦ β = ϕ.

When M0 is a von Neumann subalgebra of M, we say that M0 is sufficient for {ψ, ϕ} if there exists a unital normal 2-positive map β : M → M0 such that ψ ◦ β = ψ and ϕ ◦ β = ϕ. Here is a small historical remark on sufficiency in von Neumann algebras. The study of the subject was initiated by Umegaki [133, 134] around 1960 (rather soon after the emergence of non-commutative probability and integration [32, 118]), and the idea was further discussed later in [61, 62]. In these papers,

4

1 Introduction

the sufficiency of a subalgebra M0 was defined by the existence of the common conditional expectation from M onto M0 for {ψ, ϕ}, so discussions were necessarily quite limited because of lack of the conditional expectation onto a general von Neumann subalgebra. But a breakthrough was provided by Petz’ paper [105] in 1986 as referred to shortly. If γ is reversible for {ψ, ϕ}, then the double use of the monotonicity inequality (1.1) gives Sf (ψ ◦ γ ϕ ◦ γ ) = Sf (ψϕ)

(1.2)

for any standard f -divergence Sf with an operator convex function f . Similarly, if a subalgebra M0 is sufficient for {ψ, ϕ}, then Sf (ψ|M0 ϕ|M0 ) = Sf (ψϕ).

(1.3)

The reverse direction of the above is our concern of Chap. 6, that is, we are interested in the problem whether equality (1.2) (resp., (1.3)) with a finite value implies the reversibility of γ (resp., the sufficiency of M0 ) for {ψ, ϕ}. The problem was formerly studied by Petz [105, 106] (also in [101, Chap. 9]) and by Jenˇcová and Petz [68, 69], where the treated quantum divergences were the relative entropy and the transition probability (or equivalently, the standard f -divergence for f (t) = −t 1/2 ). It was assumed in [105, 106] that ψ, ϕ ∈ M∗+ are faithful, while the assumption of ψ being faithful was removed in [68]. Further detailed discussions are found in [63] (also [65]) on the reversibility in the finite-dimensional case via standard f -divergences under a mild assumption on f . Other discussions on sufficiency in von Neumann algebras are found in [89, 90] for example. In Sects. 6.1–6.4 we revisit the sufficiency and reversibility results in [68, 105, 106] with slight improvements and furthermore obtain the reversibility result (see Theorem 6.19) via general standard f -divergences under the assumption that the support of μf has a limit point in (0, ∞), where μf is the representing measure of the integral expression of an operator convex function f . (Note that μf is supported on the whole (0, ∞) when f (t) = t log t for the relative entropy and when f (t) = −t 1/2 for the transition probability.) In recent papers [66, 67] Jenˇcová proved the reversibility theorems via the sandwiched Rényi divergences in von Neumann algebras, which are briefly reported in Sect. 6.5. A significant point in the reversibility theorems in Chap. 6 is that when a unital normal 2-positive map γ : N → M is reversible for {ψ, ϕ}, we have a reverse map β = γϕ∗ : M → N that is canonically defined as a certain dual map of γ with respect to ϕ. The map γϕ∗ is often called Petz’ recovery map and its construction described in Sect. 6.1 is originally due to Accardi and Cecchini [1]. A more extended form of reversibility (or sufficiency) is the approximate reversibility for a sequence of quantum operations βk : Mk → M (or quantum channels with input M and outputs Mk ). When βk : Mk → M (k ∈ N) are unital normal 2-positive maps between von Neumann algebras, we say that (βk : Mk → M)k∈N is approximately reversible for {ψ, ϕ} in M∗+ if there exist unital normal

1 Introduction

5

2-positive maps βk : M → Mk (k ∈ N) such that ψ ◦ αk ◦ βk −→ ψ

and ϕ ◦ αk ◦ βk −→ ϕ

(1.4)

in the σ (M∗ , M)-topology. Then a natural problem is to characterize the approximate reversibility by the approximation of quantum f -divergences lim Sf (ψ ◦ αk ϕ ◦ αk ) = Sf (ψϕ) < +∞ k

(1.5)

for a certain f . In a quantum mechanical system we have final data by measurements (or quantum-classical channels). From this point of view it is reasonable for us to consider the reversibility for {ψ, ϕ} via a measurement channel, that is, to find unital normal positive maps α : A → M and β : M → A with a commutative von Neumann algebra A such that ψ = ψ ◦ α ◦ β and ϕ = ϕ ◦ α ◦ β. It is also meaningful to consider the approximate reversibility via measurement channels, that is, to find unital normal positive maps αk : Ak → M and βk : M → Ak (k ∈ N) with commutative Ak satisfying (1.4). In the case of measurement channels (i.e., Mk ’s are commutative) the approximation for Sf in (1.5) is equivalently stated as Sfmeas (ψϕ) = Sf (ψϕ) < +∞. The approximate reversibility problem was studied by Petz [108]. It was shown in [108] that if the approximate reversibility via measurements as in (1.4) (with commutative Mk ’s) is satisfied for faithful ψ, ϕ, then ψ, ϕ commute. This means that the approximate reversibility via measurements holds only when ψ, ϕ are in the classical position. In Chap. 7 we extend this result to ψ, ϕ with s(ψ) ≤ s(ϕ) (see Theorems 7.10 and 7.11). Furthermore, it was shown in [108] that when δϕ ≤ ψ ≤ δ −1 ϕ for some δ > 0 and Mk ’s are general von Neumann algebras, condition (1.5) for Sf = D, the relative entropy, implies the approximate reversibility in (1.4). We extend this result to Sf with a general f satisfying the same support condition as in Theorem 6.19 (see Theorem 7.8). In the proofs of the reversibility and the approximate reversibility theorems in Chaps. 6 and 7, a main ingredient is to obtain the (approximate) preservation of Connes’ cocycle derivatives, as done in Sects. 6.3 and 7.1. Both Araki’s relative modular operator ψ,ϕ ([11]) and Connes’ cocycle derivative [Dψ : Dϕ]t ([26]) are certain types of the Radon–Nikodym derivative. In the finite-dimensional M = B(H) case with the density operators Dψ , Dϕ , those are given as ψ,ϕ (X) = Dψ XDϕ−1 for X ∈ B(H) (where M = B(H) is represented by the left multiplication on B(H) with the Hilbert–Schmidt inner product) and [Dψ : Dϕ ]t = Dψit Dϕ−it for t ∈ R. So ψ,ϕ and [Dψ : Dϕ]t are essentially in the same vein (roughly speaking, [Dψ : Dϕ]t is the it-power of ψ,ϕ ). A concise account of Connes’ cocycle derivatives is included in Sect. A.7 for the convenience of the reader.

6

1 Introduction

The reversibility problem may be also considered for the maximal f -divergence  Sf , in place of Sf discussed in Sect. 6.4. But it is known [58, Example 4.8] that even in the finite-dimensional case, the equality case (with a finite value) in the monotonicity inequality for  Sf in (1.1) does not necessarily imply the reversibility of γ for {ψ, ϕ}. Even though the situation is negative in that way, it is meaningful to characterize the preservation  Sf (ψ ◦ γ ϕ ◦ γ ) =  Sf (ψϕ)

(1.6)

under a unital normal positive map γ : N → M, which was discussed in [58] in the finite-dimensional case. Our concern in Chap. 8 is to extend the discussions in [58] to the von Neumann algebra case. Among many equivalent conditions, we show that for any ψ, ϕ ∈ M∗+ , if (1.6) holds with a finite value for some nonlinear operator convex function f , then the same equality holds for all operator convex functions f (see Theorem 8.4). In the course of discussions, we are led to consider operator connections (developed by Kubo and Ando [85]) for positive functionals in M∗+ . The idea is interesting on its own, so a somewhat detailed account of operator connections of elements in M∗+ is given in Appendix D separately. The rest of the monograph contains four appendices, which are used in the main body. In the long Appendix A we give concise accounts of selected topics of von Neumann algebras, including the Tomita–Takesaki theory, the standard forms, the relative modular operators, Haagerup’s Lp -spaces, Connes’ cocycle derivatives, and so on. These might be beneficial for the reader who is not familiar with von Neumann algebras. Appendix B contains a few preliminaries on positive self-adjoint operators. Appendix C is concerned with operator convex functions on (0, 1), which turn up in the integral expression of Theorem 4.8 and may be of independent interest. Finally, Appendix D is concerned with operator connections of normal positive functionals, which is a variant of Kubo and Ando’s theory in [85].

Chapter 2

Standard f -Divergences

2.1 Definition Let M be a general von Neumann algebra, and M∗+ be the positive cone of the predual M∗ consisting of normal positive linear functionals on M. Basics of von Neumann algebras are given in Sect. A.1. Throughout this monograph, we consider M in its standard form (M, H, J, P), that is, M is faithfully represented on a Hilbert space H with a conjugate-linear involution J and a self-dual cone P called the natural positive cone, satisfying properties (a)–(d) given in Sect. A.3. Any von Neumann algebra has a standard form, which is unique up to unitary conjugation, see Theorem A.19. Let ψ, ϕ ∈ M∗+ , whose vector representatives in P are denoted by , respectively, so that ψ(x) = , x,

ϕ(x) = , x ,

x ∈ M.

The M-support projection s(ϕ) = sM (ϕ) ∈ M and the M -support projection sM (ϕ) ∈ M are the orthogonal projections onto M (the closure of M := {x : x ∈ M }) and onto M respectively, and similarly for s(ψ) = sM (ψ), sM (ψ). We have the relative modular operator ψ,ϕ introduced in [11], which is described in Sect. A.4. The support projection of ψ,ϕ is sM (ψ)sM (ϕ). We write the spectral decomposition of ψ,ϕ as 

ψ,ϕ =

[0,∞)

t dEψ,ϕ (t).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_2

(2.1)

7

8

2 Standard f -Divergences

Assume that f : (0, ∞) → R is a convex function. The limits f (t) t →∞ t

f (0+ ) := lim f (t),

f (∞) := lim

t 0

exist in (−∞, +∞]. Below we will understand the expression bf (a/b) for a = 0 or b = 0 in the following way: bf (a/b) :=

 f (0+ )b

for a = 0, b ≥ 0,

f (∞)a

for a > 0, b = 0,

(2.2)

where we use the convention that (+∞)0 := 0 and (+∞)a := +∞ for a > 0. In particular, we set 0f (0/0) = 0. The next definition is a specialization of the quasi-entropy introduced in [77, 103] with modifications. Definition 2.1 For each ϕ, ψ ∈ M∗+ and f as above, we define the standard f divergence Sf (ψϕ) of ψ, ϕ by Sf (ψϕ)  :=

f (t) dEψ,ϕ (t) 2 + f (0+ )ϕ(1 − sM (ψ)) + f (∞)ψ(1 − sM (ϕ)),

(0,∞)

(2.3) where we note that the integral above is on (0, ∞) instead of [0, ∞). The integral in (2.3) may be written as , f ( ψ,ϕ )  in terms of the selfadjoint operator f ( ψ,ϕ ) := (0,∞) f (t) dEψ,ϕ (t) on sM (ψ)sM (ϕ)H, although, to be more precise, it should be understood in the sense of a lower-bounded form (see [114]). From the convexity of f , there are a, b ∈ R such that f (t) ≥ a + bt, t > 0. Then expression (2.3) is well defined with values in (−∞, +∞] since 

 f (t) dEψ,ϕ (t)  ≥

(a + bt) dEψ,ϕ (t) 2

2

(0,∞)

(0,∞) 1/2

= asM (ψ)sM (ϕ) 2 + b ψ,ϕ 2 = asM (ψ) 2 + bsM (ϕ)2 = aϕ(sM (ψ)) + bψ(sM (ϕ)) > −∞ 1/2

thanks to J ψ,ϕ = sM (ϕ). By the above computation and (2.3) we also see that Sa+bt (ψϕ) = aϕ(1) + bψ(1).

(2.4)

2.1 Definition

9

Examples 2.2 The following are the special cases of abelian M (classical case) and M = B(H). (1) Let M = L∞ (X, μ) be an abelian von Neumann algebra, where (X, X, μ) is a σ -finite measure space. For every ψ, ϕ ∈ L1 (X, μ)+ ∼ = M∗+ , since ψ,ϕ is the multiplication of 1{ϕ>0} (ψ/ϕ) (see Examples A.14 (1) and A.23 (1)), we have  Sf (ψϕ) =

+

{ψ>0}∩{ϕ>0}

ϕf (ψ/ϕ) dμ+f (0 )





{ψ=0}

ϕ dμ+f (∞)

 {ϕ=0}

ψ dμ,

which equals the classical f -divergence Sf (ψ|ϕ) = X ψf (ψ/ϕ) dμ under the convention (2.2). (2) Let M = B(H) be a factor of type I, where H is an arbitrary Hilbert space. The standard form of B(H) is described in Example A.14 (2). For every ψ, ϕ ∈ B(H)+ ∗ the relative modular operators ψ,ϕ on C2 (H) (the Hilbert space consisting of Hilbert–Schmidt operators) A.23 (2) in is given in Example  terms of the spectral decompositions Dψ = a>0 aPa and Dϕ = b>0 bQb of the density operators of ψ, ϕ. From (A.15) the definition of Sf (ψϕ) in (2.3) is rewritten as Sf (ψϕ)

= bf (ab−1 )Tr Pa Qb + f (0+ )Tr(I − Dψ0 )Dϕ + f (∞)Tr Dψ (I − Dϕ0 ), a,b>0

where Dϕ0 is the support projection of Dϕ . The above coincides with an expression in [58, Proposition 3.2] when dim H < ∞. The next proposition summarizes simple properties of Sf (ψϕ). Proposition 2.3 ([54]) Let ψ, ϕ ∈ M∗+ . (1) Invariance: If α : N → M is a ∗-isomorphism between von Neumann algebras, then Sf (ψ ◦ αϕ ◦ α) = Sf (ψϕ). (2) In the case ψ = 0 or ϕ = 0 or ψ = ϕ, Sf (0ϕ) = f (0+ )ϕ(1),

Sf (ψ0) = f (∞)ψ(1),

(3) Homogeneity: For every λ ∈ [0, ∞), Sf (λψλϕ) = λSf (ψϕ).

Sf (ϕϕ) = f (1)ϕ(1).

10

2 Standard f -Divergences

(4) Additivity: Let M = M1 ⊕ M2 be the direct sum of von Neumann algebras M1 and M2 . If ψi , ϕi ∈ (Mi )+ ∗ for i = 1, 2, then Sf (ψ1 ⊕ ψ2 ϕ1 ⊕ ϕ2 ) = Sf (ψ1 ϕ1 ) + Sf (ψ2 ϕ2 ). (5) Transpose: Let f be the transpose of f defined by f(t) := tf (t −1 ),

t ∈ (0, ∞).

Then Sf (ψϕ) = Sf(ϕψ). (Note that f(0+ ) = f (∞) and f (∞) = f (0+ ).) Example 2.4 The relative entropy is the most used quantum divergence, which is the standard f -divergence for f (t) = t log t on (0, ∞) with the transpose f(t) = − log t. Since f (0+ ) = 0 and f (∞) = +∞, the relative entropy D(ψϕ) of ψ, ϕ ∈ M∗+ [10, 11] is  D(ψϕ) = St log t (ψϕ) =

(0,∞) t

log t dEψ,ϕ (t) 2

+∞

if s(ψ) ≤ s(ϕ), otherwise,

or D(ψϕ) = S− log t (ϕψ). When M = B(H), D(ψϕ) is Umegaki’s relative entropy [134], whose alternative familiar expression is  D(ψϕ) =

Tr Dψ (log Dψ − log Dϕ )

if Dψ0 ≤ Dϕ0 ,

+∞

otherwise,

where Dψ , Dϕ are as in Example 2.2 (2).

2.2 Variational Expression of Standard f -Divergences We extend the variational expression of the relative entropy given in [79] to standard f -divergences. The extended expression will be quite useful to verify various properties of standard f -divergences. Throughout this section, we assume that a function f : (0, ∞) → R is operator convex, i.e., the operator inequality f (λA + (1 − λ)B) ≤ λf (A) + (1 − λ)f (B),

0≤λ≤1

holds for every invertible A, B ∈ B(H)+ of any H. Also, a function h : (0, ∞) → R is said to be operator monotone if A ≤ B ⇒ h(A) ≤ h(B) for every invertible A, B ∈ B(H)+ of any H. It is well-known that an operator monotone function h on (0, ∞) is automatically operator concave (i.e., −h is operator convex). For general theory on operator monotone and operator convex functions, see, e.g., [21, 51]. Note

2.2 Variational Expression of Standard f -Divergences

11

here [58, Proposition A.1] that if f is operator convex on (0, ∞), then so is the transpose f(t) = tf (t −1 ). Recall [87] (also [39, Theorem 5.1] for a more general form) that any operator convex function f on (0, ∞) has the integral expression  f (t) = a + b(t − 1) + c(t − 1) +

(t − 1)2 dμ(s), t +s

2

[0,∞)

t ∈ (0, ∞), (2.5)

where a, b, c ∈ R with c ≥ 0 and μ is a positive measure on [0, ∞) satisfying (1 + s)−1 dμ(s) < +∞ (moreover, a, b, c and μ are uniquely determined). [0,∞) Letting d := μ({0}) ≥ 0 we also write f (t) = a +b(t −1)+c(t −1)2 +d

(t − 1)2 + t

 (0,∞)

(t − 1)2 dμ(s), t +s

t ∈ (0, ∞).

One can easily verify that 

+

s −1 dμ(s),

f (0 ) = a − b + c + (+∞)d + f (∞) = b + (+∞)c + d +

(0,∞)



dμ(s). (0,∞)

For each n ∈ N we define fn (t) := a + b(t − 1) + c  +

[1/n,n]

(t − 1)2 n(t − 1)2 +d t +n t + (1/n)

(t − 1)2 dμ(s), t +s

t ∈ (0, ∞).

(2.6)

Then fn is operator convex on (0, ∞), and we have 

+

fn (0 ) = a − b + c + nd + fn (∞)

 = b + nc + d +

fn (0+ )  f (0+ ),

[1/n,n]

[1/n,n]

s −1 dμ(s) < +∞,

dμ(s) < +∞,

fn (∞)  f (∞),

fn (t)  f (t)

(2.7) (2.8)

(2.9)

12

2 Standard f -Divergences

as n → ∞ for all t ∈ (0, ∞). Moreover, for every ψ, ϕ ∈ M∗+ we have Sfn (ψϕ)  Sf (ψϕ)

as n  ∞.

Next, we define  hn (t) := (0,∞)

t (1 + s) dνn (s), t +s

t ∈ [0, ∞),

where νn is a finite positive measure supported on [1/n, n] given by dνn (s) := c(1 + n)δn + d(1 + n)δ1/n + 1[1/n,n] (s)

1+s dμ(s) s

(2.10)

with the point masses δn at n and δ1/n at 1/n. Then hn is an operator monotone function on [0, ∞), and fn is written as fn (t) = fn (0+ ) + fn (∞)t − hn (t),

t ∈ (0, ∞).

(2.11)

Now, let L be a subspace of M containing 1, and assume that L is dense in M with respect to the strong* operator topology. Our variational expression of Sf (ψϕ) is given as follows: Theorem 2.5 ([54]) Let f be an operator convex function on (0, ∞). For each n ∈ N let fn (0+ ), fn (∞) and νn be given in (2.7), (2.8) and (2.10), respectively. Then for every ψ, ϕ ∈ M∗+ ,  Sf (ψϕ) = sup sup fn (0+ )ϕ(1) + fn (∞)ψ(1) n∈N x(·)

 −

[1/n,n]



ϕ((1 − x(s))∗ (1 − x(s))) + s −1 ψ(x(s)x(s)∗ ) (1 + s) dνn (s) , (2.12)

where the second supremum is taken over all L-valued (finitely many values) step functions x(·) on (0, ∞). To prove this, note from (2.11) and (2.4) that Sfn (ψϕ) = fn (0+ )ϕ(1) + fn (∞)ψ(1) + S−hn (ψϕ)  + = fn (0 )ϕ(1) + fn (∞)ψ(1) − hn (t) dEψ,ϕ (t) 2 . (0,∞)

2.2 Variational Expression of Standard f -Divergences

13

Since hn (0) = h n (∞) = 0, we can apply [79, Theorem 2.2] to the above integral term, so we have  Sfn (ψϕ) = sup fn (0+ )ϕ(1) + fn (∞)ψ(1) x(·)

 −

[1/n,n]



ϕ((1 − x(s))∗ (1 − x(s))) + s −1 ψ(x(s)x(s)∗ ) (1 + s) dνn (s) .

Taking supn of both sides of the above yields (2.12). Example 2.6 Consider f (t) = − log t, whose integral expression in (2.5) is  − log t = −(t − 1) + (0,∞)

(t − 1)2 ds. (t + s)(1 + s)2

Hence, in this case, a = c = d = 0,

b = −1,

dμ(s) =

1 ds, (1 + s)2

f (0+ ) = +∞,

f (∞) = 0.

Moreover, compute fn (0+ ) = 1 +



n 1/n

fn (∞)



= −1 +

1 2 + log n, ds = 2 s(1 + s) n+1 n

1/n

dνn (s) = 1[1/n,n] (s)

2 1 , ds = − 2 (1 + s) n+1

1 ds. s(1 + s)

For every ψ, ϕ ∈ M∗+ the relative entropy D(ϕψ) is St log t (ϕψ) = S− log t (ψϕ) (see Example 2.4), for which one can write expression (2.12) as  D(ϕψ) = sup sup ϕ(1) log n + (ϕ(1) − ψ(1))  −

n∈N x(·)

[1/n,n]

2 n+1



−1 ∗ −1 ∗ ϕ((1 − x(s)) (1 − x(s))) + s ψ(x(s)x(s) ) s ds .

14

2 Standard f -Divergences

This expression resembles the variational expression  D(ϕψ) = sup sup ϕ(1) log n  −

n∈N x(·)

[1/n,∞)



ϕ((1 − x(s))∗ (1 − x(s))) + s −1 ψ(x(s)x(s)∗ ) s −1 ds



in [79, Theorem 3.2].

2.3 Properties of Standard f -Divergences Most of the important properties of standard f -divergences can immediately be verified from the variational expression in Theorem 2.5, similarly to [79] where the relative entropy was treated. In the rest of this section, assume that f is an operator convex function on (0, ∞). Theorem 2.7 ([54]) Let ψ, ϕ, ψi , ϕi ∈ M∗+ for i = 1, 2. (i) Joint lower semicontinuity: The map (ψ, ϕ) ∈ M∗+ × M∗+ → Sf (ψϕ) ∈ (−∞, +∞] is jointly lower semicontinuous in the σ (M∗ , M)-topology. (ii) Joint convexity: The map in (i) is jointly convex and jointly subadditive, i.e., for every ψi , ϕi ∈ M∗+ , 1 ≤ i ≤ k, Sf



k





k  k  ψi  ϕi ≤ Sf (ψi ϕi ).

i=1

i=1

i=1

(iii) If f (0+ ) ≤ 0 and ϕ1 ≤ ϕ2 , then Sf (ψϕ1 ) ≥ Sf (ψϕ2 ). Also, if f (∞) ≤ 0 and ψ1 ≤ ψ2 , then Sf (ψ1 ϕ) ≥ Sf (ψ2 ϕ). (iv) Monotonicity: Let N be another von Neumann algebra and γ : N → M be a unital positive linear map that is normal (i.e., if {xα } is an increasing net in M+ with xα  x ∈ M+ , then γ (xα )  γ (x)) and is a Schwarz map (i.e., γ (x ∗ x) ≥ γ (x)∗ γ (x) for all x ∈ N). Then Sf (ψ ◦ γ ϕ ◦ γ ) ≤ Sf (ψϕ).

(2.13)

In particular, if N is a unital von Neumann subalgebra of M, then Sf (ψ|N ϕ|N ) ≤ Sf (ψϕ).

(2.14)

(v) Peierls–Bogolieubov inequality: Sf (ψϕ) ≥ ϕ(1)f (ψ(1)/ϕ(1)).

(2.15)

2.3 Properties of Standard f -Divergences

15

Assume that f is non-linear and ψ, ϕ = 0. Then equality holds in (2.15) if and only if ψ = (ψ(1)/ϕ(1))ϕ. (vi) Strict positivity: Assume that f is non-linear with f (1) = 0 and ψ(1) = ϕ(1) > 0. Then Sf (ψϕ) ≥ 0, and Sf (ψϕ) = 0 if and only if ψ = ϕ. (vii) Martingale convergence: If {M  increasing net of unital von Neumann  α } is an subalgebras of M such that M = M, then α α Sf (ψ|Mα ϕ|Mα )  Sf (ψϕ). For each ϕ ∈ M∗+ and any projection e ∈ M, we write eϕe for the restriction of ϕ to the reduced von Neumann algebra eMe. The next corollary is easily seen from properties in Theorem 2.7. Corollary 2.8 (1) If e ∈ M is a projection such that sM (ψ), sM (ϕ) ≤ e, then Sf (ψϕ) = Sf (eψeeϕe). (2) If ψi , ϕi ∈ M∗+ , i = 1, 2, and sM (ψ1 ) ∨ sM (ϕ1 ) ⊥ sM (ψ2 ) ∨ sM (ϕ2 ), then Sf (ψ1 + ψ2 ϕ1 + ϕ2 ) = Sf (ψ1 ϕ1 ) + Sf (ψ2 ϕ2 ). (This is a stronger version of Proposition 2.3 (4).) (3) If ω1 , ω2 ∈ M∗+ and Sf (ω1 ω2 ) < +∞, then for every ψ, ϕ ∈ M∗+ , Sf (ψϕ) = lim Sf (ψ + εω1 ϕ + εω2 ). ε 0

In particular, for every ψ, ϕ, ω ∈ M∗+ , Sf (ψϕ) = lim Sf (ψ + εωϕ + εω). ε 0

(2.16)

The relative entropy D(ψϕ) has the scaling property D(λψμϕ) = λD(ψϕ) + λψ(1) log

λ , μ

λ, μ > 0,

from which it is clear that if D(ψϕ) < +∞ then D(λψμϕ) < +∞ for all λ, μ > 0. The next proposition says that this holds for any standard f -divergence, which will be used in the proof of Theorem 7.11 of Sect. 7.2. Proposition 2.9 Let ψ, ϕ ∈ M∗+ . If Sf (ψϕ) < +∞, then Sf (λψμϕ) < +∞ and Sf (λψ + μϕϕ) < +∞ for all λ, μ > 0.

16

2 Standard f -Divergences

Proof Since Sf (λψλμ) = λSf (ψϕ), we may assume μ = 1 for the first assertion. Since λψ,ϕ = λ ψ,ϕ , we have 

λψ,ϕ =

 [0,∞)

λt dEψ,ϕ (t) =

[0,∞)

t dEψ,ϕ (λ−1 t).

Note that  Sf (ψϕ) =

f (t) dEψ,ϕ (t) 2 (0,∞)



+ f (0+ )ϕ(1 − s(ψ)) + f (+∞)ψ(1 − s(ϕ)) < +∞, f (t) dEψ,ϕ (λ−1 t) 2

Sf (λψϕ) = (0,∞)



+ f (0+ )ϕ(1 − s(ψ)) + f (+∞)λψ(1 − s(ϕ))

=

f (λt) dEψ,ϕ (t) 2 (0,∞)

+ f (0+ )ϕ(1 − s(ψ)) + f (+∞)λψ(1 − s(ϕ)), so it suffices to prove that  f (λt) dEψ,ϕ (t) 2 < +∞.

(2.17)

(0,∞)

One can write f (t) = f0 (t) + at + b for t > 0 with an operator convex function f0 on (0, ∞) such that f0 (1) = 0 and f0 (t) ≥ 0 for all t > 0. So we may assume that f (1) = 0 and f (t) ≥ 0 for all t > 0. (1) Assume that λ > 1. Note that g(t) := f (t t)−f (where g(1) = f (1) = 0) is −1 operator monotone and so operator concave on (0, ∞), see [51, Corollaries 2.7.8 ) g(λt ) and 2.5.4]. Hence for every t > 1 one has g(t t −1 ≥ λt −1 so that  f (λt) ≤ Since

 λt −1 2 t −1

λt − 1 t −1

2 f (t),

≤ 2λ2 for all sufficiently large t, one can choose a c > 0 such that f (λt) ≤ 2λ2 f (t) + c,

and so 

 f (λt) dEψ,ϕ (t)  ≤ 2

[1,∞)

t > 1.

t ≥ 1,

2

2λ f (t) + c dEψ,ϕ (t) 2 < +∞.

2.3 Properties of Standard f -Divergences

17

Moreover, for every t ∈ (0, λ−1 ) one has f (λt) ≤

f (t ) t −1

≤

1 − λt f (t) ≤ f (t), 1−t

f (λt ) λt −1

so that

0 < t < λ−1 .

Hence one can choose a d > 0 such that f (λt) ≤ f (t) + d,

0 < t ≤ 1,

and so 

 f (λt) dEψ,ϕ (t)  ≤

{f (t) + d} dEψ,ϕ (t) 2 < +∞.

2

(0,1]

(0,1]

Therefore, (2.17) follows. Next, assume that 0 < λ < 1. One can apply the above argument to the transpose f(t) := tf (t −1 ) and λ−1 > 1. Hence one can choose c , d > 0 such that f(λ−1 t) ≤ 2λ−2 f(t) + c , f(λ−1 t) ≤ f˜(t) + d ,

t ≥ 1, 0 < t ≤ 1.

By replacing t with t −1 these are rewritten as f (λt) ≤ 2λ−1 f (t) + c λt,

f (λt) ≤ λf (t) + d λt,

0 < t ≤ 1, t ≥ 1,

from which (2.17) follows as in the above case where λ > 1. For the second assertion we may assume by the first that λ + μ = 1. Then Theorem 2.7 (ii) gives Sf (λψ + μϕϕ) ≤ λSf (ψϕ) + μSf (ϕϕ) < +∞. The next continuity property is not included in the martingale convergence in Theorem 2.7, because eMe is not a unital von Neumann subalgebra of M. Proposition 2.10 ([54]) Let {eα } be an increasing net of projections in M such that eα  1. Then for every ψ, ϕ ∈ M∗+ , lim Sf (eα ψeα eα ϕeα ) = Sf (ψϕ). α

When f ≥ 0 in Proposition 2.10, from the monotonicity of Sf we see that Sf (eα ψeα eα ϕeα ) is increasing as eα  1. But this is not the case unless f ≥ 0.

Chapter 3

Rényi Divergences and Sandwiched Rényi Divergences

3.1 Rényi Divergences Let M be a general von Neumann algebra given in a standard form (M, H, J, P) as in Chap. 2. The notion of Rényi divergences Dα (ψϕ) for ψ, ϕ ∈ M∗+ where α ∈ [0, ∞) \ {1} is defined as follows: Definition 3.1 Let ψ, ϕ ∈ M∗+ . Since the vector representative of ϕ is in 1/2 1/2 α/2 D( ψ,ϕ ), the domain of ψ,ϕ , note that ∈ D( ψ,ϕ ) for any α ∈ [0, 1]. Define the quantities Qα (ψϕ) for α ≥ 0 as follows: When 0 ≤ α < 1, α/2

Qα (ψϕ) :=  ψ,ϕ 2 ∈ [0, ∞),

(3.1)

and when α > 1, Qα (ψϕ) :=

 α/2  ψ,ϕ 2 +∞

α/2

if sM (ψ) ≤ sM (ϕ) and ∈ D( ψ,ϕ ), otherwise.

(3.2)

Moreover, when α = 1, define Q1 (ψϕ) := ψ(1). Assume that ψ = 0. For each α ∈ [0, ∞) \ {1} the α-Rényi divergence (or α-Rényi relative entropy) Dα (ψϕ) is defined by Dα (ψϕ) :=

Qα (ψϕ) 1 log . α−1 ψ(1)

(3.3)

  In particular, Q0 (αϕ) = ϕ(sM (ψ)) and D0 (ψϕ) = − log ϕ(sM (ψ))/ψ(1) .

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_3

19

20

3 Rényi Divergences and Sandwiched Rényi Divergences

Define convex functions fα on [0, ∞) by  fα (t) :=

tα

if α ≥ 1,

−t α

if 0 < α < 1.

Then for every ψ, ϕ ∈ M∗+ , Qα (ψϕ) is given as Qα (ψϕ) =

 Sfα (ψϕ)

if α ≥ 1,

−Sfα (ψϕ)

if 0 < α < 1.

Indeed, when 0 < α < 1, since fα (0) = fα (∞) = 0, by (3.1),  Sfα (ψϕ) =

(−t α ) dEψ,ϕ (t) 2 = −Qα (ψϕ). (0,∞)

When α = 1, by (2.4), Sf1 (ψϕ) = ψ(1) = Q1 (ψϕ). When α > 1, since fα (0) = 0 and fα (∞) = +∞, by (3.2),  Sfα (ψϕ) =

t α dEψ,ϕ (t) 2 + (+∞)ψ(1 − sM (ϕ)) = Qα (ψϕ). (0,∞)

Therefore, Qα (ψϕ) is essentially a standard f -divergence and so Dα (ψϕ) is a variant of standard f -divergences. Properties of Qα and Dα are summarized in the next theorem, some of which are also found in, e.g., [20, 52, 88, 94, 95] though mostly in the finite-dimensional situation. Many of them are shown based on Theorem 2.7 applied to Sfα . Theorem 3.2 ([54]) Let ψ, ϕ ∈ M∗+ with ψ = 0. (1) Unless sM (ψ) ⊥ sM (ϕ), Qα (ψϕ) > 0 for all α ≥ 0 and the function α ∈ [0, ∞) → log Qα (ψϕ) is convex. (2) Limit values: The limit D1 (ψϕ) := limα1 Dα (ψϕ) exists and D1 (ψϕ) =

D(ψϕ) , ψ(1)

where D(ψϕ) is the relative entropy. If Dα (ψϕ) < +∞ for some α > 1, then limα 1 Dα (ψϕ) = D1 (ψϕ). Moreover, if ψ(1) = ϕ(1), then limα 0 α1 Dα (ψϕ) = D1 (ϕψ). (3) The function α ∈ [0, ∞) → Dα (ψϕ) is monotone increasing. (4) Joint lower semicontinuity: For every α ∈ [0, 2], the map (ψ, ϕ) ∈ (M∗+ \ {0}) × M∗+ −→ Dα (ψϕ) ∈ (−∞, +∞] is jointly lower semicontinuous in the σ (M∗ , M)-topology.

3.1 Rényi Divergences

21

(5) The map (ψ, ϕ) ∈ M∗+ × M∗+ → Qα (ψϕ) is jointly concave and jointly superadditive for 0 ≤ α ≤ 1, and jointly convex and jointly subadditive for 1 ≤ α ≤ 2. Hence, when 0 ≤ α ≤ 1, Dα (ψϕ) is jointly convex on {(ψ, ϕ) ∈ M∗+ × M∗+ : ψ(1) = c} for any fixed c > 0. (6) When 0 ≤ α ≤ 2, the map ϕ ∈ M∗+ → Dα (ψϕ) is convex for any fixed ψ ∈ M∗+ with ψ = 0. (7) Let ψi , ϕi ∈ M∗+ for i = 1, 2. If 0 ≤ α < 1, ψ1 ≤ ψ2 and ϕ1 ≤ ϕ2 , then Qα (ψ1 ϕ1 ) ≤ Qα (ψ2 ϕ2 ). If 1 ≤ α ≤ 2 and ϕ1 ≤ ϕ2 , then Qα (ψϕ1 ) ≥ Qα (ψϕ2 ). Hence, if ϕ1 ≤ ϕ2 , then Dα (ψϕ1 ) ≥ Dα (ψϕ2 ) for all α ∈ [0, 2]. (8) Monotonicity: For each α ∈ [0, 2], Dα (ψϕ) is monotone under unital normal Schwarz maps, i.e., Dα (ψ ◦ γ ϕ ◦ γ ) ≤ Dα (ψϕ) for any unital normal Schwarz map γ : N → M as in Theorem 2.7 (iv). (9) Strict positivity: Let α ∈ (0, ∞) and ψ, ϕ = 0. The inequality Dα (ψϕ) ≥ log

ψ(1) ϕ(1)

(3.4)

holds, and equality holds in (3.4) if and only if ψ = (ψ(1)/ϕ(1))ϕ. If ψ(1) = ϕ(1), then Dα (ψϕ) ≥ 0, and Dα (ψϕ) = 0 if and only if ψ = ϕ. Proof We prove (6) only since all other items are from [54, Proposition 5.3]. In view of (5) we may assume that 1 < α ≤ 2, and let ψ (= 0), ϕ1 , ϕ2 ∈ M∗+ with respective vector representatives , 1 , 2 . Let 0 < λ < 1 and ϕ := λϕ1 + (1 − λ)ϕ2 . What we need to prove is Qα (ψϕ) ≤ Qα (ψϕ1 )λ Qα (ψϕ2 )1−λ .

(3.5)

Since Qα (ψϕi ) > 0 (i = 1, 2) from (1) and (3.2), we may assume that α/2 Qα (ψϕi ) < +∞ for i = 1, 2, so s(ψ) ≤ s(ϕi ) and i ∈ D( ψ,ϕi ). By Proposition A.22 of Sect. A.4 note that s( ψ,ϕi ) = s(ψ)J s(ϕi )J and −1 ϕi ,ψ =

J ψ,ϕi J , where −1 ϕi ,ψ is the inverse with restriction to the support. Hence 1/2

s( ψ,ϕi ) ≥ s(ψ)J s(ψ)J so that  ∈ s( ψ,ϕi )H. Since ψ,ϕi i = , it follows (α−1)/2

from Proposition B.2 that  ∈ D( ψ,ϕi (see Proposition B.2 again) and α/2

(α−1)/2

Qα (ψϕi ) =  ψ,ϕi i 2 =  ψ,ϕi

(1−α)/2

). Hence we find that  ∈ D( ϕi ,ψ (1−α)/2

2 =  ϕi ,ψ

2 ,

Since Proposition A.22 (5) gives ˙ (1−λ)ϕ2 ,ψ = λ ϕ1 ,ψ + ˙ (1 − λ) ϕ2 ,ψ ,

ϕ,ψ = λϕ1 ,ψ +

i = 1, 2.

)

22

3 Rényi Divergences and Sandwiched Rényi Divergences

it follows from Proposition B.10 that (1−α)/2

 ϕ,ψ

(1−α)/2

2 ≤  ϕ1 ,ψ

(1−α)/2

2λ  ϕ2 ,ψ

2(1−λ) = Qα (ψϕ1 )λ Qα (ψϕ2 )1−λ .

As above, note that −1 ϕ,ψ = J ψ,ϕ J and  ∈ s( ψ,ϕ )H (thanks to s(ψ) ≤ (α−1)/2

s(ϕi ) ≤ s(ϕ)). Therefore, we have  ψ,ϕ 1/2

ψ,ϕ ,

 = follows.

(1−α)/2

 =  ϕ,ψ

 < +∞. Since α/2

this implies by Proposition B.2 again that ∈ D( ψ,ϕ ) and (3.5)

Remarks 3.3 (1) In Theorem 3.2 (2) the assumption that Dα (ψϕ) < +∞ for some α > 1 is essential to have limα 1 Dα (ψϕ) = D1 (ψϕ) in the infinite-dimensional case. (2) Theorem 3.2 (6) settles a question raised in [54, Remark 5.4 (3)]. (3) The convexity of Qα for 1 ≤ α ≤ 2 in Theorem 3.2 (5) cannot extend to α > 2 even in the finite-dimensional case and in separate arguments. This implies that the monotonicity property of Dα in (8) fails to hold for α > 2, because the monotonicity of Qα under unital CP maps yields its joint convexity, see Sect. 6.1 for the definition of CP (completely positive) maps. Also, the monotone decreasing of ϕ → Qα (ψϕ) for 1 ≤ α ≤ 2 in (7) cannot extend to α > 2.

3.2 Description of Rényi Divergences When M = B(H) on a finite-dimensional Hilbert space H, the main term Qα of the Rényi divergence Dα in (3.3) has a simple expression   Qα (ψϕ) = Tr Dψα Dϕ1−α

(3.6)

in terms of the density operators Dψ , Dϕ , where Dϕ1−α is defined with restriction to the support of ϕ. In this section we give a similar expression of Qα in the general von Neumann algebra case. It is convenient for us to work in the framework of Haagerup’s Lp -spaces p  of τ -measurable operators affilL (M), which are constructed inside the space N iated with the crossed product N := M σ R of M by the modular automorphism group σt (t ∈ R). Here note that N is a semifinite von Neumann algebra with the canonical trace. See the beginning of Sect. A.5 for operators affiliated with N and Sect. A.6 for Haagerup’s Lp -spaces. Also, recall that M∗ ∼ = L1 (M) by an order1 isomorphic linear bijection ϕ ∈ M∗ → hϕ ∈ L (M), the functional tr on L1 (M) is

3.2 Description of Rényi Divergences

23

defined by tr(hϕ ) = ϕ(1) for ϕ ∈ M∗ , and the standard form of M is given as   M, L2 (M), J = ∗ , L2 (M)+ , where x ∈ M (resp., J xJ ∈ M ) acts as the left (resp., the right) multiplication by x on L2 (M). In particular, for any ϕ ∈ M∗+ the support projection s(ϕ) = sM (ϕ) is equal to s(hϕ ) and it acts as the left multiplication on L2 (M), while the M -support projection sM (ϕ) = J s(ϕ)J does so as the right multiplication by s(ϕ). Let ψ, ϕ ∈ M∗+ with corresponding hψ , hϕ ∈ L1 (M)+ . For each p ≥ 0 define

p Dp (ψ, ϕ) := ξ ∈ L2 (M) : hψ ξ s(ϕ) = ηhpϕ for some η ∈ L2 (M)s(ϕ) .

(3.7)

p

We prove that the domain D( ψ,ϕ ) coincides with Dp (ψ, ϕ) for any p ≥ 0. The next lemma was first observed by Kosaki [75] (see also [54, Lemma A.3]). Lemma 3.4 For every ψ, ϕ ∈ M∗+ and 0 ≤ p ≤ 1/2, p

D( ψ,ϕ ) = Dp (ψ, ϕ). Moreover, if ξ, η ∈ L2 (M) are given as in the RHS of (3.7), then p

p

ψ,ϕ ξ = ψ,ϕ (ξ s(ϕ)) = η.

(3.8)

The next extension to the whole p ≥ 0 is due to Jenˇcová.1 Lemma 3.5 The results of Lemma 3.4 hold for all p ≥ 0. Proof First, note that η ∈ L2 (M)s(ϕ) in the RHS of (3.7) is uniquely determined for each ξ ∈ Dp (ψ, ϕ). Also, recall that s( ψ,ϕ ) = s(ψ)J s(ϕ)J by Proposition A.22 (1), so the first equality in (3.8) is clear. Define D∞ (ψ, ϕ)

:= ξ ∈ L2 (M) : t ∈ R → itψ,ϕ ξ ∈ L2 (M) extends to an entire function . By a familiar regularization technique with Gaussian kernels (which will be also used in the last part of the proof here), it is seen that D∞ (ψ, ϕ) is dense in L2 (M) p and is a core of ψ,ϕ for any p ≥ 0. Let ξ ∈ D∞ (ψ, ϕ), and show that for every p ≥ 0, ξ ∈ Dp (ψ, ϕ) and p

p

( ψ,ϕ ξ )hpϕ = hψ ξ s(ϕ).

(3.9)

1 This is a private communication “A remark on the relative modular operator” (2017, September). The author is grateful to A. Jenˇcová for permitting her proof to be included here.

24

3 Rényi Divergences and Sandwiched Rényi Divergences

When 0 ≤ p ≤ 1/2, (3.9) holds by Lemma 3.4. For every p > 1/2 write p = k/2 + p0 with k ∈ N and p0 ∈ [0, 1/2). Using Lemma 3.4 and Proposition B.2 repeatedly, one has 1/2

p

p−1/2

( ψ,ϕ ξ )hpϕ = ( ψ,ϕ ( ψ,ϕ 1/2

1/2

p−1/2

ξ ))hϕ1/2 hp−1/2 = hψ ( ψ,ϕ ϕ

1/2

p−1

k/2

p

ξ )hp−1/2 ϕ

p−1

= hψ ( ψ,ϕ ( ψ,ϕ ξ ))hϕ1/2 hp−1 = hψ ( ψ,ϕ ξ )hp−1 ϕ ϕ p

0 = · · · = hψ ( ψ,ϕ ξ )hpϕ 0 = hψ ξ s(ϕ).

Therefore, (3.9) has been shown for any p ≥ 0, which implies that D∞ (ψ, ϕ) ⊂ Dp (ψ, ϕ) and (3.8) hold. Now, let p > 1/2 and let Tp be the operator with the domain Dp (ψ, ϕ) defined by Tp ξ := η, which is clearly a linear operator on L2 (M). By the previous paragraph p note that Tp ξ = ψ,ϕ ξ for all ξ ∈ D∞ (ψ, ϕ). We show that Tp is a closed operator with a core D∞ (ψ, ϕ). Then the result follows since D∞ (ψ, ϕ) is also a core of p

ψ,ϕ . So let {ξn } be a sequence in Dp (ψ, ϕ) such that ξn → ξ and Tp ξn → η in . L2 (M). Then by [129, Chap. II, Proposition 26], ξn → ξ and Tp ξn → η in N  is a complete Hausdorff topological ∗-algebra with the measure topology Since N (see [129, Chap. I, Theorem 28]), one has p

p

hψ ξ s(ϕ) = lim hψ ξn s(ϕ) = lim(Tp ξn )hpϕ = ηhpϕ n

n

, so that ξ ∈ Dp (ψ, ϕ) and Tp ξ = η. Hence Tp is closed. To show in N that D∞ (ψ, ϕ) is a core of Tp , let ξ ∈ Dp (ψ, ϕ) and η = Tp ξ . Here, since s(hψ ) = s(ψ), one can assume that ξ = s(ψ)ξ . For n ∈ N set  ξn :=  ηn :=

n π n π

 

∞ −∞ ∞ −∞

e−nt itψ,ϕ ξ dt + ξ(1 − s(ϕ)), 2

e−nt itψ,ϕ η dt. 2

Then ξn ∈ D∞ (ψ, ϕ) and ξn → ξ , ηn → η in L2 (M). From the formula (A.18) of Sect. A.6 it follows that it p −it it p hψ itψ,ϕ ξ = hitψ hψ ξ h−it ϕ = hψ ηhϕ hϕ = ( ψ,ϕ η)hϕ , p

p

t ∈ R.

(3.10)

3.2 Description of Rényi Divergences

25

∞ For each n, to see that Tp ξn = ηn , one can take sequences {ξn,k }∞ k=1 , {ηn,k }k=1 of Riemann sums

 ξn,k :=

mk  2 (k)  (k) n

itl (k)  −n tl(k)

ψ,ϕ ξ + ξ(1 − s(ϕ)), tl − tl−1 e π l=1

 ηn,k :=

mk  2 (k)  (k) n

itl (k)  −n tl(k)

ψ,ϕ η, tl − tl−1 e π l=1

(k) with −∞ < t0(k) < t1(k) < · · · < tm k < ∞, such that ξn,k − ξn 2 → 0 and ηn,k − ηn 2 → 0 as k → ∞. Then it follows from (3.10) that p

p

hψ ξn s(ϕ) = lim hψ ξn,k s(ϕ) = lim ηn,k hpϕ = ηn hpϕ k

k

, so that Tp ξn = ηn for all n. Hence D∞ (ψ, ϕ) is a core of Tp , as desired. in N The following theorem provides an explicit description of Qα (hence the Rényi divergence Dα (ψϕ)) in terms of hψ , hϕ ∈ L1 (M)+ . Theorem 3.6 Let ψ, ϕ ∈ M∗+ . (1) When 0 ≤ α < 1, Qα (ψϕ) = tr(hαψ h1−α ϕ ).

(3.11)

(2) When s(ψ) ≤ s(ϕ) and α > 1, the following conditions are equivalent: 1/2

(i) hϕ

1/2

α/2

∈ D( ψ,ϕ ); (α−1)/2

(ii) hψ ∈ D( ψ,ϕ

); α/2

(α−1)/2

(iii) there exists an η ∈ L2 (M)s(ϕ) such that hψ = ηhϕ

.

If the above conditions hold, then η in (iii) is unique and Qα (ψϕ) = η22 . Proof α/2 1/2

(1) For 0 < α < 1, since hψ hϕ

α/2 (1−α)/2

= (hψ hϕ

α/2

α/2

)hϕ , we have by (3.8)

α/2

2 Qα (ψϕ) =  ψ,ϕ hϕ1/2 2 = hψ h(1−α)/2 ϕ = tr(h(1−α)/2 hαψ h(1−α)/2 ) = tr(hαψ h1−α ϕ ϕ ϕ ). 1/2

(2) Assume that s(ψ) ≤ s(ϕ) and α > 1. Since hϕ

1/2

1/2 1/2

∈ D( ψ,ϕ ) and ψ,ϕ hϕ

1/2 1/2 hψ s(ϕ) = hψ , it follows from Proposition B.2 that (i) α/2 1/2 (α−1)/2 1/2 case ψ,ϕ hϕ = ψ,ϕ hψ . Hence Lemma 3.5 with p

=

⇐⇒ (ii) and in this = (α − 1)/2 implies

26

3 Rényi Divergences and Sandwiched Rényi Divergences

that (ii) ⇐⇒ (iii) and in this case Qα (ψϕ) = η22 . The uniqueness of η in (iii) is obvious. α/2

Remark 3.7 In the above (2), if hψ α/2 (1−α)/2

may write η = hψ hϕ

(α−1)/2

= ηhϕ

with η ∈ L2 (M)s(ϕ), then we

in a formal sense, so that

α/2

α/2

α/2

α 1−α Qα (ψϕ) = hψ h(1−α)/2 2 = tr(hψ h1−α ϕ ϕ hψ ) = tr(hψ hϕ ),

which is the same expression as in (1). The form is the same as (3.6) in the finitedimensional case, with tr instead of the usual trace and hψ , hϕ instead of the density operators. The next corollary supplements Theorem 3.2 (4). Corollary 3.8 When 0 ≤ α < 1, the map (ψ, ϕ) ∈ (M∗+ \ {0}) × M∗+ → Dα (ψϕ) ∈ R is jointly continuous in the norm topology. Proof By (3.11) it suffices to show that (ψ, ϕ) ∈ M∗+ × M∗+ → hαψ h1−α ∈ L1 (M) ϕ + is continuous. Let ψn , ψ, ϕn , ϕ ∈ M∗ be such that ψn − ψ1 → 0 and ϕn − ϕ1 → 0. Then α 1−α α α 1−α α 1−α 1−α hαψn h1−α ϕn − hψ dϕ 1 ≤ (hψn − hψ )hϕn 1 + hψ (hϕn − hϕ )1

≤ hαψn − hαψ 1/α h1−α ϕn 1/(1−α) 1−α + hαψ 1/α h1−α ϕn − hϕ 1/(1−α)

≤ hψn − hϕ α1 hϕn 1−α + hψ α1 hϕn − hϕ 1−α 1 1 −→ 0, where the second inequality is Hölder’s inequality for Haagerup’s Lp -norms (see Theorem A.38) and the third follows from Kosaki’s generalized Powers–Størmer inequality given in the appendix of [59].

3.3 Sandwiched Rényi Divergences When H is finite-dimensional, for every ψ, ϕ ∈ B(H)+ ∗ (ψ = 0) with the density operators Dψ , Dϕ and every α ∈ (0, ∞) \ {1}, the α-sandwiched Rényi divergence introduced in [98, 136] is α (ψϕ) := D

α (ψϕ) Q 1 log , α−1 ψ(1)

3.3 Sandwiched Rényi Divergences

27

where ⎧  1−α α ⎨Tr D 2α D D 1−α 2α ϕ ψ ϕ α (ψϕ) := Q ⎩+∞

if 0 < α < 1 or s(ψ) ≤ s(ϕ),

(3.12)

otherwise.

These sandwiched variants of Rényi divergences have been in active consideration with applications, e.g., [16, 38, 96, 97]. A more general notion of the α-z-Rényi relative entropies was first introduced in [64, Sec. 4.3.3] and further discussed in [15]. In this section we will survey the α-sandwiched Rényi divergences in von Neumann algebras, recently developed in [20, 66, 67]. Definition 3.9 ([66]) Let ψ, ϕ ∈ M∗+ with ψ = 0, and 1 < α < ∞. The αsandwiched Rényi divergence due to Jenˇcová is α(J) (ψϕ) := D

(J) Q 1 α (ψϕ) log , α−1 ψ(1)

where (J) Q α (ψϕ)

:=

 hψ αα,ϕ +∞

if s(ψ) ≤ s(ϕ) and hψ ∈ Lα (M, ϕ), otherwise.

Here, Lα (M, ϕ) = Lα (M, ϕ)η=1/2 is Kosaki’s Lα -space in the symmetric case and  · α,ϕ is the norm on Lα (M, ϕ), more precisely,  · α,ϕ,η with η = 1/2, see Remark A.62 of Sect. A.8. (Here we put the subscript ϕ to specify the dependence on it.) An alternative definition of the sandwiched divergence in von Neumann algebras given in [20] is based on Araki and Masuda’s Lp -norm [13]. So here we briefly recall the definition in [13]. Let ψ, ϕ ∈ M∗+ . For 2 ≤ p ≤ ∞, Araki and Masuda’s 1/2 Lp -norm of the vector representative hψ with respect to ϕ is 1/2

hψ p,ϕ

⎧  1 − 1   ⎨ 2 p 1/2  sup  ω,ϕ hψ : ω ∈ M∗+ , ω(1) = 1 if s(ψ) ≤ s(ϕ), := ⎩ +∞ otherwise. (3.13) 1/2

For 1 ≤ p < 2, Araki and Masuda’s Lp -norm of hψ is  1 − 1   1/2 2 p 1/2  hψ p,ϕ := inf  ω,ϕ hψ : ω ∈ M∗+ , ω(1) = 1, s(ω) ≥ s(ψ) ,

(3.14)

28

3 Rényi Divergences and Sandwiched Rényi Divergences

 12 − p1 1/2   21 − p1  1/2 . (Note that the above where  ω,ψ hψ  means +∞ if hψ is not in D ω,ψ p-norm of vector representatives can be defined under any ∗-representation of M on any Hilbert space H, while here we use the standard representation on Haagerup’s L2 (M).) Definition 3.10 ([20]) For every ψ, ϕ ∈ M∗+ with ψ = 0 and α ∈ [1/2, ∞) \ {1}, the α-sandwiched Rényi divergence due to Berta–Scholz–Tomamichel (called the Araki–Masuda divergence in [20]) is α(BST) (ψϕ) := D

(BST) Q (ψϕ) 1 α log , α−1 ψ(1)

1/2 (BST) where Q (ψϕ) := hψ 2α α 2α,ϕ .

The next theorem is a main result of this section, which was proved by Jenˇcová. The proof below is different from that in [66], see also Remark 3.20 (1) below. Theorem 3.11 ([66]) Let ψ, ϕ ∈ M∗+ with ψ = 0. (1) For every α ∈ (1, ∞) we have (BST) (J) Q (ψϕ) = Q α α (ψϕ).

(3.15)

Hence α(J) (ψϕ). α(BST) (ψϕ) = D D (2) For every α ∈ [1/2, 1) we have  1−α 1−α α (BST) (ψϕ) = tr hϕ2α hψ hϕ2α . Q α

(3.16)

Hence 1−α   1−α 2α 2α α tr h h h 1 ϕ ψ ϕ (BST)  log . Dα (ψϕ) = α−1 ψ(1)

To prove the theorem, we first give two lemmas. Lemma 3.12 Assume that 1 < α < ∞. (1) (2) (3)

α(J) (ψϕ) is jointly lower semicontinuous in ψ, ϕ ∈ M∗+ in the norm topology. D + (J) Q α (ψϕ) is jointly convex in ψ, ϕ ∈ M∗ . (BST) α Q (ψϕ) is jointly convex in ψ, ϕ ∈ M∗+ .

Proof See [66] for the proofs of (1) and (2). (3) Consider a unital ∗-homomorphism (hence CP) γ : M → M ⊕ M given by γ (x) := x ⊕ x. For ψi , ϕi ∈ M∗+ (i = 1, 2) and λ ∈ (0, 1) set ψ := λψ1 ⊕

3.3 Sandwiched Rényi Divergences

29

(1 − λ)ψ2 and ϕ := λϕ1 ⊕ (1 − λ)ϕ2 . Since ψ ◦ γ = λψ1 + (1 − λ)ψ2 and α(BST) (equivalent ϕ ◦ γ = λϕ1 + (1 − λ)ϕ2 , from the monotonicity property of D (BST) α to that of Q ) given in [20, Theorem 14] it follows that (BST) (BST) Q (λψ1 + (1 − λ)ψ2 λϕ1 + (1 − λ)ϕ2 ) ≤ Q (ψϕ). α α So it suffices to prove that (BST) (BST) (BST) (ψϕ) = λQ (ψ1 ϕ1 ) + (1 − λ)Q (ψ2 ϕ2 ). Q α α α

(3.17)

Note that L2 (M ⊕ M) = L2 (M) ⊕ L2 (M) and 1/2

1/2

1/2

hψ = λ1/2 hψ1 ⊕ (1 − λ)1/2 hψ2 ,

hϕ1/2 = λ1/2 hϕ1/2 ⊕ (1 − λ)1/2 hϕ1/2 . 1 2

Let p := 2α ∈ (2, ∞). For every ω ∈ (M ⊕ M)+ ∗ with ω(1) = 1, we can write ω = tω1 ⊕ (1 − t)ω2 , where ωi ∈ M∗+ , ωi (1) = 1 and 0 ≤ t ≤ 1. Since t 1−t

ω1 ,ϕ1 ⊕

ω2 ,ϕ2 , λ 1−λ

ω,ϕ = t ω1 ,λϕ1 ⊕ (1−t )ω2 ,(1−λ)ϕ2 = we have

1 1  12 − p1 1/2   1− 2 2  12 − p1 1/2 2  1/2 2 2 −  ω,ϕ h  = t p λ p  ω ,ϕ h  + (1− t)1− p (1− λ) p  ω2 ,ϕp h1/2 2 . 1 1 ψ1 2 2 ψ2 ψ

Therefore, 1/2

hψ p,ϕ = sup

t,ω1 ,ϕ2

  1 1 1 1 2 − − 1/2 2 1/2 2 1/2 1− 2 2  1− 2 t p λ p  ω2 1 ,ϕp1 hψ1  + (1 − t) p (1 − λ) p  ω2 2 ,ϕp2 hψ2 

(the sup is over 0 ≤ t ≤ 1, ω1 , ω2 ∈ M∗+ , ω1 (1) = ω2 (1) = 1)  1/2 2 1/2 2 1− 2 2  1/2 2 1− 2 = sup t p λ p hψ1 p,ϕ + (1 − t) p (1 − λ) p hψ2 p,ϕ 1

0≤t ≤1

2

1/p  1/2 p 1.2 p = λhψ  + (1 − λ)h  , p,ϕ p,ϕ 1 2 ψ2 1 1/2 p

1/2 p

1/2 p

so that hψ p,ϕ = λhψ1 p,ϕ1 + (1 − λ)hψ2 p,ϕ2 , which is (3.17). Lemma 3.13 Let α ∈ (1, ∞) and ψ, ϕ ∈ M∗+ . Then: α(J) (ψϕ) = limε 0 D α(J) (ψϕ + εψ), and (1) D α(BST) (ψϕ) = limε 0 D α(BST) (ψϕ + εψ) if s(ψ) ≤ s(ϕ). (2) D

30

3 Rényi Divergences and Sandwiched Rényi Divergences

Proof (J) (1) It is clear that Q α (ψψ) = 1. Hence by Lemma 3.12 (2), (J) Q α (ψϕ) + ε (J) . Q α (ψ(ϕ + εψ)/(1 + ε)) ≤ 1+ε Moreover, as easily seen and noted in [66, (14)], α(J) (ψϕ + εψ) + log(1 + ε). α(J) (ψ(ϕ + εψ)/(1 + ε)) = D D Therefore, α(J) (ψϕ + εψ) = lim sup D α(J) (ψ(ϕ + εψ)/(1 + ε)) ≤ D α(J) (ψϕ). lim sup D ε 0

ε 0

α(J) (ψϕ) ≤ lim infε 0 D α(J) (ψϕ + On the other hand, by Lemma 3.12 (1), D εψ). α(BST) is the same as [66, (14)] for D α(J) . Hence as in (2) The scaling property of D the above proof of (1), by Lemma 3.12 (3) we have α(BST) (ψϕ). α(BST) (ψϕ + εψ) ≤ D lim sup D ε 0

So it suffices to prove that α(BST) (ψϕ) ≤ lim inf D α(BST) (ψϕ + εψ), D ε 0

that is, for p := 2α ∈ (2, ∞), 1/2

1/2

hψ p,ϕ ≤ lim inf hψ p,ϕ+εψ .

(3.18)

ε 0

Let εn 0 be arbitrary and let ω ∈ M∗+ . By [11, Lemma 4.1] we have  −1 1/2 1/2 −1 1 + ω,ϕ+εn ψ J s(ϕ)J −→ (1 + ω,ϕ ) J s(ϕ)J

strongly.

Since s(ψ) ≤ s(ϕ), both of ω,ϕ+εn ψ and ω,ϕ have the support projection s(ω)J s(ϕ)J (≤ J s(ϕ)J ). Hence the above strong convergence means that  −1   1/2 1/2 −1 −→ 1 + ω,ϕ 1 + ω,ϕ+εn ψ

strongly.

(3.19)

3.3 Sandwiched Rényi Divergences

31

Moreover, since s(ϕ + ε1 ψ) = s(ϕ), we find by Proposition A.22 (4) of 1/2 1/2 Sect. A.4 that ω,ϕ+ε1 ψ ≤ ω,ϕ+ε2 ψ ≤ · · · . Hence ω,ϕ+ε1 ψ ≤ ω,ϕ+ε2 ψ ≤ · · · by Lemma B.7 of Appendix B. By this and (3.19) we can apply Proposi1/2 tion B.9 to An = ω,ϕ+εn ψ ; then we have  1− 1  12 − p1 1/2 2 1/2  2 ,  ω,ϕ h  = lim  2 p ψ ω,ϕ+εn ψ hψ n→∞

which implies (3.18) in view of (3.13). Proof (Theorem 3.11) (1) If s(ψ) ≤ s(ϕ), then both sides of (3.15) are +∞ by Definitions 3.9 and 3.10. Hence by Lemma 3.13 we may prove equality (3.15) in the case where ψ ≤ 1/2 1/2 λϕ for some λ > 0. Then by Lemma A.24, hϕ = Ahϕ for some A ∈ s(ϕ)Ms(ϕ). Let p := 2α ∈ (2, ∞) and ω ∈ M∗+ with ω(1) = 1. From (A.19) of Sect. A.6 one has 1

−1

1

1/2

−1

1

2 p 2 p

ω,ϕ hψ = ω,ϕ Ahϕ1/2 = hω2

− p1

1

Ahϕp

so that  1− 1 2 1 1  12 − p1 1/2 2 −  ω,ϕ h  = tr hω2 p Ahϕp A∗ hω2 p = tr(h(p−2)/p Ah2/p A∗ ). ω ϕ ψ (p−2)/p

Apply Hölder’s inequality (Theorem A.38) to hω 2/p Ahϕ A∗ ∈ Lp/2 (M) to obtain

∈ Lp/(p−2)(M) and

  tr(h(p−2)/p Ah2/p A∗ ) ≤ Ah2/p A∗ p/2 ω

ϕ

(3.20)

ϕ

(3.21)

(p−2)/p

thanks to hω p/(p−2) = (tr hω )(p−2)/p = 1. On the other hand, set h :=  2/p ∗ p/2 2/p p/2 Ahϕ A /Ahϕ A∗ p/2 . Then h ∈ L1 (M)+ and tr(h) = 1, so that h = hω0 for some ω0 ∈ M∗+ with ω0 (1) = 1. We then have   ∗ tr h(p−2)/p Ah2/p ω0 ϕ A =

 2/p p/2  tr Ahϕ A∗ 2/p ∗  2/p p/2 (p−2)/p = Ahϕ A p/2 . ∗ tr Ahϕ A (3.22)

Combining (3.20)–(3.22) yields hψ 2p,ϕ = Ahϕ A∗ p/2 so that 1/2

2/p

 2α1 ∗ 2α1 α 1/2 1/α ∗ α   hψ 2α 2α,ϕ = Ahϕ A α = hϕ A Ahϕ α .

(3.23)

32

3 Rényi Divergences and Sandwiched Rényi Divergences

Now define 1   1 α h0 := hϕ2α A∗ Ahϕ2α ∈ L1 (M),

for which we have, with 1/α + 1/β = 1, 1

1/α

1

1

hϕ2β h0 hϕ2β = hϕ2α

1 + 2β

1

A∗ Ahϕ2α

1 + 2β

= hϕ1/2 A∗ Ahϕ1/2 = hψ .

This implies that hψ ∈ Lα (M, ϕ) by Theorem A.61 (in the η = 1/2 case) and 1   1 hψ α,ϕ = (tr h0 )1/α = hϕ2α A∗ Ahϕ2α α .

(3.24)

1/2

α By (3.23) and (3.24) we thus obtain hψ 2α 2α,ϕ = hψ α,ϕ , which yields (3.15) by Definitions 3.9 and 3.10. (2) Assume that 1/2 ≤ α < 1. First, note that the RHS of (3.16) is well defined 1−α   1−α α since hϕ2α hψ hϕ2α is in L1 (M). Since −1 ω,ϕ = J ϕ,ω J for every ω, ϕ ∈

M∗+ by Proposition A.22 (3), the definition of hψ 2α,ϕ in (3.14) is rewritten as 1/2

 1 − 1   1/2 2α 2 1/2  hψ 2α,ϕ = inf  ϕ,ω hψ : ω ∈ M∗+ , ω(1) = 1, s(ω) ≥ s(ψ) . Using Lemma 3.4 we find that  1/2 hψ 22α,ϕ = inf η22 : ω ∈ M∗+ , ω(1) = 1, s(ω) ≥ s(ψ), 1−α 1−α  1/2 η ∈ L2 (M), hϕ2α hψ = ηhω2α ,

1/2

1−α

1/2

where we note that hψ s(ω) = hψ 1−α 2α

ηs(ω)hω

(3.25)

(thanks to s(ω) ≥ s(ψ)), ηhω2α

and ηs(ω)22 ≤ η22 . Now let p := α and q := 1−α α

α 1−α

=

so that

1/p − 1/q = 1. By letting b := hω ∈ + expression (3.25) can be rewritten as  1/2 hψ 22α,ϕ = inf η22 : b ∈ Lq (M)+ , bq = 1, s(b) ≥ s(ψ), Lq (M)

 1−α 1/2 η ∈ L2 (M), hϕ2α hψ = ηb1/2 . 1−α  1−α 1/2 2 1/2 Furthermore, let a := hϕ2α hψ  and hϕ2α hψ = va 1/2 be the polar decomposition where v ∈ M is a partial isometry with v ∗ v = s(a) ≤ s(ψ).

3.3 Sandwiched Rényi Divergences

33

Since va 1/2 = ηb1/2 ⇐⇒ a 1/2 = v ∗ ηb 1/2 ⇐⇒ va 1/2 = vv ∗ ηb 1/2 and η2 ≥ v ∗ η2 = vv ∗ η2 , it follows that 1/2 hψ 22α,ϕ = inf η22 : b ∈ Lq (M)+ , bq = 1, s(b) ≥ s(ψ),

η ∈ L2 (M), a 1/2 = ηb1/2 .

(3.26)

Now we show that the RHS of (3.26) is equal to ap . For this we may assume that a = 0 (the result is immediate when a = 0). If b and η are as in the above RHS, then Hölder’s inequality (Theorem A.38) implies that 1/2

1/2

ap = a 1/22p = ηb 1/22p ≤ η2 b1/22q = η2 bq p/2q

= η2 .

−p/q

Furthermore, set η := ap a p/2 and b := ap a p/q ; then it is immediate to check that η ∈ L2 (M), b ∈ Lq (M)+ , bq = 1, a 1/2 = ηb1/2 and η22 = ap . Note that s(η) = s(b) = s(a) ≤ s(ψ). Choose a b0 ∈ Lq (M)+ with 1/2 s(b0 ) = s(ψ) − s(a) and for ε > 0 set ηε := b + εb0 q η and bε := 1/2 = η b 1/2 and η 2 → b + εb0 −1 ε ε ε 2 q (b + εb0 ). Then s(bε ) = s(ψ), a η22 = ap as ε 0. Therefore, the RHS of (3.26) is ap so that   1−α 1−α α 1/α 1/2 hψ 22α,ϕ = ap = tr hϕ2α hψ hϕ2α , which is (3.16) by Definition 3.10 when 1/2 ≤ α < 1. In view of Theorem 3.11 we simply denote the α-sandwiched Rényi divergence by α (ψϕ) = D

α (ψϕ) Q 1 log α−1 ψ(1)

for ψ, ϕ ∈ M∗+ and α ∈ [1/2, ∞) \ {1}, where ⎧ ⎨Q (J) (BST) (ψϕ) = Q α α (ψϕ)  1−α α (ψϕ) = 1−α α Q (BST) ⎩Q α (ψϕ) = tr hϕ2α hψ hϕ2α

if 1 < α < ∞, if 1/2 ≤ α < 1.

(3.27)

(3.28)

Remark 3.14 The expression for 1/2 ≤ α < 1 in (3.28) has a complete resemblance to the matrix case in (3.12). For 1 < α < ∞, if hψ ∈ Lα (M, ϕ) (Kosaki’s Lα space in the symmetric case), then there exists an x ∈ (s(ϕ)Lα (M)s(ϕ))+ such that α−1

α−1

hψ = hϕ2α xhϕ2α and hψ α,ϕ = xα . Hence we may write  1−α 1−α α α (ψϕ) = tr x α = tr hϕ2α hψ hϕ2α Q in a formal sense, which is the same expression as (3.12) again.

34

3 Rényi Divergences and Sandwiched Rényi Divergences

Remark 3.15 In particular, when α = 1/2, note that 1/2 (ψϕ) = −2 log F (ψ, ϕ) , D ψ(1)

1/2 (ψϕ) = F (ψ, ϕ), Q 1/2

1/2

(3.29)

1/2 1/2

where F (ψ, ϕ) := tr(hϕ hψ hϕ )1/2 = hψ hϕ 1 is the so-called fidelity and will be discussed in Example 5.18 of Sect. 5.3. On the other hand, as for the Rényi divergence for α = 1/2 note that Q1/2 (ψϕ) = P (ψ, ϕ), 1/2 1/2

D1/2 (ψϕ) = −2 log 1/2

P (ψ, ϕ) , ψ(1)

(3.30)

1/2

where P (ψ, ϕ) := tr(hϕ hϕ ) = hψ , hϕ  is the so-called transition probability, introduced by Raggio [112] in the von Neumann algebra setting. α are summarized in the next theorem, which are more or α and Q Properties of D less similar to those of Dα and Qα in Theorem 3.2. Theorem 3.16 ([20, 66, 67]) Let ψ, ϕ ∈ M∗+ with ψ = 0. α (ψϕ) is monotone increasing on [1/2, 1) ∪ (1, ∞). (1) The function α → D (2) Limit values: α (ψϕ) = lim D

α1

D(ψϕ) , ψ(1)

α (ψϕ) < +∞ for some where D(ψϕ) is the relative entropy. Moreover, if D  α > 1, then limα 1 Dα (ψϕ) = D(ψϕ)/ψ(1). On the other hand, α (ψϕ) = Dmax (ψϕ), lim D

α→∞

where Dmax (ψϕ) := log inf{t > 0 : ψ ≤ tϕ} is the max-relative entropy introduced in [31] (here inf ∅ = +∞ as usual). α (ψϕ) (3) Joint lower semicontinuity: The map (ψ, ϕ) ∈ (M∗+ \ {0}) × M∗+ → D is jointly lower semicontinuous in the norm topology for every α ∈ (1, ∞) and jointly continuous in the norm topology for every α ∈ [1/2, 1). (4) Monotonicity: For each α ∈ [1/2, ∞) \ {1} and for any unital normal positive map γ : N → M between von Neumann algebras, α (ψ ◦ γ ϕ ◦ γ ) ≤ D α (ψϕ). D α (ψϕ) is jointly convex for 1 < α < ∞ (5) The map (ψ, ϕ) ∈ M∗+ × M∗+ → Q α (ψϕ) is and jointly concave for 1/2 ≤ α < 1. Hence, when 1/2 ≤ α < 1, D jointly convex on {(ψ, ϕ) ∈ M∗+ × M∗+ : ψ(1) = c} for any fixed c > 0.

3.3 Sandwiched Rényi Divergences

35

α and Dα : For every α ∈ [1/2, ∞) \ {1}, (6) Relation between D α (ψϕ) ≤ Dα (ψϕ). D

(3.31)

In particular, F (ψ, ϕ) ≥ P (ψ, ϕ), see (3.29) and (3.30). If ψ(1) = 1, then for every α ∈ (1, ∞), α (ψϕ). D2− 1 (ψϕ) ≤ D

(3.32)

α

Moreover, D2 (ψϕ) ≤ Dmax (ψϕ) ≤ D∞ (ψϕ) := limα→∞ Dα (ψϕ). (7) Let ψi , ϕi ∈ M∗+ for i = 1, 2. If 1/2 ≤ α < 1, ψ1 ≤ ψ2 and ϕ1 ≤ ϕ2 , α (ψ1 ϕ1 ) ≤ Q α (ψ2 ϕ2 ). If 1 < α < ∞, ψ1 ≥ ψ2 and ϕ1 ≤ ϕ2 , then then Q   α (ψϕ1 ) ≥ D α (ψϕ2 ) Qα (ψ1 ϕ1 ) ≥ Qα (ψ2 ϕ2 ). Hence, if ϕ1 ≤ ϕ2 , then D for all α ∈ [1/2, ∞) \ {1}. (8) Strict positivity: Let α ∈ [1/2, ∞) \ {1} and ψ, ϕ = 0. The inequality α (ψϕ) ≥ log ψ(1) D ϕ(1) holds, and equality holds if and only if ψ = (ψ(1)/ϕ(1))ϕ. If ψ(1) = ϕ(1), α (ψϕ) ≥ 0, and D α (ψϕ) = 0 if and only if ψ = ϕ. then D Proof (1) See [20, Lemma 8] and [66, Proposition 3.7]. (2) See [20, Theorem 13, Lemma 9] and [66, Proposition 3.8]. α for 1 < α < ∞ was given in [66, (3) The joint lower semicontinuity of D α (or Q α ) for 1/2 ≤ α < 1, Proposition 3.10]. For the joint continuity of D 1−α

1−α

by (3.16) it suffices to show that (ψ, ϕ) ∈ M∗+ ×M∗+ → hϕ2α hψ hϕ2α ∈ Lα (M) is continuous, whose proof is similar to that of Corollary 3.8 by using a + p p p bp ≤ ap + bp for a, b ∈ Lp (M) (0 < p ≤ 1) in [37, Theorem 4.9 (iii)], Hölder’s inequality in Theorem A.38 and Kosaki’s generalized Powers–Størmer inequality in the appendix of [59]. (4) See [66, Theorem 3.14] and [67, Theorem 4.1] (see also Remark 3.20 (2) α under a unital normal CP map was shown below). The monotonicity of D in [20, Theorem 14] as well. (Note that we have used [66, Theorem 3.14] in the proof of Lemma 3.12 (3).) α for 1 < α < ∞ was given in [66, Corollary 3.16]. For (5) The joint convexity of Q α in (4) means the “reverse” monotonicity 1/2 ≤ α < 1 the monotonicity of D α . Then the joint concavity of Q α for 1/2 ≤ α < 1 can be shown similarly of Q to the proof of [66, Corollary 3.16] (or Lemma 3.12 (3)) by considering γ : α in Theorem 3.11 (2). M → M ⊕ M, γ (x) := x ⊕ x, and the expression of Q (6) See [20, Theorem 12] and [66, Corollary 3.6], and see [20, (95)] for D2 ≤ Dmax . (More information will be given in Remark 3.18 (1) for the proofs of inequalities (3.31) and (3.32).)

36

3 Rényi Divergences and Sandwiched Rényi Divergences

(7) The case 1 < α < ∞ was given in [66, Proposition 3.9]. When 1/2 ≤ α < 1,  1−α  1−α 1−α α 1−α α 2α 2α 2α h h ≤ h h h . Lemma B.7 of Appendix B implies that hϕ2α ψ ϕ ϕ ψ ϕ 1 1 1 1 2 1 1−α

1], by Lemma B.7 again one has hϕ1α Since 1−α α ∈ (0, α    1−α 1−α 1/2 1/2 1/2 1/2 α hψ2 hϕ1α hψ2 ≤ hψ2 hϕ2α hψ2 . Therefore,

1−α

≤ hϕ2α and hence

 1−α   1−α  1−α α 1−α α 1−α 1/2 α 1/2 α 2α 2α 2α tr hϕ2α h h ≤ tr h h h = tr h h h ψ ψ ϕ ϕ ϕ ϕ 1 2 1 1 1 1 1 ψ2 ψ2    1−α 1−α 1−α α 1/2 1/2 α 2α ≤ tr hψ2 hϕ2α hψ2 = tr hϕ2α , 2 hψ2 hϕ2 α (ψ1 ϕ1 ) ≤ Q α (ψ2 ϕ2 ). that is, Q (8) By (1) and (4) for γ : C1 → M one has 1/2 (ψϕ) ≥ −2 log ψ(1) α (ψϕ) ≥ D D

1/2 ϕ(1)1/2

ψ(1)

= log

ψ(1) . ϕ(1)

α (ψϕ) ≤ Dα (ψϕ) = log ψ(1)/ϕ(1) by (6) If ψ = (ψ(1)/ϕ(1))ϕ, then D α (ψϕ) = log ψ(1)/ϕ(1). Conversely, if this and Theorem 3.2 (9), so that D 1/2 (ψϕ) = log ψ(1)/ϕ(1), and ψ = (ψ(1)/ϕ(1))ϕ equality holds, then D follows as in the proof of [54, Corollary 4.2 (1)].  1−α 1−α α α in (3.27) with Q α (ψϕ) = tr hϕ2α hψ hϕ2α Remark 3.17 The expression of D in (3.28) makes sense for all α ∈ (0, 1). In the matrix case, as seen from discussions α (0 < α < 1) under CP trace-preserving maps in [38] that the monotonicity of D α . But we note by [52, Proposition 5.1] is equivalent to the joint concavity of Q α is not jointly concave (even not separately concave in the second variable) that Q α under unital normal CP maps in for 0 < α < 1/2. Thus, the monotonicity of D the above (4) is restricted to α ∈ [1/2, ∞), while that of Dα in Theorem 3.2 (8) is restricted to α ∈ [0, 2]. See also [19, Theorem 7] for violation of the monotonicity α when α < 1/2. property of D Remarks 3.18 (1) Inequality (3.31) was proved in [20, 66] by using the complex interpolation technique, while the proof of (3.32) in [66] is by Hölder’s inequality for Haagerup’s Lp -norms. In the case M = B(H) with dim H < ∞, the inequality α ≤ Dα for any α ∈ (0, ∞) \ {1} is an immediate consequence of the D ALT (Araki–Lieb–Thirring) inequality [12] (also [5]). The ALT inequality was extended to Haagerup’s Lp -norms by Kosaki [80, Theorem 4]. In view of (3.27) and (3.28) note that (3.31) for 1/2 ≤ α < 1 is equivalently written as  1/2 1−α     h hϕ2α  ≥  hα/2 h(1−α)/21/α  , ϕ ψ ψ 2α 2α

3.3 Sandwiched Rényi Divergences

37

which is a special case of Kosaki’s extension of the ALT inequality. But when α > 1, we are not able to derive (3.31) from the ALT inequality in the von Neumann algebra setting. (2) Let ψ, ϕ ∈ M∗+ with ψ = 0. If ψ, ϕ commute in the sense that hψ , hϕ commute (see Lemma 4.20 and Proposition A.56 of Sect. A.7), then α (ψϕ) = Dα (ψϕ) D

for all α ∈ (0, ∞) \ {1}.

α (ψϕ) = tr hα h1−α = Qα (ψϕ) This is obvious for α ∈ (0, 1), since Q ψ ϕ α (ψϕ) ≤ Dα (ψϕ) by (3.28) and (3.11). When α > 1, since D α (ψϕ) < +∞. Then there exists an by (3.31), we may assume that D α−1

α−1

x ∈ (s(ϕ)Lα (M)s(ϕ))+ such that hψ = hϕ2α xhϕ2α . Since hψ , hϕ commute, by taking the commuting spectral resolutions of hψ , hϕ one can show that α/2 (α−1)/2 hψ = ηhϕ with η := x α/2 ∈ s(ϕ)L2 (M)s(ϕ). Theorem 3.6 gives  Qα (ψϕ) = η22 = xαα = hψ αα,ϕ = Q(ψϕ). α for α ∈ (0, 1), At the end of the section we give a variational expression of Q whose special case for α = 1/2 will be used in Example 5.18 of Sect. 5.3. α (ψϕ) := Lemma 3.19 Let 0 < α < 1 and ψ, ϕ ∈ M∗+ . Define Q  1−α 1−α α tr hϕ2α hψ hϕ2α . Then α (ψϕ) ≤ Q

 inf

x∈M++

1−α  α   1−α αtr(hψ x) + (1 − α)tr hϕ2α x −1 hϕ2α 1−α ,

(3.33)

where M++ is the set of positive invertible operators in M. Furthermore, if 1/2 ≤ α < 1 or ψ ≤ λϕ for some λ > 0, then the above inequality becomes an equality, i.e., α (ψϕ) = Q

 inf

x∈M++

1−α  α   1−α αtr(hψ x) + (1 − α)tr hϕ2α x −1 hϕ2α 1−α .

(3.34)

Proof Write Rα (ψϕ) for the RHS of (3.33). For every x ∈ M++ we have 1−α  1−α    α (ψϕ) = h1/2 hϕ2α 2α = h1/2 x 1/2 x −1/2hϕ2α 2α Q ψ ψ 2α 2α   1−α   2α 1/2 ≤ hψ x 1/2 2 x −1/2hϕ2α  2α (by Hölder’s inequality) 1−α

  1−α 1−α  α 1−α  1/2 1/2 α = tr(hψ xhψ ) tr hϕ2α x −1 hσ2α 1−α 1−α  α  1−α ≤ αtr(hψ x) + (1 − α)tr hϕ2α x −1 hϕ2α 1−α ,

α (ψϕ) ≤ Rα (ψϕ). which implies that Q

38

3 Rényi Divergences and Sandwiched Rényi Divergences

To prove the latter assertion, first assume that λ−1 ϕ ≤ ψ ≤ λϕ for some λ > 1. We may assume that ψ, ϕ are faithful. Write β := 1−α α ∈ (0, ∞), so (1+β)(1−α) = 1+β β/2 β/2 1+β −1 β. Since λ hϕ ≤ hϕ hψ hϕ ≤ λhϕ , it follows from (A.24) of Lemma A.58 that there are a0 , b0 ∈ M such that β/2 β/2 hβ/2 ϕ = a0 (hϕ hψ hϕ )

1−α 2

β/2 (hβ/2 ϕ hψ hϕ )

,

1−α 2

= b0 hβ/2 ϕ .

It is immediate to verify that a0 b0 = b0 a0 = 1 so that b0 = a0−1 . Set x0 := a0 a0∗ so that x0−1 = (a0−1 )∗ a0−1 = b0∗ b0 . Since β/2 β/2 β/2 hβ/2 ϕ hψ hϕ = (hϕ hψ hϕ )

1−α 2

β/2 a0∗ hψ a0 (hβ/2 ϕ hψ hϕ )

1−α 2

,

one has a0∗ hψ a0 = (hϕ hψ hϕ )α so that β/2

β/2

β/2 α tr(hψ x0 ) = tr(a0∗ hψ a0 ) = tr(hβ/2 ϕ hψ hϕ ) .

(3.35)

Moreover, since −1 β/2 β/2 ∗ β/2 β/2 β/2 1−α , hβ/2 ϕ x0 hϕ = hϕ b0 b0 hϕ = (hϕ hψ hϕ )

one has −1 β/2 1/β β/2 α = tr(hβ/2 tr(hβ/2 ϕ x0 hϕ ) ϕ hψ hϕ ) .

(3.36)

β/2 β/2 α (ψϕ). Therefore, From (3.35) and (3.36), Rα (ψϕ) ≤ tr(hϕ hψ hϕ )α = Q Qα (ψϕ) = Rα (ψϕ) holds. Now, assume that 1/2 ≤ α < 1, and let ψ, ϕ ∈ M∗+ be general. From the joint α (ψϕ) in Theorem 3.16 (7) and the joint continuity monotonicity of (ψ, ϕ) → Q in Theorem 3.16 (3) we have

α (ψϕ) Q α (ψ + εϕϕ + εψ) = inf Q ε>0

= inf

 inf

ε>0 x∈M++

=

inf

αtr((hψ + εhϕ )x)  1/β  + (1 − α)tr (hϕ + εhψ )β/2 x −1 (hϕ + εhψ )β/2

  1/β inf αtr((hψ + εhϕ )x) + (1 − α)x −1/2 (hϕ + εhψ )β x −1/21/β ,

x∈M++ ε>0

where the second equality follows from the case proved first. It is obvious that tr((hψ + εhϕ )x) tr(hψ x) as ε 0. On the other hand, by Kosaki’s generalized

3.3 Sandwiched Rényi Divergences

39

Powers–Størmer inequality we have β

β

(hϕ + εhψ )β − hβϕ 1/β ≤ (hϕ + εhψ ) − hϕ 1 = εhψ 1 −→ 0 so that x −1/2(hϕ +εhψ )β x −1/2 → x −1/2 hϕ x −1/2 as ε 0 in the measure topology + , where N := M σ ω R, see Sect. A.6. Moreover, since β ∈ (0, 1], note in N by Lemma B.7 that x −1/2 (hϕ + εhψ )β x −1/2 is monotone decreasing as ε 0 + ). Hence by [37, in the sense of Proposition B.4 (equivalently, in the order in N Lemmas 3.4 and 4.8] we have β

x −1/2 (hϕ + εhψ )β x −1/2 1/β = μ1 (x −1/2 (hϕ + εhψ )β x −1/2 )

μ1 (x −1/2 hβϕ x −1/2 ) = x −1/2 hβϕ x −1/2 1/β as ε 0. Therefore, α (ψϕ) = Q =

 inf

x∈M++

inf

x∈M++



αtr(hψ x) + (1 − α)x −1/2 hβϕ x −1/2 1/β 1/β



 −1 β/2 1/β αtr(hψ x) + (1 − α)tr(hβ/2 . ϕ x hϕ )

Next, assume that ψ ≤ λϕ for some λ > 0. From the proofs of Theorem 3.16 (3) and (7) we note that the continuity property of Theorem 3.16 (3) and the monotonicα (ψϕ) hold for all α ∈ (0, 1). Hence we have ity of ψ → Q α (ψϕ) = inf Q α (ψ + εϕϕ) Q ε>0

= inf

 inf

ε>0 x∈M++

=



inf

x∈M++

   −1 β/2 1/β αtr((hψ + εhϕ )x) + (1 − α)tr hβ/2 ϕ x hϕ

   −1 β/2 1/β αtr(hψ x) + (1 − α)tr hβ/2 , ϕ x hϕ

where the second equality holds by the first case. Remarks 3.20 α for α > 1 in Definition 3.9 ([66]) is based on Kosaki’s (1) The definition of D p symmetric L -spaces. In [67] Jenˇcová introduced a sightly different definition α for all α ∈ [1/2, ∞) \ {1} based on Kosaki’s right Lp -spaces of D α(BST) in (see Sect. A.8), and proved that the new definition coincides with D α Definition 3.10 for all α ∈ [1/2, ∞) \ {1}. Therefore, all the definitions of D in [20, 66, 67] are equivalent.

40

3 Rényi Divergences and Sandwiched Rényi Divergences

(2) In [67] the variational expression in (3.34) was used to prove the monotonicity α (1/2 ≤ α < 1) under a unital normal general positive map. This and the of D α (α > 1) together give Theorem 3.16 (4). The same monotonicity in [66] of D variational expression  α−1  α   α−1 α = sup αtr(hψ x) − (α − 1)tr hϕ2α x −1 hϕ2α α−1 Q

(3.37)

x∈M+

for α > 1 was also given in [67, Proposition 3.4]. Hence (3.34) and (3.37) together are the von Neumann algebra version of [38, Lemma 4] in the finitedimensional case.

Chapter 4

Maximal f -Divergences

4.1 Definition and Basic Properties Let M be a von Neumann algebra with its standard form (M, H, J, P) as before. Throughout this chapter, we assume that f is an operator convex function on (0, ∞). For ψ, ϕ ∈ M∗+ we write ψ ∼ ϕ if δϕ ≤ ψ ≤ δ −1 ϕ for some δ > 0, and for convenience we set (M∗+ × M∗+ )∼ := {(ψ, ϕ) ∈ M∗+ × M∗+ : ψ ∼ ϕ}, (M∗+ × M∗+ )≤ := {(ψ, ϕ) ∈ M∗+ × M∗+ : ψ ≤ αϕ for some α > 0}, which are convex sets. We first define the maximal f -divergence for (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ and then extend it to general ψ, ϕ ∈ M∗+ . Definition 4.1 For (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ with respective vector representatives , ∈ P, by Lemma A.24 of Sect. A.3 we have a unique A ∈ s(ϕ)Ms(ϕ) such that  = A . We write Tψ/ϕ for A∗ A ∈ (s(ϕ)Ms(ϕ))+ . Now assume that (ψ, ϕ) ∈ (M∗+ ×M∗+ )∼ , so Tψ/ϕ is a positive invertible operator in s(ϕ)Ms(ϕ). Then we have a self-adjoint operator f (Tψ/ϕ ) in s(ϕ)Ms(ϕ) via functional calculus and define the maximal f -divergence of ψ with respect to ϕ by  Sf (ψϕ) := ϕ(f (Tψ/ϕ )) ∈ R.

(4.1)

When M = B(H) on a finite-dimensional Hilbert space H and ψ, ϕ ∈ B(H)+ ∗ are faithful with the density operators Dψ , Dϕ , the above (4.1) is given as  Sf (ψϕ) = Tr Dϕ1/2 f (Dϕ−1/2 Dψ Dϕ−1/2 )Dϕ1/2 .

(4.2)

This notion of quantum divergence was first introduced by Petz and Ruskai [110] and later developed in more detail in [58, 92]. The above expression © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_4

41

42

4 Maximal f -Divergences

1/2

−1/2

−1/2

1/2

Dϕ f (Dϕ Dψ Dϕ )Dϕ is the operator perspective [35, 36] of Dψ , Dϕ by the function f , which will appear in Sect. 8.1 again. To extend  Sf (ψϕ) to general ψ, ϕ ∈ M∗+ , we need the following: Lemma 4.2 ([55]) Let ψ, ϕ ∈ M∗+ . (1) For every ω ∈ M∗+ with ω ∼ ψ + ϕ, the limit lim  Sf (ψ + εωϕ + εω) ∈ (−∞, +∞]

ε 0

exists, and moreover the limit is independent of the choice of ω as above. (2) If (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ , then  Sf (ψ + εϕ(1 + ε)ϕ). Sf (ψϕ) = lim  ε 0

The lemma says that the RHS of (4.3) is well defined independently of the choice of ω and extends Definition 4.1 for the case ψ ∼ ϕ. Definition 4.3 For every ψ, ϕ ∈ M∗+ define the maximal f -divergence  Sf (ψϕ) by  Sf (ψϕ) := lim  Sf (ψ + εωϕ + εω) ∈ (−∞, +∞] ε 0

(4.3)

Sf (ψ + εωϕ + εω) is defined in for any ω ∈ M∗+ with ω ∼ ψ + ϕ, where  Definition 4.1. The most basic properties of  Sf (ψϕ) are the following: Theorem 4.4 ([55]) (i) Monotonicity: Let N be another von Neumann algebra and γ : N → M be a unital normal positive map. Then for every ψ, ϕ ∈ M∗+ ,  Sf (ψ ◦ γ ϕ ◦ γ ) ≤  Sf (ψϕ). (ii) Joint convexity: For every ψi , ϕi ∈ M∗+ and λi ≥ 0 (1 ≤ i ≤ n),  Sf



n i=1

 n 

n 

Sf (ψi ϕi ). λi ψi  λ ϕ λi  ≤ i i  i=1

i=1

Our proofs of Lemma 4.2 and Theorem 4.4 in [55] go as follows: The properties (i) and (ii) of Theorem 4.4 are first shown restricted to (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ , from which we can prove the convergence results in Lemma 4.2. Then from definition (4.3) the properties of Theorem 4.4 can extend to general ψ, ϕ ∈ M∗+ .

4.1 Definition and Basic Properties

43

Another significant property of  Sf (ψϕ) is the joint lower semicontinuity, which will be given later after developing a general integral formula in the next section. Some other basic properties are summarized in the next proposition. Proposition 4.5 ([55]) Let ψ, ϕ ∈ M∗+ . (1) Let f be the transpose of f (see Proposition 2.3 (5)). Then  Sf (ϕψ). Sf(ψϕ) =  (2) Let ψi , ϕi ∈ M∗+ (i = 1, 2). If s(ψ1 ) ∨ s(ϕ1 ) ⊥ s(ψ2 ) ∨ s(ϕ2 ), then  Sf (ψ1 + ψ2 ϕ1 + ϕ2 ) =  Sf (ψ1 ϕ1 ) +  Sf (ψ2 ϕ2 ). (3) When (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ ,  Sf (ψϕ) = lim  Sf (ψ + εϕϕ). ε 0

When (ϕ, ψ) ∈ (M∗+ × M∗+ )≤ ,  Sf (ψϕ) = lim  Sf (ψϕ + εψ). ε 0

(4) If s(ψ) ≤ s(ϕ) and f (∞) = +∞, then  Sf (ψϕ) = +∞. If s(ϕ) ≤ s(ψ) and f (0+ ) = +∞, then  Sf (ψϕ) = +∞. (5) If f (0+ ) < +∞, then  Sf (ψϕ) = lim  Sf (ψϕ + εψ). ε 0

If f (∞) < +∞, then  Sf (ψϕ) = lim  Sf (ψ + εϕϕ). ε 0

(6) If f (0+ ) < +∞, then expression (4.1) holds for every (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ , where f (Tψ/ϕ ) is defined for f on [0, ∞) with f (0) = f (0+ ). Examples 4.6 (1) Assume that M = L∞ (X, μ) is an abelian von Neumann algebra on a σ finite measure space (X, X, μ). Let ψ, ϕ ∈ M∗+ , identified with functions in L1 (X, μ)+ so that ψ(φ) = X φψ dμ, ϕ(φ) = X φϕ dμ for φ ∈ L∞ (X, μ). With ω = ψ + ϕ we have  Sf (ψ + εωϕ + εω) = Sf (ψ + εωϕ + εω),

44

4 Maximal f -Divergences

see Example 2.2 (1). Take the limit of the above as ε 0 to see that  Sf (ψϕ) coincides with the classical f -divergence Sf (ψϕ) = X ϕf (ψ/ϕ) dμ. (2) Assume that M = B(H) on a finite-dimensional Hilbert space H. Let ψ, ϕ ∈ B(H)+ ∗ with the density operators Dψ , Dϕ . When ψ, ϕ are faithful, (4.1) becomes (4.2). For general ψ, ϕ ∈ B(H)+ ∗ let e be the support projection of ψ + ϕ. By Proposition 4.5 (2) we have

Sf (ψ + εeϕ + εe) +  Sf (ψ + εI ϕ + εI ) = lim  lim  Sf (ε(I − e)ε(I − e))

ε 0

ε 0

= lim  Sf (ψ + εeϕ + εe) + εTr(I − e)f (1) ε 0

= lim  Sf (ψ + εeϕ + εe), ε 0

so that  Sf (ψϕ) coincides with that defined in [58, Definition 3.21]. Examples 4.7 (1) Consider a linear function f (t) = a + bt with a, b ∈ R. Let (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ with e = s(ψ) = s(ϕ). Let A ∈ eMe be as in Lemma A.24 so that  = A ; then  Sa+bt (ψϕ) = ϕ(ae + bA∗A) = aϕ(e) + bA 2 = aϕ(1) + bψ(1). Hence by Definition 4.3 and (2.4) we have for every ψ, ϕ ∈ M∗+ ,  Sa+bt (ψϕ) = Sa+bt (ψϕ) = aϕ(1) + bψ(1).

(4.4)

(2) Consider f (t) = t 2 and show that for every ψ, ϕ ∈ M∗+ ,  St 2 (ψϕ) = St 2 (ψϕ).

(4.5)

Let (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ with corresponding , ∈ P. With A in Lemma A.24 we have  St 2 (ψϕ) = , (A∗ A)2  = , AA∗  = A∗ 22 . On the other hand, St 2 (ψϕ) =  ψ,ϕ 22 =  ψ,ϕ 22 =  ψ,ϕ A 22 = A∗ 22 , 1/2

1/2

so (4.5) follows in this case. Therefore, in view of (4.3) and (2.16), this holds for all ψ, ϕ ∈ M∗+ . By (4.4) and (4.5) together,  Sf = Sf if f is a quadratic polynomial.

4.2 Further Properties of Maximal f -Divergences

45

4.2 Further Properties of Maximal f -Divergences Let ψ, ϕ ∈ M∗+ and let ,  ∈ P be the vector representatives of ψ and ω := ψ + ϕ. We have a unique A ∈ s(ω)Ms(ω) such that  = A. Let Tψ/ω := A∗ A in s(ω)Ms(ω) as in Definition 4.1. Since 0 ≤ Tψ/ω ≤ 1, we write the spectral decomposition of Tψ/ω as 

1

Tψ/ω =

t dEψ/ω (t)

(4.6)

0

with the spectral resolution {Eψ/ω (t) : 0 ≤ t ≤ 1} in s(ω)Ms(ω). In the next theorem we present a general integral formula of  Sf (ψϕ), which may serve as the second definition of maximal f -divergences. Theorem 4.8 ([55]) Let ψ, ϕ ∈ M∗+ and ω := ψ +ϕ with the vector representative . Then for every operator convex function f on (0, ∞),  Sf (ψϕ) =



1

 (1 − t)f

0

where (1 − t)f



t 1−t



t 1−t

 dEψ/ω (t)22 ,

(4.7)

is understood as f (0+ ) for t = 0 and f (∞) for t = 1.

Corollary 4.9 If f (0+ ) < +∞ and f (+∞) < +∞, then  Sf (ψϕ) is finite for every ψ, ϕ ∈ M∗+ .  t  is Proof If f (0+ ) < +∞ and f (+∞) < +∞, then the function (1 − t)f 1−t bounded on [0, 1]. Hence the result is obvious from expression (4.7). The proof of Theorem 4.8 in [55] is based on the integral representation (2.5) of f . Here it is worth noting that the integrand of (4.7) enjoys a special theoretical meaning as follows. Let f be a real function on (0, ∞) and g be given as g(t) :=  t (1 − t)f 1−t , t ∈ (0, 1). Then f is operator convex on (0, ∞) if and only if g is operator convex on (0, 1). Moreover, in this case, g(0+ ) = f (0+ ) and g(1− ) = f (∞). We include the proof of this fact in Appendix C. Remark 4.10 Similarly to the definition of the standard f -divergence Sf (ψϕ) in (2.3), one can write (4.7) as the sum of three terms  Sf (ψϕ) =



 (1 − t)f (0,1)

t 1−t

 dEψ/ω (t)22

! ! + f (0+ ) , Eψ/ω ({0}) + f (∞) , Eψ/ω ({1}) ,

(4.8)

where Eψ/ω ({0}) and Eψ/ω ({1}) are the spectral projections of Tψ/ω for {0} and {1}. One might expect that the above boundary terms including f (0+ ) and f (∞) are equal to the corresponding terms f (0+ )ϕ(1 − s(ψ)) and f (∞)ψ(1 − s(ϕ)), respectively, in (2.3). But it is not true, as shown in [55, Example 4.6].

46

4 Maximal f -Divergences

In the following we state the joint lower semicontinuity in the norm and the martingale convergence for  Sf (ψϕ). Theorem 4.11 ([55]) The function (ψ, ϕ) ∈ M∗+ × M∗+ →  Sf (ψϕ) is jointly lower semicontinuous in the norm topology. Theorem 4.12 ([55]) Let {M α } be an increasing net of unital von Neumann  subalgebras of M such that = M. Then for every ψ, ϕ ∈ M∗+ , α Mα  Sf (ψ|Mα ϕ|Mα )   Sf (ψϕ). The proofs of these properties are based on the integral formula (4.7). For each n ∈ N consider the approximation fn of f in (2.6) with integral on the cut-off interval [1/n, n]. From (2.9) we have as n → ∞,  (1 − t)fn

t 1−t



  (1 − t)f

 t , 1−t

t ∈ [0, 1].

Hence from the monotone convergence theorem applied to the integral formula in (4.7) for fn and f we have for every ψ, ϕ ∈ M∗+ ,  Sf (ψϕ) = lim  Sfn (ψϕ) n→∞

increasingly,

so that  Sf (ψϕ) = supn≥1  Sfn (ψϕ). Moreover, since fn (t) − fn (0+ ) > −∞, t 0 t

lim fn (t) = lim

t 0

to prove Theorems 4.11 and 4.12, we may assume that f (0+ ) < +∞, f (∞) < +∞ and limt 0 f (t) > −∞. See [55, Sec. V] for the details of the proofs of the theorems. Problem 4.13 Theorem 2.7 (i) says that the standard f -divergence Sf (ψϕ) is jointly lower semicontinuous in the σ (M∗ , M)-topology. It follows from Theorem 4.12 that this property (stronger than Theorem 4.11) holds for  Sf (ψϕ) as well when M is injective, or equivalently, approximately finite dimensional (AFD), that is, there is an increasing {Mα } of finite-dimensional unital subalgebras of net  M such that M = M , see Sect. A.1. In fact, in this case,  Sf (ψϕ) = α α supα  Sf (ψ|Mα ϕ|Mα ) is lower Sf (ψ|Mα ϕ|Mα ) by Theorem 4.12 and (ψ, ϕ) →  semicontinuous in the σ (M∗ , M)-topology. However, it is unknown whether or not  Sf (ψϕ) is jointly lower semicontinuous in the σ (M∗ , M)-topology for general M. Another martingale convergence like Proposition 2.10 for Sf holds for  Sf as well.

4.3 Minimal Reverse Test

47

Proposition 4.14 Let {eα } be an increasing net of projections in M such that eα  1. Then for every ψ, ϕ ∈ M∗+ , Sf (eα ψeα eα ϕeα ) =  lim  Sf (ψϕ), α

where eα ϕeα is the restriction of ϕ to the reduced von Neumann algebra eα Meα .

4.3 Minimal Reverse Test For every ψ, ϕ ∈ M∗+ let ω := ψ + ϕ with the vector representative  of ω as 1 well as  of ψ. We have a positive operator Tψ/ω = A∗ A = 0 t dEψ/ω (t) in s(ω)Ms(ω) as in (4.6), where A ∈ s(ω)Ms(ω) satisfies  = A. Define a finite Borel measure ν on [0, 1] by ν := ω(Eψ/ω (·)) = Eψ/ω (·)2 ,

(4.9)

and consider an abelian von Neumann algebra L∞ ([0, 1], ν) = L1 ([0, 1], ν)∗ . Lemma 4.15 ([55]) Define λ0 : M → L∞ ([0, 1], ν) by λ0 (x) =

d J x, Eψ/ω (·) (the Radon–Nikodym derivative), dν

x ∈ M.

Then λ0 is a unital normal positive map and its predual map λ0∗ : L1 ([0, 1], ν) → M∗ is given by "  λ0∗ (φ)(x) = J x,

1

 # φ dEψ/ω 

0

for every φ ∈ L∞ ([0, 1], ν) (⊂ L1 ([0, 1], ν)) and x ∈ M. The above λ0 satisfies 



1

λ0∗ (φ)(1) = ω

φ dEψ/ω

 =

0

1

φ dν 0

for all φ ∈ L∞ ([0, 1], ν) (hence all φ ∈ L1 ([0, 1], ν)). Moreover, for every x ∈ M, λ0∗ (1)(x) = J x,  = , x = ω(x), λ0∗ (t)(x) = J x, Tψ,ω  = J xJ , A∗A = J xJ A, A = J x,  = , x = ψ(x),

48

4 Maximal f -Divergences

where t denotes the identity function t → t on [0, 1]. Hence λ0∗ (t) = ψ and λ0∗ (1 − t) = ω − ψ = ϕ. Now, following Matsumoto’s idea [92], we introduce a key notion below. Definition 4.16 Let (X, X, μ) be a σ -finite measure space and λ : M → L∞ (X, μ) be a positive linear map which is unital and normal. Then the predual map λ∗ : L1 (X, μ) → M∗ is trace-preserving in the sense that X φ dμ = λ∗ (φ)(1) (= λ∗ (φ)) for every φ ∈ L1 (X, μ)+ . We call a triplet (λ, p, q) of such a map λ and p, q ∈ L1 (X, μ)+ a reverse test for ψ, ϕ if λ∗ (p) = ψ and λ∗ (q) = ϕ, i.e., p ◦ λ = ψ and q ◦ λ = ϕ under identification L1 (X, μ)+ = L∞ (X, μ)+ ∗. The next variational formula of  Sf (ψϕ) may serve as the third definition of maximal f -divergences. Theorem 4.17 ([55]) For every ψ, ϕ ∈ M∗+ ,  Sf (ψϕ) = min{Sf (pq) : (λ, p, q) a reverse test for ψ, ϕ}.

(4.10)

Moreover, λ : M → L∞ (X, μ) in (4.10) can be restricted to those with a standard Borel probability space (X, X, μ) or more specifically to those with a Borel probability space on [0, 1]. In fact, (λ0 , t, 1 − t) given in Lemma 4.15 is a reverse test, for which the equality  Sf (ψϕ) = Sf (t1 − t) holds since from (4.7) and (4.9) we have  Sf (ψϕ) =

 0

1



t (1 − t)f 1−t

 dν(t) = Sf (t1 − t).

Therefore, the reverse test (λ0 , t, 1−t) is a minimizer for expression (4.10), which is considered as the von Neumann algebra version of Matsumoto’s minimal or optimal reverse test [92] for ψ, ϕ. Apply the monotonicity property of Sf (Theorem 2.7 (iv)) to this λ0 (that is a unital and normal CP map) to have Sf (ψϕ) ≤ Sf (t1 − t) =  Sf (ψϕ), so we have the following: Theorem 4.18 ([55]) For every ψ, ϕ ∈ M∗+ , Sf (ψϕ) ≤  Sf (ψϕ). The following corollary is worth giving, which is immediate from the above theorem and Definition 4.1.1

1

A simple direct proof of Sf (ψϕ) < +∞ for (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ is unknown to us.

4.3 Minimal Reverse Test

49

Corollary 4.19 For every (ψ, ϕ) ∈ (M∗+ × M∗+ )∼ , Sf (ψϕ) < +∞. Sf (ψϕ) ≤  When H is finite-dimensional, it is easy to verify that if ψ, ϕ ∈ B(H)+ ∗ commute in the sense that the density operators Dψ , Dϕ commute, then Sf (ψϕ) =  Sf (ψϕ) for every operator convex (even simply convex) function on (0, ∞). We extend this to the general von Neumann algebra setting. To do this, we first state the next lemma whose proof was given in [55, Appendix B]. Lemma 4.20 For every ψ, ϕ ∈ M∗+ the following conditions are equivalent: ψ+ϕ

(i) ψ ◦ σt = ψ on s(ψ + ϕ)Ms(ψ + ϕ) for all t ∈ R (note that the modular ψ+ϕ automorphism group σt is defined on s(ψ + ϕ)Ms(ψ + ϕ)); ψ+ϕ (ii) ϕ ◦ σt = ϕ on s(ψ + ϕ)Ms(ψ + ϕ) for all t ∈ R; (iii) hψ hϕ = hϕ hψ , where hψ , hϕ are the corresponding elements of Haagerup’s L1 (M) (see Sect. A.6). ϕ

When s(ψ) ≤ s(ϕ), the above are also equivalent to ψ ◦ σt = ψ for all t ∈ R. More equivalent conditions on the commutativity of ψ, ϕ are provided in Proposition A.56 of Sect. A.7. Proposition 4.21 ([55]) If ψ, ϕ ∈ M∗+ commute in the sense that the equivalent conditions of Lemma 4.20 hold, then Sf (ψϕ) =  Sf (ψϕ) for any operator convex function f on (0, ∞). Example 4.22 In [17] Belavkin and Staszewski introduced a type of relative entropy for states on a C ∗ -algebra. In the von Neumann algebras case, their relative entropy is realized as  Sf for f (t) = t log t (see [55, Example 3.5] for a detailed explanation). So Belavkin and Staszewski’s relative entropy DBS (ψϕ) for every ψ, ϕ ∈ M∗+ is defined to be DBS (ψϕ) :=  St log t (ψϕ). Since D(ψϕ) = St log t (ψϕ), by Theorem 4.18 and Proposition 4.21 we have for every ψ, ϕ ∈ M∗+ , D(ψϕ) ≤ DBS (ψϕ), and D(ψϕ) = DBS (ψϕ) if ψ, ϕ commute (in the sense of Lemma 4.20).

50

4 Maximal f -Divergences

Remark 4.23 In Theorem 7.11 of Sect. 7.2 we will prove that the converse direction of Proposition 4.21 is also true, in such a way that ψ, ϕ ∈ M∗+ commute if Sf (ψϕ) =  Sf (ψϕ) < +∞ under s(ψ) ≤ s(ϕ) and a certain mild assumption on f . (This was shown in [58, Theorem 4.3] in the finite-dimensional case.) In particular, for ψ, ϕ ∈ M∗+ with D(ψϕ) < +∞, it will follow that ψ, ϕ commute if and only if D(ψϕ) = DBS (ψϕ).

Chapter 5

Measured f -Divergences

5.1 Definition Let f be a convex function on (0, ∞), not necessarily operator convex unless we specify that. We use the convention in (2.2). Let M be a general von Neumann algebra. A measurement M in M is given by M = (Aj )1≤j ≤n for some n ∈ N, where Aj ∈ M+ for 1 ≤ j ≤ n and nj=1 Aj = 1. The measurement M is identified with a unital positive map α : Cn → M, determined by α(δj ) := Aj , 1 ≤ j ≤ n. For ψ, ϕ ∈ M∗+ we write M(ψ) := ψ ◦α, i.e., M(ψ) := (ψ(Aj ))1≤j ≤n , and similarly for M(ϕ). Then the classical f -divergence Sf (M(ψ)M(ϕ)) is defined by Sf (M(ψ)M(ϕ)) :=

n

j =1

 ϕ(Aj )f

ψ(Aj ) ϕ(Aj )



under the convention (2.2). A measurement E =(Ej )1≤j ≤n in M is said to be projective if Ej ’s are orthogonal projections with nj=1 Ej = 1. Definition 5.1 For ψ, ϕ ∈ M∗+ we define the measured f -divergence and its projective variant as follows: Sfmeas (ψϕ) := sup{Sf (M(ψ)M(ϕ)) : M a measurement in M},

(5.1)

pr

Sf (ψϕ) := sup{Sf (E(ψ)E(ϕ)) : E a projective measurement in M}. (5.2) pr

Obviously, Sf (ψϕ) ≤ Sfmeas (ψϕ). Moreover, Sfmeas (ψϕ) ≤ Sf (ψϕ) whenever f is operator convex on (0, ∞), due to the monotonicity of Sf , see Theorem 2.7 (iv).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_5

51

52

5 Measured f -Divergences

Proposition 5.2 For every ψ, ϕ ∈ M∗+ , Sfmeas (ψϕ) = sup{Sf (ψ ◦ αϕ ◦ α) : α : A → M a unital normal positive map, A a commutative von Neumann algebra}.

(5.3)

Proof Since Sfmeas (ψϕ) ≤ the RHS of (5.3) is obvious, we may prove that Sfmeas (ψϕ) ≥ Sf (ψ ◦ αϕ ◦ α)

(5.4)

for every α : A → M as in (5.3). Let e0 := s(ψ ◦ α) ∨ s(ϕ ◦ α) and define β : e0 Ae0 → A by β(x) := x + ψ(x)(1 − e0 ) for x ∈ e0 Ae0 . Then β is a unital normal positive map and it is obvious that Sf (ψ ◦αϕ ◦α) = Sf (ψ ◦α ◦βϕ ◦α ◦β). Replacing α with α ◦ β : e0 Ae0 → M, we may assume that A = L∞ (X, X, μ), where (X, X, μ) is a finite measure space. Identifying ψ ◦ α, ϕ ◦ α as functions ˆ ϕˆ ∈ L1 (X, X, μ)+ we have ψ, Sf (ψ ◦ αϕ ◦ α)   ˆ ϕ) = ϕf ˆ (ψ/ ˆ dμ + f (0+ ) ˆ {ψ>0}∩{ ϕ>0} ˆ

ˆ {ψ=0}

ϕˆ dμ + f (+∞)

 {ϕ=0} ˆ

ψˆ dμ.

Hence, to prove (5.4), it suffices to prove that  $

  n p(Xj ) dp dq ≤ sup : f q(Xj )f dq q(Xj ) X0





j =1

% {X1 , . . . , Xn } a finite measurable partition of X0 , (5.5)

ˆ ϕ, where X0 := {ψˆ > 0} ∩ {ϕˆ > 0}, dp := ψˆ dμ, dq := ϕˆ dμ, and dp/dq = ψ/ ˆ the Radon–Nikodym derivative. To show (5.5), we may and do assume that f ≥ 0. Indeed, there exist a, b ∈ R and a convex function f0 ≥ 0 on (0, ∞) such that f (t) = f0 (t)+at +b, t ∈ (0, ∞). Then       dp dp f f0 dq = dq + ap(X0 ) + bq(X0), dq dq X0 X0 n

j =1

 q(Xj )f

p(Xj ) q(Xj )

 =

n

j =1

 q(Xj )f0

p(Xj ) q(Xj )

 + ap(X0 ) + bq(X0),

5.1 Definition

53

so that it suffices to show (5.5) for f0 . For every n ∈ N consider a finite measurable partition {X1 , . . . , Xn2n +1 } given by % $ j j −1 dp Xj := x ∈ X : (x) < , 1 ≤ j ≤ n2n , ≤ 2n dq 2n $ % dp (x) ≥ n2n , Xn2n +1 := x ∈ X : dq  n +1 j −1 and define a simple function κn := n2 j =1 λj 1Xj , where λj := inf f (t) : 2n ≤

t ≤ 2jn (with f (0) = f (0+ )), 1 ≤ j ≤ n2n , and λn2n +1 := inf{f (t) : t ≥ n2n }. Then, sincef is convex and so continuous on (0, ∞), it is immediate to see that  κn (x) → f dp (x) for q-a.e. x ∈ X. By Fatou’s lemma we have dq 



n +1   n2

dp f κn dq = lim inf λj q(Xj ) dq ≤ lim inf n→∞ X n→∞ dq X0 0

j =1



n2n +1

≤ lim inf n→∞

j =1

f

 p(Xj ) q(Xj ) q(Xj )

≤ the RHS of (5.5), where the second inequality above follows since   j −1 j  p(Xj ) 1 dp 1 ≤ j ≤ n2n , 2n , 2n , = dq ∈ q(Xj ) q(Xj ) Xj dq [n2n , +∞), j = n2n + 1. Hence (5.4) is obtained. Let H be a self-adjoint operator affiliated with M and  ∞ H = λ dEH (λ) −∞

be the spectral decomposition of H with the spectral resolution {EH (λ) : λ ∈ R} in M, see the beginning of Sect. A.5. Write EH (ψ) for the finite positive measures dψ(EH (λ)) on the Borel space (R, B(R)). Then similarly to the proof of Proposition 5.2 we have the following: Proposition 5.3 For every ψ, ϕ ∈ M∗+ , pr

Sf (ψϕ) = sup{Sf (EH (ψ)EH (ϕ)) : H a self-adjoint operator affiliated with M}. (5.6)

54

5 Measured f -Divergences pr

In this way, Sf is the supremum of the classical f -divergences over the von Neumann (or projective) measurements, while Sfmeas is that over all measurements of quantum-classical channels. pr The next proposition summarizes basic properties of Sfmeas and Sf . Proposition 5.4 pr

(1) Joint lower semicontinuity: Sfmeas (ψϕ) and Sf (ψϕ) are jointly lower semicontinuous in (ψ, ϕ) ∈ M∗+ × M∗+ in the σ (M∗ , M)-topology. pr (2) Joint convexity: Sfmeas (ψϕ) and Sf (ψϕ) are jointly convex and jointly subadditive on M∗+ × M∗+ . (3) Monotonicity: Let γ : N → M be a unital normal positive map between von Neumann algebras. For every ψ, ϕ ∈ M∗+ , Sfmeas (ψ ◦ γ ϕ ◦ γ ) ≤ Sfmeas (ψϕ). (4) For every ψ, ϕ, ω ∈ M∗+ , Sfmeas (ψϕ) = lim Sfmeas (ψ + εωϕ + εω), ε 0

pr Sf (ψϕ)

pr

= lim Sf (ψ + εωϕ + εω). ε 0

 Proof Note that the classical f -divergence Sf (ab) = nj=1 bj f (aj /bj ) for a = (aj )nj=1 , b = (bj )nj=1 ∈ [0, ∞)n is jointly lower semicontinuous and jointly convex. Hence (1) and (2) immediately follow from Definition 5.1. (3) Let γ : N → M be as stated. For any measurement M = (Bj )nj=1 in N, since γ (M) := (γ (Bj ))nj=1 is a measurement in M, we have Sf (M(ψ ◦ γ )M(ϕ ◦ γ )) = Sf (γ (M)(ψ)γ (M)(ϕ)) ≤ Sfmeas (ψϕ), which implies that Sfmeas (ψ ◦ γ ϕ ◦ γ ) ≤ Sfmeas (ψϕ). (4) From (1) it follows that Sfmeas (ψϕ) ≤ lim inf Sfmeas (ψ + εωϕ + εω). ε 0

On the other hand, by (2) we have Sfmeas (ψ + εωϕ + εω) ≤ Sfmeas (ψϕ) + Sfmeas (εωεω) = Sfmeas (ψϕ) + εω(1)f (1). pr

Hence the assertion for Sfmeas follows. The proof for Sf is similar.

5.2 Variational Expressions

55

This is a good place for us to explain an abstract approach to quantum f divergences. Assume that f is operator convex on (0, ∞). We say that a function q Sf : M∗+ × M∗+ → (−∞, +∞], where M varies over all von Neumann algebras, is a monotone quantum f -divergence if the following two properties are satisfied: q

q

(a) Sf (ψ ◦ γ ϕ ◦ γ ) ≤ Sf (ψϕ) for any unital normal CP map γ : N → M between von Neumann algebras and for every ψ, ϕ ∈ M∗+ . (b) When M is an abelian von Neumann algebra with M = L∞ (X, μ) on a σ -finite q measure space (X, X, μ), Sf (ψϕ) coincides with the classical f -divergence of ψ, ϕ ∈ L1 (X, μ)+ as in Example 2.2 (1). q

If Sf is a monotone quantum f -divergence, then it is readily seen from Definition 5.1 and Theorem 4.17 that for every ψ, ϕ ∈ M∗+ , Sfmeas (ψϕ) ≤ Sf (ψϕ) ≤  Sf (ψϕ), q

which justifies the name maximal f -divergence for  Sf and suggests that we call Sfmeas the minimal f -divergence. Since the standard f -divergence Sf is typical among monotone quantum f -divergences, we have the following: Proposition 5.5 For every ψ, ϕ ∈ M∗+ , Sfmeas (ψϕ) ≤ Sf (ψϕ) ≤  Sf (ψϕ).

(5.7)

5.2 Variational Expressions We consider the following classes of functions on (0, ∞): 

(a) Fcv (0, ∞) is the set of non-decreasing convex real functions f on (0, ∞) such that f (∞) = +∞,  (b) Fcc (0, ∞) is the set of non-decreasing concave real functions g on (0, ∞) such that g (∞) = 0. The following variational formulas were shown in [53, Appendix A], which provides a theory of conjugate functions (or the Legendre transform) on the halfline (0, ∞). Lemma 5.6 

(1) For every f ∈ Fcv (0, ∞) define f  (t) := sup{st − f (s)}, s>0

t ∈ (0, ∞).

56

5 Measured f -Divergences 



Then f  ∈ Fcv (0, ∞) and f → f  is an involutive bijection on Fcv (0, ∞),  i.e., a bijection such that f  = f for all f ∈ Fcv (0, ∞). Therefore, f (s) = sup{st − f  (t)},

s ∈ (0, ∞),

t >0

and moreover f  (0+ ) = −f (0+ ) (∈ R).  (2) For every g ∈ Fcc (0, ∞) define g  (t) := inf {st − g(s)}, s>0

t ∈ (0, ∞).





Then g  ∈ Fcc (0, ∞) and g → g  is an involutive bijection on Fcc (0, ∞), i.e.,  a bijection such that g  = g for all g ∈ Fcc (0, ∞). Therefore, g(s) = inf {st − g  (t)}, t >0

s ∈ (0, ∞),

and moreover g  (0+ ) = −g(∞) and g(0+ ) = −g  (∞). For a, b with −∞ ≤ a < b ≤ ∞, we write M(a,b) for the set of self-adjoint operators A ∈ M whose spectrum is included in (a, b). In particular, M(0,∞) is the the set M++ of positive invertible operators in M, and M(−∞,∞) is the set Msa of self-adjoint operators in M. For −∞ < a < b < ∞ we also write M[a,b] for the set of self-adjoint operators A ∈ M whose spectrum is in [a, b], i.e., a1 ≤ A ≤ b1. pr From Lemma 5.6 we can show variational expressions of Sf in the following theorem. Theorem 5.7 Let ψ, ϕ ∈ M∗+ . 

(1) Assume that f ∈ Fcv (0, ∞). Let a := f (0+ ) ∈ [0, ∞) and b := −f (0+ ) ∈ (−∞, ∞). Then the inverse (f  )−1 : (b, ∞) → (a, ∞) exists and pr

Sf (ψϕ) = sup {ψ(A) − ϕ(f  (A))}

(5.8)

A∈M++

=

sup {ψ(A) − ϕ(f  (A))}

(5.9)

A∈M(a,∞)

=

sup {ψ((f  )−1 (A)) − ϕ(A)}.

(5.10)

A∈M(b,∞) 

(2) Assume that g ∈ Fcc (0, ∞) and g is non-constant. Let f := −g, and a := g (0+ ), b := −g(0+ ), c := −g(∞). Then a ∈ (0, ∞], −∞ ≤ c < b ≤ ∞, the inverse (g  )−1 : (c, b) → (0, a) exists, and pr

Sf (ψϕ) = sup {−ψ(A) + ϕ(g  (A))} A∈M++

(5.11)

5.2 Variational Expressions

57

=

sup {−ψ(A) + ϕ(g  (A))}

(5.12)

A∈M(0,a)

sup {−ψ((g  )−1 (A)) + ϕ(A)}.

=

(5.13)

A∈M(c,b)

Proof (1) Let E = (Ej )1≤j ≤n be a projective measurement in M. For each j we prove that   ψ(Ej ) ϕ(Ej )f (5.14) = sup{tψ(Ej ) − f  (t)ϕ(Ej )}. ϕ(Ej ) t >0 When ψ(Ej ) > 0 and ϕ(Ej ) > 0, by Lemma 5.6 (1) the LHS of (5.14) is % $ ψ(Ej )  − f (t) = sup{tψ(Ej ) − f  (t)ϕ(Ej )}. ϕ(Ej ) sup t ϕ(Ej ) t >0 t >0 When ψ(Ej ) = 0 and ϕ(Ej ) > 0, the LHS of (5.14) is f (0+ )ϕ(Ej ) = −f  (0+ )ϕ(Ej ) = sup{−f  (t)ϕ(Ej )} t >0

by the convention (2.2) and f (0+ ) = −f  (0+ ) stated in Lemma 5.6 (1). When ψ(Ej ) > 0 and ϕ(Ej ) = 0, the LHS of (5.14) is f (∞)ψ(Ej ) = +∞ = sup{tψ(Ej )}. t >0

When ψ(Ej ) = ϕ(Ej ) = 0, both sides of (5.14) are 0. Hence (5.14) holds in all the cases. Summing up (5.14) for all j we have n

j =1



ψ(Ej ) ϕ(Ej )f ϕ(Ej )

 =

sup

n

{tj ψ(Ej ) − f  (tj )ϕ(Ej )}

(tj ):tj >0 j =1

=

sup

$ 

 

% ψ tj Ej − ϕ f  (tj )Ej .

(tj ):tj >0

j

j

(5.15) Therefore, we have pr Sf (ψϕ)

$ 

  

%  = sup sup ψ tj Ej − ϕ f tj Ej (Ej ) (tj )

j

j

= sup {ψ(A) − ϕ(f (A))}, 

A∈M++

58

5 Measured f -Divergences

since ψ(A) − ϕ(f  (A)) for  any A ∈ M++ is approximated by ψ(A0 ) − ϕ(f  (A0 )) for discrete A0 := nj=1 tj Ej . Hence (5.8) follows. 

By Lemma 5.6 (1) note that f  ∈ Fcv (0, ∞) and f  (0+ ) = −f (0+ ) = b so that f  (t) = −f (0+ ) for 0 < t ≤ a. Hence we can restrict tj to tj > a in the supremum in (5.15), so the supremum in (5.8) is the same as that taken over A ∈ M(a,∞) . Furthermore, we easily see that f  is strictly increasing from (a, ∞) onto (b, ∞). Hence the inverse (f  )−1 : (b, ∞) → (a, ∞) exists. Since X → (f  )−1 (X) is a bijective map from M(b,∞) onto M(a,∞) , we have (5.10) as well. (2) Let E = (Ej )1≤j ≤n be a projective measurement in M. For each j we prove that   ψ(Ej ) = sup{−tψ(Ej ) + g  (t)ϕ(Ej )}. ϕ(Ej )f (5.16) ϕ(Ej ) t >0 When ψ(Ej ) > 0 and ϕ(Ej ) > 0, by Lemma 5.6 (2) the LHS of (5.16) is  −ϕ(Ej )g

ψ(Ej ) ϕ(Ej )



$ = −ϕ(Ej ) inf

t >0

ψ(Ej ) t − g  (t) ϕ(Ej )

%

= sup{−tψ(Ej ) + g  (t)ϕ(Ej )}. t >0

When ψ(Ej ) = 0 and ϕ(Ej ) > 0, the LHS of (5.16) is −g(0+ )ϕ(Ej ) = g  (∞)ϕ(Ej ) = sup{g  (t)ϕ(Ej )} t >0

from (2.2) and g(0+ ) = −g  (∞) stated in Lemma 5.6 (2). When ψ(Ej ) > 0 and ϕ(Ej ) = 0, the LHS of (5.16) is −g (∞)ψ(Ej ) = 0 = sup{−tψ(Ej )}. t >0

When ψ(Ej ) = ϕ(Ej ) = 0, both sides of (5.16) are 0. Hence (5.16) has been shown. Since g is non-constant, a ∈ (0, ∞] and −∞ ≤ c < b ≤ ∞ are obvious. Moreover, when a < ∞, since −b = g(0+ ) > −∞ and g  (t) = −b for t ≥ a, we find that the supremum in (5.16) is the same as that taken over t ∈ (0, a). Thus, (5.11) and (5.12) follow as in the proof of (1). Furthermore, we easily see that g  is strictly increasing from (0, a) onto (c, b). Hence the inverse (g  )−1 : (c, b) → (0, a) exists. Since X → (g  )−1 (X) is a bijective map from M(c,b) onto M(0,a), we have (5.13) as well. pr

Based on Theorem 5.7 we can show the equality Sf = Sfmeas in the following cases.

5.2 Variational Expressions

59

Theorem 5.8 Let f be a convex function on (0, ∞), and f be the transpose of f , i.e., f(t) := tf (t −1 ), t > 0. Let g := −f and  g := −f (hence  g is the transpose of g). Assume that one of the following conditions (1)–(4) is satisfied: 

(1) f ∈ Fcv (0, ∞) and either f  is operator convex on (f (0+ ), ∞) or (f  )−1 is operator concave on (−f (0+ ), ∞),  (2) f ∈ Fcv (0, ∞) and either f is operator convex on (f (0+ ), ∞) or (f )−1 is operator concave on (−f(0+ ), ∞),  (3) g ∈ Fcc (0, ∞) and either g  is operator concave on (0, g (0+ )) or (g  )−1 is operator convex on (−g(∞), −g(0+ )),  (4)  g ∈ Fcc (0, ∞) and either  g  is operator concave on (0,  g (0+ )) or ( g  )−1 is + operator convex on (− g (∞), − g (0 )). Then for every ψ, ϕ ∈ M∗+ , pr

Sf (ψϕ) = Sfmeas (ψϕ). pr

Proof Since Sf (ψϕ) ≤ Sfmeas (ψϕ) holds trivially, we may prove the converse inequality. 

Case (1). Assume that f ∈ Fcv (0, ∞) and f  is operator convex on (f (0+ ), ∞). For any measurement (Aj )1≤j ≤n in M, as in the proof of Theorem 5.7 (1) with ψ(Aj ), ϕ(Aj ) in place of ψ(Ej ), ϕ(Ej ), we have, with a := f (0+ ),

 ϕ(Aj )f

j

ψ(Aj ) ϕ(Aj )

$ 

 

% tj Aj − ϕ f  (tj )Aj ψ

 =

sup (tj ):tj >a

≤

j

(tj ):tj >a

=

j

$ 

  

%  tj Aj − ϕ f tj Aj sup ψ j

j

sup {ψ(A) − ϕ(f (A))} = 

A∈M(a,∞)

pr Sf (ψϕ).

In the above, the last equality holds by Theorem 5.7 (1), and the inequality is seen from the operator convexity of f  on (a, ∞) as follows: f



j

 tj Aj

= f



j

1/2

1/2

Aj tj Aj

 ≤

j

1/2

1/2

Aj f  (tj )Aj

=



f  (tj )Aj

j

by [25, Theorem 2.1] (also [51, Theorem 2.5.7]). Therefore, we have pr Sfmeas (ψϕ) ≤ Sf (ψϕ).

60

5 Measured f -Divergences

Next, assume that (f  )−1 is operator concave on (−f (0+ ), ∞). We have, with b := −f (0+ ),

j



ψ(Aj ) ϕ(Aj )f ϕ(Aj )



$ 

 

%  sup ψ tj Aj − ϕ f (tj )Aj

=

(tj ):tj >a

=

sup

j

(sj ):sj >b

≤

sup

j

$ 

 

% (f  )−1 (sj )Aj − ϕ sj Aj ψ j

j

$  

 

% sj Aj sj Aj ψ (f  )−1 −ϕ

(sj ):sj >b

=

j

sup A∈M(b,∞)

)

pr ψ(f  )−1 (A) − ϕA = Sf (ψϕ).

In the above, the last equality holds by Theorem 5.7 (1), and the inequality follows from the operator concavity of (f  )−1 on (b, ∞). Case (2). We readily see that pr

pr

Sf (ψϕ) = Sf (ϕψ),

Sfmeas (ψϕ) = Sfmeas  (ϕψ).

Hence the result is immediate from case (1).  Case (3). Assume that g ∈ Fcc (0, ∞). Since the assertion is obvious when g is a constant function, we may assume that g is non-constant; then g (0+ ) > 0 and g(0+ ) < g(∞). Assume that g  is operator concave on (0, g (0+ )). For any measurement (Aj )1≤j ≤n in M, as in the proof of Theorem 5.7 (2), we have, with a := g (0+ ),

j

 ϕ(Aj )f

ψ(Aj ) ϕ(Aj )

$

 =

−ψ

sup (tj ):0 t0 .

Therefore, ψ((f  )−1 (A0 )) − ϕ(A0 ) ≥ ψ((f  )−1 (A)) − ϕ(A). By Theorem 5.7 (1) and case (1) of Theorem 5.8, this implies that pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup



ψ((f  )−1 (A)) − ϕ(A) ,

A∈M[b,t0 ]

where (f  )−1 is operator concave on [b, ∞) with (f  )−1 (b) := a. By pr Lemma 5.12 (2) there exists an H0 ∈ M[b,t0 ] such that Sf (ψϕ) = ψ((f  )−1 (H0 )) − ϕ(H0 ). Let H := (f  )−1 (H0 ); then H ∈ M[a,t1 ] with pr t1 := (f  )−1 (t0 ) and H0 = f  (H ), so we have Sf (ψϕ) = ψ(H ) − ϕ(f  (H )). pr Hence, similarly to the previous case, we have Sf (ψϕ) = Sf (EH (ψ)EH (ϕ)). 

Theorem 5.14 Assume that g ∈ Fcc (0, ∞) and either g  is operator concave on (0, g (0+ )) or (g  )−1 is operator convex on (−g(∞), −g(0+ )). Let f := −g and ψ, ϕ ∈ M∗+ . Assume that one of the following conditions (1)–(4) is satisfied: (1) (2) (3) (4)

g(∞) < ∞, g (0+ ) < ∞ (with no condition on ψ, ϕ), g(∞) = ∞, g (0+ ) < ∞, and ψ ≤ λϕ for some λ > 0, g(∞) < ∞, g (0+ ) = ∞, and ϕ ≤ λψ for some λ > 0, g(∞) = ∞, g (0+ ) = ∞, and λ−1 ϕ ≤ ψ ≤ λϕ for some λ > 1.

5.3 Optimal Measurements

69

Then there exists an H ∈ Msa such that pr

Sfmeas (ψϕ) = Sf (ψϕ) = Sf (EH (ψ)EH (ϕ)). 

Proof Let g ∈ Fcc (0, ∞) and f := −g. We may assume that g is non-constant; the conclusion is obviously true when g and hence f are constant. Case (1). Assume that g  is operator concave on (0, a), where a := g (0+ ). Since g  (0+ ) = −g(∞) > −∞ (by Lemma 5.6 (2)) and a = g (0+ ) < ∞, g  is an operator concave function on [0, a], where g  (0) = −g(∞) and g  (a) = −g(0+ ) < ∞ (since g (0+ ) < ∞). By Theorem 5.7 (2) and case (3) of Theorem 5.8, we have pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup {−ψ(A) + ϕ(g  (A))}. A∈M[0,a] pr

Hence by Lemma 5.12 (2), there exists an H ∈ M[0,a] such that Sf (ψϕ) = −ψ(H ) + ϕ(g  (H )), and the remaining proof is similar to that of Theorem 5.13. Next, assume that (g  )−1 is operator convex on (c, b), where b := −g(0+ ) and c := −g(∞), so (g  )−1 extends to an operator convex function on [c, b] with (g  )−1 (c) = 0 and (g  )−1 (b) = g (0+ ). By Theorem 5.7 (2) and case (3) of Theorem 5.8 we have pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup {−ψ((g  )−1 (A)) + ϕ(A)}. A∈M[c,b]

Hence the proof proceeds as above by using Lemma 5.12 (1). Case (2). Assume that g  is operator concave on (0, a), where a := g (0+ ) < ∞. Since g  (0+ ) = −g(∞) = −∞ and hence (g  ) (0+ ) = ∞, there is a t0 ∈ (0, a) such that (g  ) (t0 ) > λ. For any A ∈ M(0,a) with the spectral decomposition a A = 0 t dEA (t), define an A0 ∈ M[t0 ,a] by  A0 := t0 EA ((0, t0 )) +

[t0 ,a]

t dEA (t).

Then {−ψ(A0 ) + ϕ(g  (A0 ))} − {−ψ(A) + ϕ(g  (A))}   =− (t0 − t) dψ(EA (t)) + (g  (t0 ) − g  (t)) dϕ(EA (t)) 

(0,t0 )

(t0 − t)

≥ (0,t0 )



(0,t0 )

g  (t0 ) − g  (t) t0 − t

 − λ dϕ(EA (t)) ≥ 0,

70

5 Measured f -Divergences

since g  (t0 ) − g  (t) ≥ (g  ) (t0 ) > λ, t0 − t

0 < t < t0 .

Therefore, by Theorem 5.7 (2) and case (3) of Theorem 5.8, we have pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup {−ψ(A) + ϕ(g  (A))}. A∈M[t0 ,a]

Since g  is operator concave on [t0 , a], it follows from Lemma 5.12 (2) that the above supremum is attained by some H ∈ M[t0 ,a] , for which we have pr

Sf (ψϕ) = Sf (EH (ψ)EH (ϕ)) similarly to the proof of Theorem 5.13. Next, assume that (g  )−1 is operator convex on (−∞, b), where b := −g(0+ ) < ∞. Since g  (0+ ) = −g(∞) = −∞ and hence (g  ) (0+ ) = ∞, note that 

 (g  )−1 (t) =

1 (g  ) ((g  )−1 (t))

−→

1 (g  ) (0+ )

=0

as b > t → −∞.

  So there is a t0 ∈ (−∞, b) such that (g  )−1 (t0 ) < 1/λ. For any A ∈ M(−∞,b) b with A = −∞ t dEA (t), define an A0 ∈ M[t0 ,b] by  A0 := t0 EA ((−∞, t0 )) +

b

t dEA (t). t0

Then {−ψ((g  )−1 (A0 ) + ϕ(A0 )} − {−ψ((g  )−1 (A)) + ϕ(A)}   ((g  )−1 (t0 ) − (g  )−1 (t)) dψ(EA (t)) + =−

(t0 − t) dϕ(EA (t))

(−∞,t0 )

  1 (g  )−1 (t0 ) − (g  )−1 (t) (t0 − t) − dψ(EA (t)) ≥ 0, λ t0 − t (−∞,t0 )

 ≥

(−∞,t0 )

since (g  )−1 (t0 ) − (g  )−1 (t)   −1  1 ≤ (g ) (t0 ) < , t0 − t λ

−∞ < t < t0 .

5.3 Optimal Measurements

71

Hence we have sup {−ψ((g  )−1 (A)) + ϕ(A)}

pr

Sfmeas (ψϕ) = Sf (ψϕ) =

A∈M[t0 ,b]

= −ψ((g  )−1 (H0 )) + ϕ(H0 ) for some H0 ∈ M[t0 ,b] . Letting H := (g  )−1 (H0 ) ∈ M[t1 ,a] , where t1 := g  (t0 ), we have pr

Sf (ψϕ) = −ψ(H ) + ϕ(g  (H )) = Sf (EH (ψ)EH (ϕ)) as before. Case (3). Assume that g  is operator concave on (0, ∞), where g (0+ ) = ∞. Since g  (0+ ) = −g(∞) > −∞, g  extends to [0, ∞) with g  (0) = −g(∞).  Since (g  ) (∞) = 0 (due to g  ∈ Fcc (0, ∞)), there is a t0 ∈ (0, ∞) such ∞ that (g  ) (t0 ) < 1/λ. For any A ∈ M++ with A = 0 t dEA (t), define an A0 ∈ M[0,t0 ] by 

t0

A0 :=

t dEA (t) + t0 EA ((t0 , ∞)).

0

Then, from ϕ ≤ λψ we have {−ψ(A0 ) + ϕ(g  (A0 ))} − {−ψ(A) + ϕ(g  (A))}   = (t − t0 ) dψ(EA (t)) − (g  (t) − g  (t0 )) dϕ(EA (t)) 

(t0 ,∞)



1 ≥ − (t − t0 ) λ (t0 ,+∞)

(t0 ,∞)

 − g  (t0 ) dϕ(EA (t)) ≥ 0, t − t0

g  (t)

since 1 g  (t) − g  (t0 ) ≤ (g  ) (t0 ) < , t − t0 λ

t > t0 .

Hence by Theorem 5.7 (2) and case (3) of Theorem 5.8, we have pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup {−ψ(A) + ϕ(g  (A))}, A∈M[0,t0 ]

and the remaining proof is the same as before.

72

5 Measured f -Divergences

Next, assume that (g  )−1 is operator convex on (c, b), where b := −g(0+ ) ∈ (−∞, ∞] and c := −g(∞) > −∞. Since g  (∞) = b and (g  ) (∞) = 0, note that   −1  (g ) (t) =

1

−→

(g  ) ((g  )−1 (t))

1 (g  ) (∞)

=∞

as c < t  b.

  So there is a t0 ∈ (c, b) such that (g  )−1 (t0 ) > λ. For any A ∈ M(c,b) with b A = c t dEA (t), define an A0 ∈ M[c,t0 ] by 

t0

A0 :=

t dEA (t) + t0 EA ((t0 , b)).

c

Then {−ψ((g  )−1 (A0 )) + ϕ(A0 )} − {−ψ((g  )−1 (A)) + ϕ(A)}    −1  −1 = ((g ) (t) − (g ) (t0 )) dψ(EA (t)) + (t − t0 ) dϕ(EA (t)) 

(t0 ,b)

 (t0 − t)

≥

(t0 ,b)

(g  )−1 (t)

− (g  )−1 (t0 )

t − t0

(t0 ,b)

 − λ (t − t0 ) dψ(EA (t)) ≥ 0,

since (g  )−1 (t) − (g  )−1 (t0 ) ≥ [(g  )−1 ] (t0 ) > λ, t − t0

t0 < t < b.

Since (g  )−1 is operator convex on [c, t0 ] with (g  )−1 (c) = 0, we have pr

Sfmeas (ψϕ) = Sf (ψϕ) =

sup {−ψ((g  )−1 (A)) + ϕ(A)} A∈M[c,t0 ]

= −ψ((g  )−1 (H0 )) + ϕ(H0 ) for some H0 ∈ M[c,t0 ] , and the remaining proof is the same as before. Case (4). When g  is operator concave on (0, ∞), we may combine the proofs of cases (2) and (3). In this case, we can choose t0 , t1 with 0 < t0 < t1 < ∞ such that (g  ) (t0 ) > λ and (g  ) (t1 ) < 1/λ. Then for any A ∈ M++ , define an A0 ∈ M[t0 ,t1 ] by  A0 := t0 EA ((0, t0 )) +

[t0 ,t1 ]

t dEA (t)) + t1 EA ((t1 , ∞)).

5.3 Optimal Measurements

73

Then the proof proceeds as before, while we omit the details. When (g  )−1 is operator convex on (−∞, b) where b := −g(0+ ) ∈ (−∞, ∞], the proof is again the combination of those of cases (2) and (3); we omit the details. 

Remark 5.15 As in Theorem 5.8, when the transpose  g is in Fcc (0, ∞) and the pr assumption in Theorem 5.14 is satisfied for  g , the same attainability result for Sf as in Theorem 5.14 holds under exchanging the roles of ψ, ϕ. Examples 5.16 

(1) When 1 < α < ∞, consider fα (t) := t α ∈ Fcv (0, ∞). From Example 5.9 (1),  (fα )−1 is operator concave on (0, ∞). Hence by Theorem 5.13, if ψ ≤ λϕ for some λ > 0, then there exists an H ∈ Msa such that pr

Sfmeas (ψϕ) = Sfα (ψϕ) = Sfα (EH (ψ)EH (ϕ)), α so that pr

Dαmeas (ψϕ) = Dα (ψϕ) = Dα (EH (ψ)EH (ϕ)). 

(2) When 0 < α < 1, consider gα (t) := t α ∈ Fcc (0, ∞) and fα := −gα . Then gα (0+ ) = 0, gα (∞) = ∞, and gα (0+ ) = ∞. From Example 5.9 (2), when  g α (0+ )) = (0, ∞). When 0 < α ≤ 1/2 ≤ α < 1,  gα is operator concave on (0,   −1 gα (∞), − gα (0+ )) = (−∞, 0). Hence by 1/2, ( gα ) is operator convex on (− −1 case (4) of Theorem 5.14 and Remark 5.15, if λ ϕ ≤ ψ ≤ λϕ for some λ > 1, there exists an H ∈ Msa such that pr

Sfmeas (ψϕ) = Sfα (ψϕ) = Sfα (EH (ψ)EH (ϕ)), α so that pr

Dαmeas (ψϕ) = Dα (ψϕ) = Dα (EH (ψ)EH (ϕ)). 

(3) Consider g0 (t) := log t ∈ Fcc (0, ∞) and f0 := −g0 . Then g0 (0+ ) = −∞,  g0 (∞) = ∞, and g0 (0+ ) = ∞. From Example 5.10, g0 is operator concave + on (0, g0 (0 )) = (0, ∞). Hence by case (4) of Theorem 5.14, for every ψ, ϕ ∈ M∗+ with λ−1 ϕ ≤ ψ ≤ λϕ, there exists an H ∈ Msa such that D meas (ψϕ) = D pr (ψϕ) = D(EH (ψ)EH (ϕ)). (4) For each λ ∈ (0, ∞), consider gλ (t) :=

t t +λ



∈ Fcc (0, ∞) and fλ := −gλ .

Then gλ (0+ ) = 0, gλ (∞) = 1, and gλ (0+ ) = 1/λ. From Example 5.11, (gλ )−1 is operator convex on (−gλ (∞), −gλ (0+ )) = (−1, 0). Hence by case (1) of 

74

5 Measured f -Divergences

Theorem 5.14, for every ψ, ϕ ∈ M∗+ (with no condition), there exists an H ∈ Msa such that pr

Sfmeas (ψϕ) = Sfλ (ψϕ) = Sfλ (EH (ψ)EH (ϕ)). λ In the two examples below we will give the descriptions of optimal measurements for the particular Rényi divergences Dα when α = 2 and 1/2. Similar discussions on optimal measurements for those Rényi divergences in the finite-dimensional case are found in [91]. In the following, let ψ, ϕ ∈ M∗+ with ψ(1) = 1 for simplicity. 

Example 5.17 (Optimal Measurement for D2 ) Let f2 (t) := t 2 and so f2 (t) = t 2 /4 as given in Example 5.9 (1). By (5.17) we have (ψϕ) = Sfmeas (ψϕ) = sup {2ψ(x) − ϕ(x 2 )}. Qmeas 2 2 x∈M+

Assume that ψ ≤ λϕ for some λ > 0. In this case, we may assume that ϕ is faithful. Then by the linear Radon–Nikodym theorem due to Sakai [116, 1.24.4], there exists an h0 ∈ M+ such that ψ(x) =

1 ϕ(h0 x + xh0 ), 2

x ∈ M.

(5.24)

For every h ∈ Msa we find that 2ψ(h0 + h) − ϕ((h0 + h)2 ) = 2ψ(h0 ) − ϕ(h20 ) + {2ψ(h) − ϕ(h0 h + hh0 )} − ϕ(h2 ) ≤ 2ψ(h0 ) − ϕ(h20 ). Therefore, Qmeas (ψϕ) = 2ψ(h0 ) − ϕ(h20 ) = 2ψ(h0 ) − ψ(h0 ) = ψ(h0 ). 2 Consequently, the optimal measurement for D2 is induced from the spectral resolution of h0 . According to Kosaki [76, Theorem 1.6], a linear Radon–Nikodym derivative h0  ≤ λϕ for some λ > 0, where satisfying (5.24) exists if and only if ψ  := ψ



∞ −∞

1 ϕ ψ ◦ σt dt cosh πt

 ≤ λϕ), and in this case, h0 is determined as (of course, ψ ≤ λϕ ⇒ ψ  : Dϕ]∗−i/2 [D ψ  : Dϕ]−i/2 , h0 = [D ψ

(5.25)

5.3 Optimal Measurements

75

 : Dϕ]t , t ∈ R, is Connes’ cocycle derivative and [D ψ  : Dϕ]−i/2 is the where [D ψ analytic continuation at t = −i/2, see Lemma A.58 of Sect. A.7. Therefore,    : Dϕ]∗−i/2 [D ψ  : Dϕ]−i/2 . D2meas (ψϕ) = log ψ [D ψ Assume that M = B(H) on a finite-dimensional Hilbert space H and ψ, ϕ ∈  B(H)+ ∗ have the density operators Dψ , Dϕ . Then we note that ψ in (5.25) has the density  Dψ =

∞ −∞

1 D −it Dψ Dϕit dt. cosh πt ϕ

Therefore, h0 = (Dψ Dϕ−1/2 )∗ (Dψ Dϕ−1/2 ) = Dϕ−1/2 Dψ Dϕ−1/2  ∞ 1 D −it Dϕ−1/2 Dψ Dϕ−1/2 Dϕ−it dt = cosh πt ϕ −∞ 1/2

1/2

and  Qmeas (ψϕ) 2

= Tr Dψ h0 =

∞ −∞

1 Tr Dψ Dϕ−it Dϕ−1/2 Dψ Dϕ−1/2 Dϕ−it dt. cosh πt

Since (5.24) is nothing but the familiar Lyapunov equation Dψ = 12 (h0 Dϕ +Dϕ h0 ), the alternative well-known formula of h0 is  h0 = 2

∞

e−t Dϕ Dψ e−t Dϕ dt.

0

Example 5.18 (Optimal Measurement for D1/2 ) We first show that for every ψ, ϕ ∈ M∗+ (with ψ(1) = 1), meas 1/2 (ψϕ) = −2 log F (ψ, ϕ). D1/2 (ψϕ) = D

Indeed, the latter equality is (3.29). By Example 5.9 (2) we note that Qmeas 1/2 (ψϕ) =

1 inf {ψ(x −1 ) + ϕ(x)}. 2 x∈M++

On the other hand, by (3.34) of Lemma 3.19 we have 1/2 (ψϕ) = 1 inf {ψ(x −1 ) + ϕ(x)} Q 2 x∈M++

(5.26)

76

5 Measured f -Divergences

so that (5.26) holds, which also says that the measured version of the fidelity F is the same as F itself and F (ψ, ϕ) = 12 supx∈M++ {ψ(x −1 ) + ϕ(x)}. This and other similar expressions for F were given in [2, 3]. Now, assume that hδψ ≤ λhδϕ for some δ, λ > 0, or equivalently, s(ψ) ≤ s(ϕ) and Connes’ cocycle derivative [Dψ : Dϕ]t has the analytic continuation to the strip −δ/2 ≤ Im z ≤ 0, see Lemma A.58 of Sect. A.7. Then by the p = 1 case of Lemma A.59 (Kosaki’s extension [81] of Sakai’s quadratic Radon–Nikodym theorem), there exists a unique k1 ∈ (s(ϕ)Ms(ϕ))+ such that hψ = k1 hϕ k1 , i.e., ψ(x) = ϕ(k1 xk1 ) for all x ∈ M. Moreover, in this case, k1 satisfies 1/2 1/2 1/2 1/2 (hϕ hψ hϕ )1/2 = hϕ k1 hϕ so that F (ψ, ϕ) = tr(hϕ1/2 hψ hϕ1/2 )1/2 = tr hϕ k1 = ϕ(k1 ).

(5.27)

r With the spectral decomposition k1 = 0 t dEk1 (t) with r := k1 , let p := dψ(Ek1 (t)) and q := dϕ(Ek1 (t)). Then p  q (absolutely continuous) since s(ψ) ≤ s(ϕ), and we have   r  1/2 dp 1/2 dp ≤ dq = lim dq ε 0 dq dq 0 ε    1 r −1 dp (t) + t dq(t) t ≤ lim ε 0 2 ε dq   r  r 1 = lim t −1 dp(t) + t dq(t) . ε 0 2 ε 0 

Qmeas 1/2 (ψϕ)

Furthermore, since 

r

t

−1

r 0

r



t dq(t) = ϕ(k1 ) and 

dp(t) = ψ

ε

=ϕ

r

t

ε  r ε

−1

  dEk1 (t) = ϕ k1 

 −→ ϕ(k1 )

t dEk1 (t)

r

t ε

−1

  dEk1 (t) k1

as ε 0,

it follows that  meas (ψϕ) ≥ D1/2 (pq) = −2 log D1/2

r 0



dp dq

1/2 dq ≥ −2 log ϕ(k1 ).

(5.28)

By (5.26)–(5.28) we find that meas D1/2 (ψϕ) = −2 log F (ψ, ϕ) = −2 log ϕ(k1 )

and the optimal measurement for D1/2 is induced from the spectral resolution of k1 .

5.3 Optimal Measurements

77

In the finite-dimensional case with the density operators Dψ , Dϕ with ϕ faithful, the optimal measurement is given by the spectral decomposition of k1 := Dϕ−1/2 (Dϕ1/2 Dψ Dϕ1/2 )1/2Dϕ−1/2 = Dψ #Dϕ−1 , the geometric mean of Dψ and Dϕ−1 , as explained in [100, Sec. 9.2.2]. Problems 5.19 pr

(1) An example of an operator convex function on (0, ∞) for which Sf (ψϕ) < Sfmeas (ψϕ) for some ψ, ϕ ∈ M∗+ is not known. Such an example might exist in the finite-dimensional setting if any. (2) In the attainability results in Theorems 5.13 and 5.14 (as well as Examples 5.17 and 5.18) we have proved the existence of von Neumann measurements induced by bounded self-adjoint operators in M, under such an assumption as ψ ≤ λϕ or more strongly λ−1 ϕ ≤ ψ ≤ λϕ for some λ > 0. These dominance assumptions are rather too strong in the infinite-dimensional setting, so it is desirable to remove them by considering measurements induced by unbounded self-adjoint operators. However, the problem seems difficult because the compactness argument has been used in the proof of the theorems. (3) It is also interesting to find a more explicit form of optimal measurements, as in Examples 5.17 and 5.18, in the attainability results of Theorems 5.13 and 5.14. Since an optimal measurement is given as the maximizer of a certain strictly convex function on a certain set M[a,b] , the maximizer might explicitly be specified.

Chapter 6

Reversibility and Quantum Divergences

6.1 Petz’ Recovery Map Let M and N be von Neumann algebras, whose standard forms are (M, H, J, P) and (N, H0 , J0 , P0 ), respectively. For convenience, we first summarize basic properties of positive linear maps between von Neumann algebras, although those have already been used in previous chapters. For a positive linear map γ : N → M, γ is unital if γ (1) = 1. The γ is normal if γ (yj )  γ (y) for any net {yj } in N+ with yj  y ∈ N+ , or equivalently, γ is continuous with respect to the σ -weak topologies (i.e., the σ (N, N∗ ), σ (M, M∗ )-topologies) on N and M. We call γ a Schwarz map if γ (y)∗ γ (y) ≤ γ (y ∗ y) for all y ∈ N. The γ is said to be n-positive if γ ⊗ idn : N ⊗ Mn → M ⊗ Mn is positive, where Mn is the n × n matrix algebra and (γ ⊗ idn )([yij ]) := [γ (yij )] for [yij ]ni,j =1 ∈ Mn (N) = N ⊗ Mn . Furthermore, γ is said to be completely positive (CP) if it is n-positive for all n ≥ 1. As is well-known, CP ⇒ 2-positive ⇒ Schwarz map ⇒ simply positive. In quantum information in the finite-dimensional setting, quantum channels (implementing quantum operations) are typically TPCP (trace-preserving CP) maps, whose dual maps are unital CP maps. But we may consider, depending on problems, weaker notions of positivity such as Schwarz positivity or 2-positivity or even plain positivity for quantum operations. In the von Neumann algebra setting, normality is essential for a channel γ : N → M, which means that γ is the dual of the predual map γ∗ : M∗ → N∗ . The next theorem was proved in [1, 106] under the assumption that both of ϕ and ϕ ◦ γ are faithful, but the faithfulness of ϕ is not essential. Theorem 6.1 ([1, 106]) Let γ : N → M be a unital normal positive map, and ϕ ∈ M∗+ . Assume that ϕ ◦ γ is faithful. Let ∈ P and 0 ∈ P0 be the vector representatives of ϕ and ϕ ◦γ , respectively. Then there exists a unique unital normal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_6

79

80

6 Reversibility and Quantum Divergences

positive map β : M → N such that

J0 β(x) 0 , y 0  = J x , γ (y) 

(6.1)

for all x ∈ M and all y ∈ N. Proof First, assuming that for each x ∈ M, β(x) ∈ N exists so that (6.1) holds for all y ∈ N, we prove that β : M → N is a unital normal positive map. Since 0 is a cyclic vector for N (i.e., N 0 := {y 0 : y ∈ N} is dense in H0 ), it is obvious that β(x) ∈ N is uniquely determined by (6.1) and β : M → N is a linear map. Since

J , γ (y)  = (ϕ ◦ γ )(y) = J0 0 , y 0  for all y ∈ N, we have β(1) = 1. For every x ∈ M+ and y ∈ N, it follows from (6.1) that

(J0 β(x)J0 )y 0 , y 0  = J0 β(x) 0, y ∗ y 0  = J x , γ (y ∗ y)  = (J xJ )γ (y ∗y)1/2 , γ (y ∗ y)1/2  ≥ 0, showing that J0 β(x)J0 ≥ 0 and so β(x) ∈ N+ . Hence β is a unital positive map. To prove the normality of β, let {xj } ⊂ M+ be a net with xj  x ∈ M+ ; then β(xj )  y0 for some y0 ∈ N+ . For every y ∈ N we have

J0 y0 0 , y 0  = lim J0 β(xj ) 0 , y 0  = lim J xj , γ (y)  = J x , γ (y) , j

j

which implies that y0 = β(x) so that β(xj )  β(x). Hence β is normal. It remains to prove, for each x ∈ M, the existence of β(x) satisfying (6.1) for all y ∈ N. For this, by linearity we may assume that x ∈ M+ . Define ψx ∈ N∗ by ψx (y) := J x , γ (y) , y ∈ N. For any y ∈ N+ one has ψx (y) = γ (y)1/2 (J xJ )γ (y)1/2 ,  ≤ x γ (y) ,  = x(ϕ ◦ γ )(y) so that 0 ≤ ψx ≤ x(ϕ ◦ γ ). Hence by Lemma A.24 of Sect. A.3 there exists a B ∈ N such that x = B 0 , where x ∈ P0 is the vector representative of ψx . Since x = J0 x = J0 B 0 = J0 BJ0 0 , one has for every y ∈ N, ψx (y) = B 0 , yJ0 BJ0 0  = J0 B ∗ J0 B 0 , y 0  = J0 B ∗ x , y 0  = J0 B ∗ B 0 , y 0 . Letting β(x) := B ∗ B ∈ N+ one obtains ψx (y) = J0 β(x) 0, y 0 , y ∈ N, which gives (6.1). Definition 6.2 For γ : N → M and ϕ ∈ M∗+ as in Theorem 6.1, we refer to β : M → N satisfying (6.1) as Petz’ recovery map of γ with respect to ϕ, and denote it by γϕ∗ . The γϕ∗ is defined by (6.1) independently of the choice of the standard forms of M, N due to Theorem A.19 of Sect. A.3.

6.1 Petz’ Recovery Map

81

Proposition 6.3 Let γ : N → M and ϕ ∈ M∗+ be as in Theorem 6.1. (1) ϕ ◦ γ ◦ γϕ∗ = ϕ. (2) Assume that ϕ is faithful as well as ϕ ◦ γ . Then (γϕ∗ )∗ϕ◦γ = γ , i.e., γ is Petz’ recovery map of γϕ∗ with respect to ϕ ◦ γ . (3) If γ is 2-positive (resp., CP), then γϕ∗ is 2-positive (resp., CP). Proof We write β for γϕ∗ . (1) Letting y = 1 in (6.1) gives ϕ(x) = J x ,  = J0 β(x) 0 , 0  = ϕ ◦ γ (β(x)),

x ∈ M,

so that ϕ = ϕ ◦ γ ◦ β. (2) immediately follows from (1) and (6.1) since the roles of γ and β are symmetric in (6.1). (3) Assume that γ is 2-positive. Consider the von Neumann algebra M (2) := M ⊗ M2 , the tensor product of M with the 2 × 2 matrix algebra M2 (C), whose standard form is described in Example A.20 of Sect. A.3. In fact, the standard representation of M (2) is given in (A.5) on H (2) := H ⊕ H ⊕ H ⊕ H with the (2) (2) standard involution J (2) in (A.7). Similarly, we take the standard H0 and J0   x11 x12 (2) for N (2) := N ⊗ M2 . Set ϕ (2) ∈ (M (2))+ := ϕ(x11) + ∗ by ϕ x21 x22 ϕ(x22 ). Since γ is 2-positive, we have ϕ (2) ◦(γ ⊗id2 ) ∈ (N (2) )+ ∗ , which is given   y y 11 12 as ϕ (2) ◦ (γ ⊗ id2 ) = ϕ ◦ γ (y11 ) + ϕ ◦ γ (y22 ). Moreover, note y21 y22   0 that the vector representatives of ϕ (2) and ϕ (2) ◦ (γ ⊗ id2 ) are (2) = 0   0 0 (2) (= 0 ⊕ 0 ⊕ 0 ⊕ 0 ), respectively, (= ⊕ 0 ⊕ 0 ⊕ ) and 0 = 0 0     x11 x12 y11 y12 (2) see Example A.20. Then for x = ∈ M and y = ∈ N (2) , x21 x22 y21 y22 with β = γϕ∗ , we have (2)

J0 (β

(2) (2) ⊗ id2 )(x) 0 , y 0 

=

2



J0 β(xij ) 0 , yij 0 

i,j =1

=

2



J xij , γ (yij ) 

i,j =1

= J (2) x (2) , (γ ⊗ id2 )(y) (2), so that β ⊗ id2 is Petz’ recovery map of γ ⊗ id2 with respect to ϕ (2) . Hence β ⊗ id2 is positive (as seen from the proof of Theorem 6.1), i.e., β is 2-positive.

82

6 Reversibility and Quantum Divergences

For any n ≥ 1, if γ is n-positive, then one can prove that so is β, in a similar way by taking the standard forms of M ⊗ Mn and N ⊗ Mn (while the details are omitted). Thus, if γ is CP, then so is β. Remark 6.4 Petz’ recovery map γϕ∗ of a Schwarz map γ is not necessarily a Schwarz map, as shown by [65, Proposition 2] even in the finite-dimensional setting. This is the reason why we often need to assume in the reversibility problem that a quantum operation is 2-positive rather than a Schwarz map. Let M0 be a von Neumann subalgebra of M and (M0 , H0 , J0 , P0 ) be the standard form of M0 . For a faithful ϕ ∈ M∗+ let 0 ∈ P0 be the vector representative of ϕ|M0 as well as ∈ P of ϕ. Then the generalized conditional expectation Eϕ = EϕM0 : M → M0 (due to Accardi and Cecchini [1]) is given as Eϕ (x) := J0 Vϕ∗ J xJ Vϕ J0 ,

x ∈ M,

which is also defined by Eϕ (x) 0 = J0 Vϕ∗ J x ,

x ∈ M,

where Vϕ : H0 → H is the isometry defined by Vϕ y 0 := y , y ∈ M0 . In this case, since

J0 Eϕ (x) 0 , y 0  = J x , Vϕ y 0  = J x , y ,

x ∈ M, y ∈ M0 ,

we have the following: Proposition 6.5 Let M0 be a von Neumann subalgebra of M. If ϕ ∈ M∗+ is faithful, then Petz’ recovery map of the inclusion map M0 → M with respect to ϕ is the generalized conditional expectation Eϕ : M → M0 . We can extend Petz’ recovery map γϕ∗ to arbitrary ϕ ∈ M∗+ as follows: Proposition 6.6 Let γ be as in Theorem 6.1. For every ϕ ∈ M∗+ let ∈ P and 0 ∈ P0 be the vector representatives of ϕ and ϕ ◦ γ , respectively. Let e := s(ϕ) ∈ M and e0 := s(ϕ ◦ γ ) ∈ N. Then there exists a unique unital (i.e., β(1) = e0 ) normal positive map β : M → e0 Ne0 such that (6.1) holds for all x ∈ M and all y ∈ e0 Ne0 . Furthermore, β(1 − e) = 0 and β(x) = β(exe) for all x ∈ M, and we have: (1) ϕ ◦ γ ◦ β = ϕ. (2) If γ is 2-positive (resp., CP), then β is 2-positive (resp., CP). Proof Let e0 := J0 e0 J0 = sN (ϕ ◦ γ ) and q0 := e0 e0 . By Proposition A.16 note that e0 Ne0 ∼ = q0 Nq0 (by y ∈ e0 Ne0 → yq0 ∈ q0 Nq0 ) and the standard form of q0 Nq0 is (q0 Nq0 , q0 H0 , q0 J0 q0 , q0 P0 ). Moreover, 0 = q0 0 ∈ q0 P0 . Although γ |e0 Ne0 : e0 Ne0 → M is not necessarily unital, we can apply the proof of Theorem 6.1 to γ |e0 Ne0 , so we have a unique normal map β : M → e0 Ne0 such

6.1 Petz’ Recovery Map

83

that

(q0 J0 q0 )(β(x)q0) 0 , (yq0 ) 0  = J x , γ (y) ,

x ∈ M, y ∈ e0 Ne0 .

Since q0 J0 q0 = e0 e0 J0 , the above LHS is equal to

J0 β(x) 0, e0 e0 y 0  = J0 β(x) 0 , y 0 . Hence β is determined by (6.1) for all x ∈ M and y ∈ e0 Ne0 . Then β(1) = e0 follows since J , γ (y)  = (ϕ ◦ γ )(y) = J0 e0 0 , y 0  for y ∈ e0 Ne0 . Moreover, for every x ∈ M, since J x(1 − e) = 0, one has β(x(1 − e)) = 0 and β((1 − e)x) = β(x ∗ (1 − e))∗ = 0. Therefore, β(1 − e) = 0 and β(x) = β(exe) for all x ∈ M. Furthermore, since ϕ ◦ γ (e0 ) = ϕ ◦ γ (1) = ϕ(1), one has ϕ(1 − γ (e0 )) = 0 so that (1 − γ (e0 )) = 0. Letting y = e0 in (6.1) gives the equality in (1) as in the proof of Proposition 6.3 (1). The assertion in (2) is seen similarly to the proof of Proposition 6.3 (3). Remark 6.7 Here we give a slightly different description of β than that given in Proposition 6.6. In the situation of the proposition define γˆ := eγ (·)e|e0 Ne0 : e0 Ne0 −→ eMe.

(6.2)

Since ϕ(1 − γ (e0 )) = 0 as seen in the proof of Proposition 6.6, e(1 − γ (e0 ))e = 0 so that γˆ (e0 ) = e. Hence γˆ is a unital normal positive map, so we can define Petz’ recovery map βˆ of γˆ with respect to ϕ|eMe (Definition 6.2). Now we show that β is essentially the same as βˆ in such a way that ˆ β(x) = β(exe) = β(exe),

x ∈ M.

(6.3)

Indeed, consider the standard form (qMq, qH, qJ q, qP) of eMe ∼ = qMq where q := eJ eJ , as well as (q0 Nq0 , q0 H0 , q0 J0 q0 , q0 P0 ) of e0 Ne0 ∼ = q0 Nq0 in the proof of Proposition 6.6. Then βˆ is determined by ˆ

(q0 J0 q0 )(β(x)q 0 ) 0 , (yq0 ) 0  = (qJ q)(xq) , γˆ (y) ,

x ∈ eMe, y ∈ e0 Ne0 ,

which can easily be reduced to ˆ

J0 β(x) 0 , y 0  = J x , γ (y) ,

x ∈ eMe, y ∈ e0 Ne0 .

Hence for every x ∈ M and y ∈ e0 Ne0 we have ˆ

J0 β(exe) 0 , y 0  = J exe , γ (y)  = J x , J eJ γ (y)  = J x , γ (y) , showing (6.3) by Proposition 6.6.

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6 Reversibility and Quantum Divergences

We denote β : M → e0 Ne0 in Proposition 6.6 by the same γϕ∗ as in Definition 6.2, which may be called again Petz’ recovery map of γ with respect to ϕ. This extended version will be used later in Sects. 6.4, 7.1 and 8.1. Example 6.8 Here we consider the finite-dimensional case M = B(H) and N = B(K) with dim H, dim K < ∞. Let T : B(H) → B(K) be a trace-preserving positive map, so the dual map γ = T ∗ : B(K) → B(H) is a unital positive map. ∗ For every ϕ ∈ B(H)+ ∗ with density Dϕ , the density of ϕ ◦ γ ∈ B(K)+ is T (Dϕ ) and E0 = s(ϕ ◦ γ ) is the support projection of T (Dϕ ). Let β : B(H) → E0 B(K)E0 be as given in Proposition 6.6. Recall that the standard representation of B(H) is the left multiplication on B(H) with the Hilbert–Schmidt inner product and J = ∗ . Hence for every X ∈ B(H), β(X) is given as Tr T (Dϕ )1/2 β(X)T (Dϕ )1/2 Y = Tr T (Dϕ1/2 XDϕ1/2 )Y, 1/2

1/2

1/2

Y ∈ E0 B(K)E0 .

1/2

1/2

1/2

Since E0 T (Dϕ )E0 = T (Dϕ ) and E0 T (Dϕ XDϕ )E0 = T (Dϕ XDϕ ), we have T (Dϕ )1/2 β(X)T (Dϕ )1/2 = T (Dϕ1/2 XDϕ1/2 ).

(6.4)

Therefore, β(X) = T (Dϕ )−1/2 T (Dϕ1/2 XDϕ1/2 )T (Dϕ )−1/2 , where T (Dϕ )−1/2 is defined under restriction to the support E0 K. This expression of β is the same as in [58, (3.19)].

6.2 A Technical Lemma The next notion of multiplicative domains plays an important role not only in operator algebras but also in quantum information. Definition 6.9 Let γ : N → M be a unital normal Schwarz map between von Neumann algebras. The multiplicative domain of γ is defined as Mγ := {y ∈ N : γ (y ∗ y) = γ (y)∗ γ (y), γ (yy ∗ ) = γ (y)γ (y ∗ )}. The next result was first shown by Choi [25] when γ is a unital 2-positive map. See [63, Lemma 3.9] for the case of a Schwarz map.

6.2 A Technical Lemma

85

Proposition 6.10 Let γ : N → M be as in Definition 6.9. For any y ∈ N, γ (y ∗ y) = γ (y)∗ γ (y) if and only if γ (zy) = γ (z)γ (y) for all z ∈ N. Consequently, Mγ = {y ∈ N : γ (zy) = γ (z)γ (y), γ (yz) = γ (y)γ (z) for all z ∈ N}, and hence Mγ is a von Neumann subalgebra of N. When γ is a unital normal Schwarz map from M into itself, we also consider the fixed-point set Fγ := {x ∈ M : γ (x) = x}. The set Fγ is not generally a subalgebra of M, and there are no general inclusion relations between Fγ and Mγ , see [58, Appendix B]. But we have the next result, which seems to have been first observed in [86] (see also [14, Theorem 2.3], [24, Lemma 3.4] and [65, Theorem 1 (i)]). We give a proof for convenience. Lemma 6.11 Let γ be a unital normal Schwarz map from M into itself. Assume that there exists a faithful ϕ ∈ M∗+ such that ϕ ◦ γ = ϕ. Then Fγ = {x ∈ M : γ (zx) = γ (z)x, γ (xz) = xγ (z) for all z ∈ M} ⊂ Mγ , and hence Fγ is a von Neumann subalgebra of M. Proof Since the latter inclusion assertion is obvious by Proposition 6.10, it suffices to prove the first equality. The inclusion ⊃ is obvious. Conversely, let x ∈ Fγ . Then x ∗ x = γ (x)∗ γ (x) ≤ γ (x ∗ x) and ϕ(γ (x ∗ x) − x ∗ x) = ϕ(x ∗ x) − ϕ(x ∗ x) = 0, implying γ (x ∗ x) = x ∗ x = γ (x)∗ γ (x). Similarly, γ (xx ∗ ) = xx ∗ = γ (x)γ (x)∗ . Therefore, x ∈ Mγ and by Proposition 6.10 we have γ (zx) = γ (z)x and γ (xz) = xγ (z) for all z ∈ M. The next lemma due to Petz [106] will play a crucial role in the next section. Lemma 6.12 ([106]) Let γ : N → M be a unital normal 2-positive map, and ϕ ϕ ∈ M∗+ . Assume that both of ϕ and ϕ ◦ γ are faithful. Let σt be the modular ϕ◦γ automorphism group of M associated with ϕ and σt be that of N associated with ϕ ◦ γ (see Sect. A.2). Then for any v ∈ N the following conditions are equivalent: (i) γ (v ∗ v) = γ (v)∗ γ (v) and γ (σt (ii) γϕ∗ ◦ γ (v) = v.

ϕ◦γ

ϕ

(v)) = σt (γ (v)) for all t ∈ R;

Moreover, N1 := Fγϕ∗ ◦γ and M1 := Fγ ◦γϕ∗ are von Neumann subalgebras of N and M respectively, and γ |N1 is an isomorphism from N1 onto M1 , whose inverse is γϕ∗ |M1 . Proof (i) ⇒ (ii). Let ∈ P and 0 ∈ P0 be as in Theorem 6.1. Recall that the operator x → x ∗ (x ∈ M) is closable and its closure has the polar decomposition J 1/2 with the modular operator with respect to ϕ. Similarly, the

86

6 Reversibility and Quantum Divergences

closure of y 0 → y ∗ 0 (y ∈ N) is J0 0 with the modular operator 0 with ϕ◦γ ϕ respect to ϕ ◦ γ . Then σt (x) = it x −it (x ∈ M) and σt (y) = it0 y −it 0 (y ∈ N), see Sect. A.2. Since γ is a Schwarz map, one has 1/2

γ (y) 2 = ϕ(γ (y)∗ γ (y)) ≤ ϕ ◦ γ (y ∗ y) = y 0 2 ,

y ∈ N,

so that a contraction Vϕ : H0 → H can be defined by extending y 0 → γ (y) ϕ◦γ ϕ (y ∈ N). By the second assumption of (i) one has γ (σt (v ∗ )) = σt (γ (v ∗ )) so ϕ◦γ ∗ ϕ ∗ that Vϕ σt (v ) 0 = σt (γ (v )) for all t ∈ R. This means that Vϕ Δit0 v ∗ 0 = Δit γ (v ∗ ) for all t ∈ R. Hence, in view of Theorem B.1 of Appendix B, the analytic continuation of those at z = −i/2 gives Vϕ 0 v ∗ 0 = 1/2γ (v ∗ ) , 1/2

that is, Vϕ J0 v 0 = J γ (v) ,

(6.5)

which implies that Vϕ J0 v 0 2 = γ (v) 2 = ϕ(γ (v)∗ γ (v)) = ϕ(γ (v ∗ v)) = v 0 2 = J0 v 0 2 , where we have used the first assumption of (i). Letting ζ0 := J0 v 0 ∈ H0 , since Vϕ is a contraction, one has Vϕ∗ Vϕ ζ0 − ζ0 2 = Vϕ∗ Vϕ ζ0 2 − 2Vϕ ζ0 2 + ζ0 2 = Vϕ∗ Vϕ ζ0 2 − ζ0 2 ≤ 0, so that Vϕ∗ Vϕ ζ0 = ζ0 . This and (6.5) give J0 v 0 = Vϕ∗ Vϕ J0 v 0 = Vϕ∗ J γ (v) .

(6.6)

Therefore,

J0 v 0 , y 0  = Vϕ∗ J γ (v) , y 0  = J γ (v) , γ (y) ,

y ∈ N,

(6.7)

which, compared with (6.1), is equivalent to the equality in (ii). Before proving (ii) ⇒ (i), we prove the latter assertion. By Proposition 6.3 (1) and (2), ϕ ◦ (γ ◦ γϕ∗ ) = ϕ,

ϕ ◦ γ ◦ (γϕ∗ ◦ γ ) = ϕ ◦ γ .

(6.8)

6.2 A Technical Lemma

87

Since γ is 2-positive, so is γϕ∗ by Proposition 6.3 (3). Hence γ ◦ γϕ∗ and γϕ∗ ◦ γ are 2positive. Since ϕ and ϕ ◦ γ are faithful by assumption, it follows from Lemma 6.11 that the fixed-point sets M1 and N1 are von Neumann subalgebras of M and N, respectively. Since γ (y) = γ ◦ γϕ∗ ◦ γ (y) for all y ∈ N1 and γϕ∗ (x) = γϕ∗ ◦ γ ◦ γϕ∗ (x) for all x ∈ M1 , it is immediate to see that γ |N1 maps N1 to M1 bijectively and γϕ∗ |M1 is the inverse of γ |N1 . Next, assume that v ∈ N1 . Then

J0 v 0 , y 0  = J0 γϕ∗ (γ (v)) 0 , y 0  = J γ (v) , γ (y)  = Vϕ∗ J γ (v) , y 0 ,

y ∈ N,

so that (6.7) holds, which in turn gives the equality J0 v 0 = Vϕ∗ J γ (v) in (6.6). Hence ϕ ◦ γ (v ∗ v) = v 0 2 = Vϕ∗ J γ (v) 2 ≤ γ (v) 2 = ϕ(γ (v)∗ γ (v)). Since γ is a Schwarz map and ϕ is faithful, γ (v ∗ v) = γ (v)∗ γ (v) holds. Since v ∗ ∈ N1 , γ (vv ∗ ) = γ (v)γ (v)∗ holds as well, so that v ∈ Mγ . Thus, by Proposition 6.10 we have γ (yv) = γ (y)γ (v) and γ (vy) = γ (v)γ (y) for all y ∈ N, showing that γ |N1 is an isomorphism. (ii) ⇒ (i). Assume (ii), i.e., v ∈ N1 . The first equality in (i) was already shown above. As for the second, since (ϕ ◦ γ )|N1 = (ϕ|M1 ) ◦ (γ |N1 ), we find by using the KMS condition (see Sect. A.2) that1 (ϕ◦γ )|N1

γ (σt

ϕ|M1

(y)) = σt

(γ (y)),

y ∈ N1 , t ∈ R.

(6.9)

The mean ergodic theorem says that the norm-limit 1

(γ ◦ γϕ∗ )k (x) n→∞ n n−1

E(x) := lim

k=0

exists for every x ∈ M and E is a norm one projection from M onto M1 . By Lemma 6.11 with (6.8) it is immediate that E is a conditional expectation onto M1 such that ϕ ◦ E = ϕ. (This is also a consequence of Tomiyama’s theorem [131].) The normality of E is also immediate from ϕ ◦ E = ϕ since ϕ is faithful. Therefore, ϕ|M ϕ we have σt 1 = σt |M1 (t ∈ R) by Takesaki’s theorem (see Theorem A.10 of Sect. A.2). Similarly, replacing ϕ and γ ◦ γϕ∗ in the above argument with ϕ ◦ γ and (ϕ◦γ )|N1

γϕ∗ ◦ γ , we have σt second equality in (i).

ϕ◦γ

= σt

|N1 (t ∈ R). Inserting these into (6.9) gives the

1 More explicitly speaking, by the KMS condition based on Theorem A.7, one can easily see that if γ : N → M is a ∗-isomorphism between von Neumann algebras and ω ∈ M∗+ is faithful, then ω◦γ = γ −1 ◦ σtω ◦ γ for all t ∈ R. σt

88

6 Reversibility and Quantum Divergences

Remark 6.13 In the above proof of Lemma 6.12 we need the 2-positivity of γ to guarantee γϕ∗ being a Schwarz map (see Remark 6.4).

6.3 Preservation of Connes’ Cocycle Derivatives Throughout this section, let f be an operator convex function on (0, ∞), having the integral expression in (2.5). We write μf for the representing measure in (2.5) to specify its dependence on f , and denote its support by supp μf , i.e., the topological support of μf consisting of t ∈ [0, ∞) such that μf ((t − ε, t + ε)) > 0 for any ε > 0. Let M, N be von Neumann algebras and γ : N → M be a unital normal Schwarz map. Furthermore, let ψ, ϕ ∈ M∗+ and assume that both of ϕ and ϕ ◦ γ are faithful (but ψ is general). There are two main ingredients of the discussions below. One is the standard f -divergence Sf (ψϕ) and another is Connes’ cocycle derivative [Dψ : Dϕ]t (t ∈ R), whose concise survey is in Sect. A.7. In the section we prove two lemmas stating that the preservation of a standard f divergence implies that of Connes’ cocycle derivatives. Those lemmas are the most substantial parts of the proof of the reversibility/sufficiency theorems in Sect. 6.4. Lemma 6.14 Assume that supp μf has a limit point in (0, ∞). If Sf (ψ ◦ γ ϕ ◦ γ ) = Sf (ψϕ) < +∞,

(6.10)

then we have γ ([D(ψ ◦ γ ) : D(ϕ ◦ γ )]t ) = [Dψ : Dϕ]t ,

t ∈ R.

(6.11)

Proof We are working in the standard forms (M, H, J, P) and (N, H0 , J0 , P0 ) as before. Write ψ0 := ψ ◦ γ and ϕ0 := ϕ ◦ γ . Let , ∈ P and 0 , 0 ∈ P0 be the vector representatives of ψ, ϕ and ψ0 , ϕ0 , respectively. Below we divide the proof into several steps. Step 1. A contraction Vϕ : H0 → H is defined by extending y 0 → γ (y) (y ∈ N) as in the first part of the proof of Lemma 6.12. For every y ∈ N one has  ψ,ϕ Vϕ (y 0 )2 =  ψ,ϕ γ (y) 2 = γ (y)∗ 2 1/2

1/2

= ψ(γ (y)γ (y)∗ ) ≤ ψ(γ (yy ∗ )) = ψ0 (yy ∗) = y ∗ 0 2 =  ψ0 ,ϕ0 (y 0 )2 . 1/2

(6.12)

6.3 Preservation of Connes’ Cocycle Derivatives

89

1/2

Note that the operator ψ,ϕ Vϕ |N 0 is closable. Indeed, let yn ∈ N (n ∈ N) be 1/2

1/2

such that yn 0 → 0 and ψ,ϕ Vϕ (yn 0 ) = ψ,ϕ γ (yn ) → ζ ∈ H. Since γ (yn ) 2 = ϕ(γ (yn )∗ γ (yn )) ≤ ϕ(γ (yn∗ yn )) = yn 0 2 , 1/2

we have γ (yn ) → 0. Since ψ,ϕ is a closed operator, ζ = 0 follows. 1/2

Thus, ψ,ϕ Vϕ |N 0 has the closure T and (6.12) means that T (y 0 ) ≤ 1/2

1/2

 ψ0 ,ϕ0 (y 0 ) for all y ∈ N. Since N 0 is a core of ψ0 ,ϕ0 , we see by Proposition B.5 of Appendix B that T ∗ T ≤ ψ0 ,ϕ0

(6.13)

∞ in the sense of Proposition B.4. Let ψ,ϕ = 0 t dEψ,ϕ (t) be the spectral n decomposition of ψ,ϕ , and for each n ∈ N let Hn := 0 t dEψ,ϕ (t). Since N 0 is a core of T and 1/2

1/2

1/2

Hn Vϕ (y 0 ) = Hn (γ (y) ) ≤  ψ,ϕ (γ (y) ) = T (y 0 ),

y ∈ N,

we have Vϕ∗ Hn Vϕ ≤ T ∗ T by Proposition B.5 again. This and (6.13) show that Vϕ∗ Hn Vϕ ≤ ψ0 ,ϕ0 . Therefore, for every s > 0, (s1 + ψ0 ,ϕ0 )−1 ≤ (s1 + Vϕ∗ Hn Vϕ )−1 ≤ Vϕ∗ (s1 + Hn )−1 Vϕ , where the latter inequality above is a consequence of Hansen’s inequality [49] since t → (s + t)−1 is an operator monotone decreasing function on [0, ∞). Letting n → ∞ gives (s1 + ψ0 ,ϕ0 )−1 ≤ Vϕ∗ (s1 + ψ,ϕ )−1 Vϕ , Step 2.

s > 0.

(6.14)

Write the expression in (2.5) as  f (t) = a + b(t − 1) + cf2 (t) +

[0,∞)

gs (t) dμf (s),

t ∈ (0, ∞), (6.15)

where f2 (t) := (t − 1)2 and gs (t) := (t − 1)2 (t + s)−1 for s ∈ [0, ∞). Note that Sf (ψϕ) = (a − b)ϕ(1) + bψ(1) + cSf2 (ψϕ)  Sgs (ψϕ) dμf (s) < +∞, + [0,∞)

(6.16)

90

6 Reversibility and Quantum Divergences

Sf (ψ0 ϕ0 ) = (a − b)ϕ0 (1) + bψ0 (1) + cSf2 (ψ0 ϕ0 )  + Sgs (ψ0 ϕ0 ) dμf (s), [0,∞)

(6.17)

and ϕ0 (1) = ϕ(1), Sf2 (ψ0 ϕ0 ) ≤ Sf2 (ψϕ),

ψ0 (1) = ψ(1), Sgs (ψ0 ϕ0 ) ≤ Sgs (ψϕ)

(6.18)

by the monotonicity property in Theorem 2.7 (iv). Therefore, assumption (6.10) implies that Sgs (ψ0 ϕ0 ) = Sgs (ψϕ)

for μf -a.e. s ∈ [0, ∞).

Write gs (t) =

((t + s) − (1 + s))2 = t − (2 + s) + (1 + s)2 hs (t), t +s

t ∈ (0, ∞),

where hs (t) := (t + s)−1 for s ∈ [0, ∞). Since Sgs (ψϕ) = ψ(1) − (2 + s)ϕ(1) + (1 + s)2 Shs (ψϕ), Sgs (ψ0 ϕ0 ) = ψ0 (1) − (2 + s)ϕ0 (1) + (1 + s)2 Shs (ψ0 ϕ0 ),

(6.19) (6.20)

we have Shs (ψ0 ϕ0 ) = Shs (ψϕ)

for μf -a.e. s ∈ [0, ∞),

that is,

0 , (s1 + ψ0 ,ϕ0 )−1 0  = , (s1 + ψ,ϕ )−1  for μf -a.e. s ∈ [0, ∞). (6.21) We easily see that

, (z1 + ψ,ϕ )

−1



∞

 = 0

1 dEψ,ϕ (t) 2 z+t

is analytic in {z ∈ C : Re z > 0}, and similarly for 0 , (z1 + ψ0 ,ϕ0 )−1 0 . From the assumption on supp μf and (6.21) it follows that the set of s > 0 for which equality (6.21) holds has a limit point in (0, ∞). Thus, the coincidence theorem for analytic functions yields that for all s > 0,

0 , (s1 + ψ0 ,ϕ0 )−1 0  = , (s1 + ψ,ϕ )−1 

6.3 Preservation of Connes’ Cocycle Derivatives

91

and hence  

0 , Vϕ∗ (s1 + ψ,ϕ )−1 Vϕ − (s1 + ψ0 ,ϕ0 )−1 0  = 0 thanks to Vϕ 0 = . By (6.14) this implies that Vϕ∗ (s1 + ψ,ϕ )−1 Vϕ 0 = (s1 + ψ0 ,ϕ0 )−1 0 , and therefore Vϕ∗ (s1 + ψ,ϕ )−1 = (s1 + ψ0 ,ϕ0 )−1 0 ,

s > 0.

(6.22)

Step 3. Let C0 [0, ∞) denote the Banach space of continuous complex functions φ on the locally compact space [0, ∞) vanishing at infinity (i.e., limt →∞ φ(t) = 0) with the sup-norm. Noting that hs ∈ C0 [0, ∞) for s ∈ (0, ∞), we define A to be the norm-closed complex linear span of hs , s ∈ (0, ∞), in C0 [0, ∞). Since hs1 hs2 =

1 (hs − hs2 ) s2 − s1 1

for s1 = s2 ,

1 h2s = lim (hs − hs+ε ) (in the norm) for s > 0, ε 0 ε we see that A is a closed subalgebra of C0 [0, ∞). Obviously, hs (t) > 0 for all t ∈ [0, ∞), and hs (t1 ) = hs (t2 ) for every t1 , t2 ∈ [0, ∞) with t1 = t2 . Hence the Stone–Weierstrass theorem implies that A = C0 [0, ∞). Since φ ∈ C0 [0, ∞) −→ Vϕ∗ φ( ψ,ϕ ) , φ( ψ0 ,ϕ0 ) 0 ∈ H0 are bounded linear maps with respect to the sup-norm on C0 [0, ∞) and the norm on H0 , it follows from (6.22) that Vϕ∗ φ( ψ,ϕ ) = φ( ψ0 ,ϕ0 ) 0 , Step 4.

φ ∈ C0 [0, ∞).

(6.23)

For every s > 0 one has by (6.22) Vϕ∗ (s1 + ψ,ϕ )−1 2 = 0 , (s1 + ψ0 ,ϕ0 )−2 0  = 0 , Vϕ∗ (s1 + ψ,ϕ )−2  (by (6.23)) = , (s1 + ψ,ϕ )−2  = (s1 + ψ,ϕ )−1 2 .

(since Vϕ 0 = )

92

6 Reversibility and Quantum Divergences

Letting ζ := (s1 + ψ,ϕ )−1 ∈ H, since the above means Vϕ∗ ζ 2 = ζ 2 , one has Vϕ Vϕ∗ ζ − ζ 2 = Vϕ Vϕ∗ ζ 2 − 2Vϕ∗ ζ 2 + ζ 2 = Vϕ Vϕ∗ ζ 2 − ζ 2 ≤ 0, so that Vϕ Vϕ∗ ζ = ζ , i.e., Vϕ Vϕ∗ (s1 + ψ,ϕ )−1 = (s1 + ψ,ϕ )−1 . By (6.22) this implies that Vϕ (s1 + ψ0 ,ϕ0 )−1 0 = (s1 + ψ,ϕ )−1 ,

s > 0.

Hence, similarly to Step 3 we have Vϕ φ( ψ0 ,ϕ0 ) 0 = φ( ψ,ϕ ) , Step 5.

φ ∈ C0 [0, ∞).

(6.24)

φ(0) := 0.

(6.25)

For each t ∈ R set φ(x) := x it

(x > 0),

For n ∈ N let ⎧ ⎪ nx ⎪ ⎪ ⎪ ⎨1 κn (x) := ⎪ 1 − n(x − n) ⎪ ⎪ ⎪ ⎩ 0

(0 ≤ x ≤ 1/n), (1/n ≤ x ≤ n), (n ≤ x ≤ n + n1 ),

(6.26)

(x ≥ n + n1 ),

and φn (x) := κn (x)φ(x) for x ∈ [0, ∞). Let s(ψ), s(ψ0 ) be the support projections of ψ, ψ0 , respectively. Note that the support projection of ψ,ϕ is s(ψ) and that of ψ0 ,ϕ0 is s(ψ0 ), because ϕ, ϕ0 are faithful. Then it is immediate to verify that φn ∈ C0 [0, ∞) and φn ( ψ,ϕ ) −→ φ( ψ,ϕ ) = s(ψ) itψ,ϕ

strongly,

φn ( ψ0 ,ϕ0 ) −→ φ( ψ0 ,ϕ0 ) = s(ψ0 ) itψ0 ,ϕ0

strongly.

By (6.24), Vϕ φn ( ψ0 ,ϕ0 ) 0 = φn ( ψ,ϕ ) . Letting n → ∞ gives Vϕ (s(ψ0 ) itψ0 ,ϕ0 0 ) = s(ψ) itψ,ϕ .

6.3 Preservation of Connes’ Cocycle Derivatives

93

−it Since −it ϕ = and ϕ0 0 = 0 , where ϕ = ϕ,ϕ is the modular operator associated with ϕ, we have it −it Vϕ (s(ψ0 ) itψ0 ,ϕ0 −it ϕ0 0 ) = s(ψ) ψ,ϕ ϕ .

(6.27)

From Proposition A.47 and Remark A.48 we write [Dψ : Dϕ]t = s(ψ) itψ,ϕ −it ϕ ,

[Dψ0 : ϕ0 ]t = s(ψ0 ) itψ0 ,ϕ0 −it ϕ0 ,

t ∈ R.

(Normally, s(ψ) and s(ψ0 ) are removed in the above formulas, because the support projection of ψ,ϕ is s(ψ), see Proposition A.22 (1), and itψ,ϕ is defined with restriction on s(ψ)H.) Hence (6.27) means that Vϕ ([Dψ0 : Dϕ0 ]t 0 ) = [Dψ : Dϕ]t , that is, γ ([Dψ0 : Dϕ0 ]t ) = [Dψ : Dϕ]t . Since is separating for M (i.e., x ∈ M, x = 0 at (6.11).

⇒ x = 0), we arrive

Lemma 6.15 Assume that γ is 2-positive. If (6.11) holds, then we have γϕ∗ ([Dψ : Dϕ]t ) = [D(ψ ◦ γ ) : D(ϕ ◦ γ )]t , γ

◦ γϕ∗ ([Dψ

: Dϕ]t ) = [Dψ : Dϕ]t ,

ψ ◦γ

◦ γϕ∗

t ∈ R,

t ∈ R,

(6.28) (6.29)

= ψ.

(6.30)

Proof We write ϕ0 := ϕ ◦ γ , ψ0 := ψ ◦ γ , ut := [Dψ : Dϕ]t and vt := [Dψ0 : Dϕ0 ]t (t ∈ R). The assumption says that γ (vt ) = ut for all t ∈ R. From properties of Connes’ cocycle derivatives in Theorem A.52 (i) we have ϕ(γ (vt )∗ γ (vt )) = ϕ(u∗t ut ) = ϕ(σt (u0 )) = ϕ(u0 ) = ϕ(γ (v0 )) = ϕ0 (v0 ), ϕ

ϕ(γ (vt∗ vt )) = ϕ0 (vt∗ vt ) = ϕ0 (σt 0 (v0 )) = ϕ0 (v0 ), ϕ

t ∈ R.

Since γ is 2-positive (hence a Schwarz map) and ϕ is faithful, the above equalities imply that γ (vt∗ vt ) = γ (vt )∗ γ (vt ),

t ∈ R.

(6.31)

94

6 Reversibility and Quantum Divergences

From the cocycle identity of vt , ut (Theorem A.52 (ii)) and Proposition 6.10 we find that γ (σsϕ0 (vt )) = γ (vs∗ vs+t ) = γ (vs )∗ γ (vs+t ) = u∗s us+t = σsϕ (ut ) = σsϕ (γ (vt )),

s, t ∈ R.

(6.32)

By (6.31) and (6.32) it follows from Lemma 6.12 that vt ∈ N1 so that ut = γ (vt ) ∈ M1 and γϕ∗ (ut ) = vt for all t ∈ R, where N1 , M1 are as given in Lemma 6.12. Therefore, γ ◦ γϕ∗ (ut ) = γ (vt ) = ut for all t ∈ R, so (6.28) and (6.29) have been shown. Moreover, since γ |N1 and γϕ∗ |M1 are ∗-isomorphisms by Lemma 6.12, note that vt ∈ Mγ and ut ∈ Mγϕ∗ (see Definition 6.9). Now, let , 0 be the vector representatives of ϕ, ϕ0 , respectively, as in Theorem 6.1. For every x ∈ M, y ∈ N and s, t ∈ R, since itψ0 ,ϕ0 = vt itϕ0 and itψ,ϕ = ut itϕ (Proposition A.47 and Remark A.48), we obtain ∗ it

J0 y is ψ0 ,ϕ0 0 , γϕ (x) ψ0 ,ϕ0 0 

= J0 yvs 0 , γϕ∗ (x)vt 0  = J0 yvs 0 , γϕ∗ (x)γϕ∗ (ut ) 0  = J0 yvs 0 , γϕ∗ (xut ) 0 

(by Proposition 6.10, since ut ∈ Mγϕ∗ )

= J0 γϕ∗ (xut ) 0 , yvs 0  = J xut , γ (yvs )  = J γ (yvs ) , xut  = J γ (y)us , xut 

(by (6.1)) (since vs ∈ Mγ )

it = J γ (y) is ψ,ϕ , x ψ,ϕ .

By analytic continuation of the above both sides twice at s = −i/2 and then at t = −i/2, it follows that

J0 y ψ0 ,ϕ0 0 , γϕ∗ (x) ψ0 ,ϕ0 0  = J γ (y) ψ,ϕ , x ψ,ϕ , 1/2

1/2

1/2

1/2

that is,

J0 y0 , γϕ∗ (x)0  = J γ (y), x,

x ∈ M, y ∈ N,

(6.33)

where , 0 are the vector representatives of ψ, ψ0 , respectively. Taking y = 1 in (6.33) yields ψ0 ◦ γϕ∗ (x) = ψ(x) for all x ∈ M, showing (6.30). Remark 6.16 Assume in Lemma 6.15 that ψ is faithful too. Then from Theorem A.52 (i) one has u0 = s(ψ) = 1 and s(ψ0 ) = v0 = γϕ∗ (u0 ) = 1. Hence ψ0 = ψ ◦ γ is also faithful automatically. Therefore, equality (6.33) means that γϕ∗ is Petz’ recovery map of γ with respect to ψ, so we have γψ∗ = γϕ∗ .

6.4 Reversibility via Standard f -Divergences

95

6.4 Reversibility via Standard f -Divergences Throughout the section, assume that γ : N → M is a unital normal 2-positive map between von Neumann algebras. For given ψ, ϕ ∈ M∗+ we say that γ is reversible for {ψ, ϕ} if there exists a unital normal 2-positive map β : M → N such that ψ ◦γ ◦β =ψ

and ϕ ◦ γ ◦ β = ϕ.

When M0 is a von Neumann subalgebra of M, we say that M0 is sufficient for {ψ, ϕ} if the inclusion map M0 → M is reversible for {ψ, ϕ}, i.e., there exists a unital normal 2-positive map β : M → M0 such that ψ ◦ β = ψ and ϕ ◦ β = ϕ. Let , ∈ P be the vector representatives of ψ, ϕ, respectively, in the standard form (M, H, J, P). The transition probability of ψ, ϕ ∈ M∗+ is P (ψ, ϕ) := ,  = −S−t 1/2 (ψϕ),

(6.34)

which already appeared in (3.30). Note that P (ψ, ϕ) has the (reverse) monotonicity property under unital normal Schwarz maps (by Theorem 2.7 (iv)). The next theorem was originally proved by Petz [106, Theorem 5] under the assumption that ϕ, ϕ ◦ γ , ψ and ψ ◦ γ are all faithful. The faithfulness assumption of ψ and ψ ◦ γ was later removed in Jenˇcová and Petz [68, Theorem 3]. Theorem 6.17 ([68, 106]) Let ψ, ϕ ∈ M∗+ and assume that ϕ and ϕ ◦γ are faithful. Then the following conditions (i)–(v) are equivalent: (i) (ii) (iii) (iv) (v)

ψ ◦ γ ◦ γϕ∗ = ψ (ϕ ◦ γ ◦ γϕ∗ = ϕ is automatic by Proposition 6.3 (1)); γ is reversible for {ψ, ϕ}; P (ψ, ϕ) = P (ψ ◦ γ , ϕ ◦ γ ); [Dψ : Dϕ]t = γ ([D(ψ ◦ γ ) : D(ϕ ◦ γ )]t ) for all t ∈ R; [Dψ : Dϕ]t = γ ◦ γϕ∗ ([Dψ : Dϕ]t ) for all t ∈ R.

Moreover, if ψ and ψ ◦ γ are faithful too, then the above (i)–(v) are equivalent to the following: (vi) γψ∗ = γϕ∗ . Proof (i) ⇒ (ii) is obvious. In view of (6.34), (ii) ⇒ (iii) is immediate from the monotonicity property of the standard f -divergence. (iii) ⇒ (iv). Condition (iii) is rephrased as S−t 1/2 (ψϕ) = S−t 1/2 (ψ ◦ γ ϕ ◦ γ ), whose value is finite. Recall the well-known integral expression of −t 1/2 −t

1/2

1 = −t + π



 (0,∞)

t t − 1+s t +s



s −1/2 ds,

96

6 Reversibility and Quantum Divergences

which is rewritten2 in the form of (2.5) as −t

1/2

1 1 = −1 − (t − 1) + 2 π

 (0,∞)

(t − 1)2 s 1/2 ds, t + s (1 + s)2

so that the support of the representing measure of −t 1/2 is (0, ∞). Hence (iii) ⇒ (iv) is a special case of Lemma 6.14. (iv) ⇒ (i) and (iv) ⇒ (v) are contained in Lemma 6.15. (v) ⇒ (iii). Let M be the von Neumann subalgebra of M generated by {[Dψ : Dϕ]t : t ∈ R}. From the cocycle identity of [Dψ : Dϕ]t (see Theorem A.52 (ii)), M is globally invariant under the modular automorphism σ ϕ . Hence by Theorem A.10 (Sect. A.2) says that there exists the conditional expectation E from M onto M with ϕ ϕ| respect to ϕ. Furthermore, since σt |M = σt M for all t ∈ R by Theorem A.10, ϕ| it follows that [Dψ : Dϕ]t is a σ M -cocycle in M. Therefore, Connes’ inverse theorem (Theorem A.54) implies that there exists a (unique) normal semifinite weight ψ˜ on M such that [Dψ : Dϕ]t = [D ψ˜ : D(ϕ|M )]t ,

t ∈ R.

(6.35)

On the other hand, one has [D(ψ˜ ◦ E) : Dϕ]t = [D(ψ˜ ◦ E) : D((ϕ|M ) ◦ E)]t = [D ψ˜ : D(ϕ|M )]t ,

t ∈ R,

(6.36)

due to Proposition A.55. Combining (6.35) and (6.36) implies by Proposition A.53 (3)3 that ψ = ψ˜ ◦ E and hence ψ ◦ E = ψ. Now, let N1 := Fγϕ∗ ◦γ and M1 := Fγ ◦γϕ∗ (see Lemma 6.12) and assume (v), that is, M ⊂ M1 . Then one has P (ψ|M1 , ϕ|M1 ) ≤ P (ψ|M , ϕ|M ) ≤ P (ψ, ϕ)

(since ψ = (ψ|M ) ◦ E, ϕ = (ϕ|M ) ◦ E)

≤ P (ψ ◦ γ , ϕ ◦ γ ) ≤ P ((ψ ◦ γ )|N1 , (ϕ ◦ γ )|N1 )

2 This

can be checked by a direct computation  0

3 This

∞ 

  (t − 1)2 s 1/2 t t ds − s −1/2 − 1+s t +s t + s (1 + s)2  ∞  ∞ π ds 2ds = (t − 1) = (t − 1). = (t − 1) 2 1/2 2 )2 2 (1 + s) s (1 + s 0 0

holds also for normal semifinite weights.

(6.37)

6.4 Reversibility via Standard f -Divergences

97

= P ((ψ|M1 ) ◦ (γ |N1 ), (ϕ|M1 ) ◦ (γ |N1 )) = P (ψ|M1 , ϕ|M1 ), where the last equality holds since γ |N1 : N1 → M1 is an isomorphism (Lemma 6.12). Therefore, (iii) follows. Finally, assume that ψ and ψ ◦ γ are also faithful. Then (iv) ⇒ (vi) was shown in the proof of Lemma 6.15 as noted in Remark 6.16. (vi) ⇒ (i) is obvious by Proposition 6.3 (1). The following is the specialization of Theorem 6.17 to the case of a von Neumann subalgebra. The corollary was first given in [106] under the assumption that both of ϕ, ψ are faithful. The faithfulness assumption of ψ was removed in [68, Theorem 1]. Corollary 6.18 ([68, 106]) Let M0 be a von Neumann subalgebra of M, and ψ, ϕ ∈ M∗+ with ϕ faithful. Then the following conditions (i)–(v) are equivalent: (i) ψ ◦ Eϕ = ψ, where Eϕ : M → M0 is the generalized conditional expectation with respect to ϕ (see the paragraph just before Proposition 6.5); (ii) M0 sufficient for {ψ, ϕ}; (iii) P (ψ, ϕ) = P (ψ|M0 , ϕ|M0 ); (iv) [Dψ : Dϕ]t = [D(ψ|M0 ) : D(ϕ|M0 )]t for all t ∈ R; (v) [Dψ : Dϕ]t ∈ M0 for all t ∈ R. Moreover, if ψ is faithful too, then the above (i)–(v) are also equivalent to the following: (vi) Eψ = Eϕ , where Eψ : M → M0 is the generalized conditional expectation with respect to ψ. Proof In view of Proposition 6.5, all the conditions of the corollary, except (v), exactly correspond to those in Theorem 6.17 in this specialized case. Condition (v) of Theorem 6.17 means that [Dψ, Dϕ]t = Eϕ ([Dψ, Dϕ]t ) for all t ∈ R, which obviously implies (v) of the corollary. Conversely, assume that (v) of the corollary holds, and let M be the von Neumann subalgebra of M generated by {[Dψ : dϕ]t : t ∈ R}; then M ⊂ M0 . In a similar way to the proof of (v) ⇒ (iii) of Theorem 6.17, there exists the conditional expectation E from M to M with respect to ϕ, and we have ψ ◦ E = ψ. We hence have P (ψ|M , ϕ|M ) ≤ P (ψ, ϕ) ≤ P (ψ|M0 , ϕ|M0 ) ≤ P (ψ|M , ϕ|M ), where the first inequality above is as in (6.37). Therefore, (iii) of the corollary follows. We now present the main reversibility theorem via standard f -divergences.

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6 Reversibility and Quantum Divergences

Theorem 6.19 Let ψ, ϕ ∈ M∗+ with ψ = 0 and assume that s(ψ) ≤ s(ϕ). Then the following conditions are equivalent: (i) ψ ◦ γ ◦ γϕ∗ = ψ (ϕ ◦ γ ◦ γϕ∗ = ϕ is automatic), where γϕ∗ is Petz’ recovery map in the extended sense of Proposition 6.6; (ii) γ is reversible for {ψ, ϕ}; (iii) Sf (ψϕ) = Sf (ψ ◦ γ ϕ ◦ γ ) for every operator convex function f on (0, ∞); (iv) Sf (ψϕ) = Sf (ψ ◦ γ ϕ ◦ γ ) < +∞ for some operator convex function f on (0, ∞) such that supp μf has a limit point in (0, ∞); (v) Dα (ψϕ) = Dα (ψ ◦ γ ϕ ◦ γ ) for some α ∈ (0, 1), where Dα (ψϕ) is the α-Rényi divergence (see Sect. 3.1); (vi) P (ψ, ϕ) = P (ψ ◦ γ , ϕ ◦ γ ). Proof Write ψ0 := ψ ◦ γ , ϕ0 := ϕ ◦ γ , e := s(ϕ) (∈ M) and e0 := s(ϕ0 ) (∈ N). (i) ⇒ (ii). Let β := γϕ∗ : M → e0 Ne0 be given in Proposition 6.6. Assume (i); then ψ ◦ γ ◦ β = ψ as well as ϕ ◦ γ ◦ β = ϕ by Proposition 6.6 (1). We extend  : M → N when e0 = 1. Choose a normal state ρ on M and β to a unital map β define (x) := β(x) + ρ(x)(1 − e0 ), β

x ∈ M.

 is a unital normal 2-positive map. Since s(ψ) ≤ s(ϕ) implies Then it is clear that β that s(ψ0 ) ≤ s(ϕ0 ) = e0 , we have for every x ∈ M, (x) = ψ ◦ γ ◦ β(x) = ψ(x) ψ ◦γ ◦β  = ϕ(x). Hence (ii) holds. as well as ϕ ◦ γ ◦ β(x) (ii) ⇒ (iii) is immediate from the monotonicity property of Sf . (iii) ⇒ (iv) is obvious. When α ∈ (0, 1), note that −t α is an operator convex function on (0, ∞), the support of whose representing measure is (0, ∞) similarly to that of −t 1/2 mentioned in the proof of (iv) ⇒ (v) of Theorem 6.17. By the definition of Dα note that condition (v) is equivalent to S−fα (ψϕ) = S−fα (ψ ◦ γ ϕ ◦ γ ), whose value is always finite. Hence it is clear that (iii) ⇒ (v) and (v) ⇒ (iv). Moreover, (vi) is equivalent to (v) with α = 1/2. After all, it only remains to prove that (iv) implies (i). (iv) ⇒ (i). Let γˆ : e0 Ne0 → eMe be defined by (6.2), which is a unital normal 2-positive map. Since s(ψ) ≤ s(ϕ) = e and s(ψ0 ) ≤ s(ϕ0 ) = e0 , note that ψ0 |e0 Ne0 = (ψ|eMe ) ◦ γˆ ,

ϕ0 |e0 Ne0 = (ϕ|eMe ) ◦ γˆ .

(6.38)

Hence, for any operator convex function f on (0, ∞), we have Sf (ψ0 ϕ0 ) = Sf (ψ0 |e0 Ne0 ϕ0 |e0 Ne0 ) = Sf ((ψ|eMe ) ◦ γˆ (ϕ|eMe ) ◦ γˆ )

(6.39)

6.4 Reversibility via Standard f -Divergences

99

as well as Sf (ψϕ) = Sf (ψ|eMe ϕ|eMe ).

(6.40)

Now, assume (iv). Then by (6.39) and (6.40), for f in condition (iv) we obtain Sf (ψ|eMe ϕ|eMe ) = Sf ((ψ|eMe ) ◦ γˆ (ϕ|eMe ) ◦ γˆ ) < +∞. Since ϕ|eMe and (ϕ|eMe ) ◦ γˆ are faithful, it follows from Lemmas 6.14 and 6.15 that ˆ ψ|eMe = (ψ|eMe ) ◦ γˆ ◦ β,

(6.41)

where βˆ : eMe → e0 Ne0 is Petz’ recovery map of γˆ with respect to ϕ|eMe . For ˆ every x ∈ M, since γϕ∗ (x) = β(exe) as given in (6.3) of Remark 6.7, we find by (6.38) and (6.41) that ˆ ψ0 ◦ γϕ∗ (x) = (ψ|eMe ) ◦ γˆ ◦ γϕ∗ (x) = (ψ|eMe ) ◦ γˆ ◦ β(exe) = ψ(exe) = ψ(x), showing (i). For every ψ, ϕ ∈ M∗+ , it is easy to see that γ is reversible for {ψ, ϕ} if and only if γ is reversible for {ψ, ψ + ϕ}. Hence the next theorem follows from Theorem 6.19. This way of presentation of the reversibility theorem was given in [68, Theorem 3] for a family {ψθ } in M∗+ . Theorem 6.20 For every ψ, ϕ ∈ M∗+ with ψ = 0 the following conditions are equivalent: ∗ (i) ψ ◦ γ ◦ γψ+ϕ = ψ; (ii) γ is reversible for {ψ, ϕ}; (iii) Sf (ψψ + ϕ) = Sf (ψ ◦ γ (ψ + ϕ) ◦ γ ) for every operator convex function f on (0, ∞); (iv) Sf (ψψ + ϕ) = Sf (ψ ◦ γ (ψ + ϕ) ◦ γ ) < +∞ for some operator convex function f on (0, ∞) such that supp μf has a limit point in (0, ∞); (v) Dα (ψψ + ϕ) = Dα (ψ ◦ γ (ψ + ϕ) ◦ γ ) for some α ∈ (0, 1); (vi) P (ψ, ψ + ϕ) = P (ψ ◦ γ , (ψ + ϕ) ◦ γ ).

Remark 6.21 When M = B(H) with dim H < ∞, the assumption on supp μf in condition (iv) of Theorems 6.19 and 6.20 can be relaxed to |supp μf | ≥ (dim H)2 , see [63, Theorem 5.1]. But the support condition on μf cannot completely be removed, so f cannot be a general nonlinear operator convex function in (iv) of the theorems. In fact, some counter-examples in the finite-dimensional case are known when f (t) = t 2 (see [70], [58, Example 4.8]) and when f (t) = (1 + t)−1 (see [65, Example 1]), in which Sf is preserved under a CP map γ and yet reversibility fails to hold.

100

6 Reversibility and Quantum Divergences

Problem 6.22 It is interesting to find whether or not Theorem 6.19 holds true without the assumption s(ψ) ≤ s(ϕ) on supports. In particular, assume that s(ψ) and s(ϕ) are orthogonal. In this case, condition (v) (also (vi)) implies that s(ψ ◦ γ ) and s(ϕ ◦ γ ) are also orthogonal. Then one can easily see that γ is reversible for {ψ, ϕ}. So it seems that the problem is not so hopeless.

6.5 Reversibility via Sandwiched Rényi Divergences α (discussed in Sect. 3.3) The reversibility via the sandwiched Rényi divergence D in the von Neumann algebra setting has recently been obtained by Jenˇcová [66, 67], which is reported below without proofs. This reversibility theorem was proved in [66] for the case α > 1 and in [67] for the case 1/2 < α < 1. Theorem 6.23 ([66, 67]) Let γ : N → M be a unital normal 2-positive map between von Neumann algebras. Let ψ, ϕ ∈ M∗+ with ψ = 0 and s(ψ) ≤ s(ϕ). Let α ∈ (1/2, ∞) \ {1}. If α (ψ ◦ γ ϕ ◦ γ ) = D α (ψϕ) < +∞ D then γ is reversible for {ψ, ϕ}. 1/2 (ψϕ) = −2 log(F (ψ, ϕ)/ψ(1)) Remark 6.24 Recall the two limit cases D  in (3.29) and limα→∞ Dα (ψϕ) = Dmax (ψϕ) in Theorem 3.16 (2). In these limit cases, the reversibility as in the above theorem for a CP map γ fails to hold even in the finite-dimensional case, see [96, Corollary A.9] (also [58, Remark 5.15]).

Chapter 7

Reversibility and Measurements

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate Reversibility This chapter is concerned with the approximate reversibility (sufficiency) for a sequence of quantum operations αk : Mk → M (or quantum channels with input M and outputs Mk ). Our main problem is to characterize the approximate + reversibility of (αk : Mk → M)∞ k=1 for ψ, ϕ ∈ M∗ in terms of the convergence Sf (ψ ◦ αk ϕ ◦ αk ) → Sf (ψϕ). In particular, we are concerned with the case where αk ’s are measurement operations with commutative Mk ’s (or quantumclassical channels). The problem was formerly investigated by Petz [108] (also [101, Chap. 9]), and we will revisit Petz’ results with some refinements. Throughout the chapter, let f be an operator convex function on (0, ∞), and ψ, ϕ ∈ M∗+ be as before. The aim of this section is to prove the approximate versions of the two lemmas of Sect. 6.3 and of the main result of Sect. 6.4. The first lemma is the approximate version of Lemma 6.14, where the equalities in (6.10) and (6.11) are replaced with the convergences in (7.1) and (7.2). The lemma improves [108, Lemma 3.1] (also [101, Lemma 9.7]) where the case Sf = S (the relative entropy) was treated under an additional assumption that λ−1 ϕ ≤ ψ ≤ λϕ for some λ > 0. Lemma 7.1 Let αk : Mk → M (k ∈ N) be a sequence of unital normal Schwarz maps with von Neumann algebras Mk . Assume that supp μf has a limit point in (0, ∞) and that ψ, ϕ ∈ M∗+ and αk (k ∈ N) are all faithful. If lim Sf (ψ ◦ αk ϕ ◦ αk ) = Sf (ψϕ) < +∞, k

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_7

(7.1)

101

102

7 Reversibility and Measurements

then we have lim αk ([D(ψ ◦ αk ) : D(ϕ ◦ αk )]t ) = [Dψ : Dϕ]t strongly*, k

t ∈ R.

(7.2)

Proof We are working in the standard forms (M, H, J, P) and (Mk , Hk , Jk , Pk ). Write ϕk := ϕ ◦ αk and ψk := ψ ◦ αk , and let , k be the vector representatives of ϕ, ϕk , respectively. We show first the strong convergence in (7.2), whose proof is divided into several steps similarly to that of Lemma 6.14. The convergence will be strengthened into the strong* one in the final step. Step 1. For each k, a contraction Vk : Hk → H is defined by extending y k → αk (y) (y ∈ Mk ). As in Step 1 of the proof of Lemma 6.14, replacing γ , N, ψ0 , ϕ0 with αk , Mk , ψk , ϕk , we have (s1 + ψk ,ϕk )−1 ≤ Vk∗ (s1 + ψ,ϕ )−1 Vk ,

s > 0.

(7.3)

Step 2. Writing f as in (6.15) we have equalities (6.16), (6.17) and inequalities (6.18), where ψ0 , ϕ0 are replaced with ψk , ϕk for any k. Therefore, assumption (7.1) implies that  [0,∞)

  Sgs (ψϕ) − Sgs (ψk ϕk ) dμf (s) −→ 0

(as well as Sf2 (ψk ϕk ) → Sf2 (ψϕ) if c > 0), and so we can choose a subsequence {k(l)} of {k} such that Sgs (ψk(l) ϕk(l) ) −→ Sgs (ψϕ)

for μf -a.e. s ∈ [0, ∞).

Thus, we may assume that Sgs (ψk ϕk ) −→ Sgs (ψϕ)

for μf -a.e. s ∈ [0, ∞).

(7.4)

(Indeed, suppose that (7.2) does not hold; then there are a t0 ∈ R and a neighborhood V of [Dψ : Dϕ]t0 in the strong topology, and a subsequence {mk } of {k} such that αk ([D(ψ ◦ αmk ) : D(ϕ ◦ αmk )]t ) ∈ V for all k. From the above argument we can choose a subsequence {mk(l)} of {mk } such that Sgs (ψmk(l) ϕmk(l) ) −→ Sgs (ψϕ)

for μf -a.e. s ∈ [0, ∞).

(7.5)

Then a contradiction occurs since the proof below under (7.5) shows the strong convergence in (7.2) for the subsequence {mk(l)}.) With hs (t) := (t + s)−1 , s ∈ [0, ∞), we have (6.19) and (6.20), where ψ0 , ϕ0 are replaced with ψk , ϕk for any k, so that we have by (7.4) Shs (ψk ϕk ) −→ Shs (ψϕ)

for μf -a.e. s ∈ [0, ∞),

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

103

that is,

k , (s1 + ψk ,ϕk )−1 k  −→ , (s1 + ψ,ϕ )−1  for μf -a.e. s ∈ [0, ∞). (7.6) Note that , (z1 + ψ,ϕ )−1  and k , (z1 + ψk ,ϕk )−1 k  are analytic in {z ∈ C : Re z > 0}, as seen just after (6.21). From the assumption on supp μf and (7.6) it follows that the set of s > 0 for which the convergence in (7.6) holds has a limit point in (0, ∞). Moreover, it is clear that the set of analytic functions k , (z1 + ψk ,ϕk )−1 k  on {z ∈ C : Re z > 0} is locally bounded, i.e., uniformly bounded on each compact subset of {z ∈ C : Re z > 0}. Thus, Vitali’s theorem for analytic functions (see, e.g., [115, Sec. 7.3, p. 156]) yields that for all s > 0,

k , (s1 + ψk ,ϕk )−1 k  −→ , (s1 + ψ,ϕ )−1  and hence  

k , Vk∗ (s1 + ψ,ϕ )−1 Vk − (s1 + ψk ,ϕk )−1 k  −→ 0, thanks to Vk k = . By (7.3) this implies that Vk∗ (s1 + ψ,ϕ )−1 Vk k − (s1 + ψk ,ϕk )−1 k  −→ 0, and therefore Vk∗ (s1 + ψ,ϕ )−1 − (s1 + ψk ,ϕk )−1 k  −→ 0,

s > 0.

(7.7)

Step 3. As shown in Step 3 of the proof of Lemma 6.14, note that the linear span of hs , s ∈ [0, ∞), is dense in C0 [0, ∞) with the sup-norm. Since φ ∈ C0 [0, ∞) −→ Vk∗ φ( ψ,ϕ ) , φ( ψk ,ϕk ) k ∈ Hk

(k ∈ N)

are uniformly bounded linear maps with respect to the sup-norm on C0 [0, ∞) and the norm on Hk , it follows from (7.7) that Vk∗ φ( ψ,ϕ ) − φ( ψk ,ϕk ) k  −→ 0, Step 4.

φ ∈ C0 [0, ∞).

For every s > 0 one has

lim Vk∗ (s1 + ψ,ϕ )−1 2 = lim k , (s1 + ψk ,ϕk )−2 k  k

(7.8)

k

= , (s1 + ψ,ϕ )−2  = (s1 + ψ,ϕ )−1 2 .

(by (7.7))

(by (7.8) and Vk k = )

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7 Reversibility and Measurements

Letting ζ := (s1 + ψ,ϕ )−1 ∈ H, since the above means that limk Vk∗ ζ 2 = ζ 2 , one has Vk Vk∗ ζ − ζ 2 = Vk Vk∗ ζ 2 − 2Vk∗ ζ 2 + ζ 2 ≤ 2ζ 2 − 2Vk∗ ζ 2 −→ 0 so that Vk Vk∗ ζ − ζ  → 0, i.e., Vk Vk∗ (s1 + ψ,ϕ )−1 − (s1 + ψ,ϕ )−1  −→ 0. By (7.7) this implies that Vk (s1 + ψk ,ϕk )−1 k − (s1 + ψ,ϕ )−1  −→ 0,

s > 0.

Hence, similarly to Step 3 we have Vk φ( ψk ,ϕk ) k − φ( ψ,ϕ )  −→ 0,

φ ∈ C0 [0, ∞).

(7.9)

Step 5. Here we may assume that ϕ is a state, so   = 1 and  k  = 1 for all k. For each t ∈ R let φ(x), x ≥ 0, be the same as (6.25). For n ∈ N with n ≥ 2 let κn be the same as (6.26) and define χn by

χn (x) :=

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨nx − 1 1 ⎪ ⎪ ⎪ ⎪ ⎪n(n − x) ⎪ ⎪ ⎩0

(0 ≤ x ≤ 1/n), (1/n ≤ x ≤ 2/n), (2/n ≤ x ≤ n − n1 ), . (n −

1 n

≤ x ≤ n),

(x ≥ n).

Moreover, let φn (x) := κn (x)φ(x) for x ∈ [0, ∞). Then it is obvious that κn , χn , φn ∈ C0 [0, ∞) and χn ≤ 1[1/n,n] ≤ κn . Since s(ψ) = s(ϕ) = 1 and s(ψk ) = s(ϕk ) = 1Mk , note that ψ,ϕ and ψk ,ϕk are non-singular and so χn ( ψ,ϕ ) −  → 0 as n → ∞. For any ε ∈ (0, 1) one can choose an n0 such that χn0 ( ψ,ϕ )  ≥ 1 − ε.

(7.10)

By (7.9) there exists a k0 such that Vk χn0 ( ψk ,ϕk ) k − χn0 ( ψ,ϕ )  ≤ ε,

k ≥ k0 .

(7.11)

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

105

It then follows from (7.10) and (7.11) that for every n ≥ n0 and k ≥ k0 , χn ( ψk ,ϕk ) k  ≥ χn0 ( ψk ,ϕk ) k  ≥ Vk χn0 ( ψk ,ϕk ) k  ≥ χn0 ( ψ,ϕ )  − Vk χn0 ( ψk ,ϕk ) k − χn0 ( ψ,ϕ )  ≥ 1 − 2ε. Therefore, for every n ≥ n0 and k ≥ k0 , (1 − κn )( ψk ,ϕk ) k 2 ≤ (1 − Eψk ϕk ([1/n, n]) k 2 = 1 − Eψk ,ϕk ([1/n, n]) k 2 ≤ 1 − χn ( ψk ,ϕk ) k 2 ≤ 1 − (1 − 2ε)2 ≤ 4ε,

(7.12)

∞ where ψk ,ϕk = 0 t dEψk ,ϕk (t) is the spectral decomposition of ψk ,ϕk . Furthermore, one can choose an n1 ≥ n0 such that (1 − κn1 )( ψ,ϕ )  ≤ ε. From this and (7.12) it follows that for every k ≥ k0 , Vk itψk ,ϕk k − itψ,ϕ  ≤ Vk φ( ψk ,ϕk ) k − Vk φn1 ( ψk ,ϕk ) k  + Vk φn1 ( ψk ,ϕk ) k − φn1 ( ψ,ϕ )  + φn1 ( ψ,ϕ ) − φ( ψ,ϕ )  ≤ φ( ψk ,ϕk )(1 − κn1 )( ψk ,ϕk ) k  + Vk φn1 ( ψk ,ϕk ) k − φn1 ( ψ,ϕ )  + φ( ψ,ϕ )(1 − κn1 )( ψ,ϕ )  ≤ 2ε1/2 + Vk φn1 ( ψk ,ϕk ) k − φn1 ( ψ,ϕ )  + ε. Therefore, by (7.9) there exists a k1 ≥ k0 such that Vk itψk ,ϕk k − itψ,ϕ  ≤ 2(ε1/2 + ε),

k ≥ k1 ,

which implies that Vk itψk ,ϕk k − itψ,ϕ  −→ 0 as k → ∞. −it Since −it ϕk k = k and ϕ = , the above implies by Proposition A.47 (and Remark A.48) that

Vk [Dψk : Dϕk ]t k − [Dψ : Dϕ]t  −→ 0

as k → ∞.

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7 Reversibility and Measurements

Since [Dψk : Dϕk ]t ∈ Mk so that Vk [Dψk : Dϕk ]t k = αk ([Dψk : Dϕk ]t ) , it follows that lim αk ([Dψk : Dϕk ]t ) − [Dψ : Dϕ]t  = 0

k→∞

for each t ∈ R. This shows that the strong convergence in (7.2) holds since M = H. Step 6. We replace f with the transpose f. It is immediate to see that dμf(s) = sdμf (s −1 ) for s ∈ (0, ∞); hence f satisfies the same assumption as f . Moreover, by Proposition 2.3 (5), condition (7.1) is rewritten as lim Sf(ϕ ◦ αk ψ ◦ αk ) = Sf(ϕψ) < +∞. k

Hence one can apply the assertion proved above to f with exchanging the roles of ψ, ϕ, so that one has lim αk ([D(ϕ ◦ αk ) : D(ψ ◦ αk )]t ) = [Dϕ : Dψ]t strongly, k

t ∈ R.

Therefore, the strong* convergence in (7.2) has been shown in view of Proposition A.53 (1). The next lemma is the approximate version of Lemma 6.15, which was given in [108] (also [101, Chap. 9]). We will make the proof more readable than that in [101, 108] while essentially the same. Lemma 7.2 Let αk : Mk → M and ψ, ϕ ∈ M∗+ be as in Lemma 7.1. Assume that ∗ : M → M be Petz’ recovery map αk ’s are all 2-positive and (7.2) holds. Let αk,ϕ k −1 of αk with respect to ϕ. If λ ϕ ≤ ψ ≤ λϕ for some λ > 0, then ∗ lim αk ◦ αk,ϕ ([Dψ : Dϕ]t ) = [Dψ : Dϕ]t k

∗ =ψ lim ψ ◦ αk ◦ αk,ϕ k

strongly*,

t ∈ R,

in σ (M∗ , M).

(7.13) (7.14)

To prove the lemma, we first prepare three technical lemmas. Lemma 7.3 Let Vk : Hk → Kk (k ∈ N) be a sequence of contractions between Hilbert spaces. Let ξk ∈ Hk and ηk ∈ Kk . If Vk ξk − ηk  → 0 and limk ξk  = limk ηk  < +∞, then Vk∗ ηk − ξk  → 0. Proof We have Vk∗ ηk − ξk 2 = Vk∗ ηk 2 − 2Re Vk∗ ηk , ξk  + ξk 2 ≤ ηk 2 − 2Re ηk , Vk ξk  + ξk 2 .

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

107

Since    ηk , Vk ξk  − ηk 2  = | ηk , Vk ξk − ηk | ≤ ηk  Vk ξk − ηk  −→ 0, we have ηk , Vk ξk  → limk ηk 2 . Therefore, lim sup Vk∗ ηk − ξk 2 ≤ lim ηk 2 − 2 lim ηk 2 + lim ξk 2 = 0, k

k

k

k

as asserted. Lemma 7.4 Let αk : Ak → B(H) be a sequence of unital Schwarz maps, where Ak ’s are unital C ∗ -algebras. Let uk , ak ∈ Ak be given such that uk ’s are unitaries and supk ak  < +∞. Assume that αk (uk ) → u in the strong* operator topology and αk (ak ) → a in the weak operator topology for some u, a ∈ B(H), where u is a unitary. Then αk (ak uk ) → au and αk (uk ak ) → ua in the weak operator topology. Remark 7.5 The lemma is in [108, Lemma 3.2] and [101, Lemma 9.8] without proof, where the convergence αk (ak uk ) → au in the strong* topology is claimed if αk (uk ) → u and αk (ak ) → a in the strong* topology. In fact, we are not able to see this assertion in the strong* convergence, and the above modification will be enough for discussions below. Proof (Lemma 7.4) For every λ ∈ R we have λ{αk (ak )αk (uk ) + αk (uk )∗ αk (ak )∗ } = αk (ak + λu∗k )αk (ak∗ + λuk ) − αk (ak )αk (ak )∗ − λ2 αk (uk )∗ αk (uk ) ≤ αk ((ak + λu∗k )(ak∗ + λuk )) − αk (ak )αk (ak )∗ − λ2 αk (uk )∗ αk (uk ) = αk (ak ak∗ ) − αk (ak )αk (ak )∗ + λ{αk (ak uk ) + αk (u∗k ak∗ )} + λ2 {αk (u∗k uk ) − αk (uk )∗ αk (uk )}.

(7.15)

For any vector ξ ∈ H the above implies that for every λ > 0,

ξ, {αk (ak )αk (uk ) + αk (uk )∗ αk (ak )∗ }ξ  ≤

1

ξ, {αk (ak ak∗ ) − αk (ak )αk (ak )∗ }ξ  + ξ, {αk (ak uk ) + αk (u∗k ak∗ )}ξ  λ + λ ξ, {1 − αk (uk )∗ αk (uk )}ξ .

(7.16)

Since αk (uk )∗ αk (uk ) → u∗ u = 1 strongly, one can choose λk > 0, k ∈ N, such that λk −→ +∞,

λk ξ, {1 − αk (uk )∗ αk (uk )}ξ  −→ 0.

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7 Reversibility and Measurements

Then using (7.16) to λ = λk and letting k → ∞ one has

ξ, (au + u∗ a ∗ )ξ  ≤ lim inf ξ, {αk (ak uk ) + αk (u∗k ak∗ )}ξ . k

(7.17)

Similarly, for every λ < 0,

ξ, {αk (ak )αk (uk ) + αk (u∗k )αk (ak∗ )}ξ  ≥

1

ξ, {αk (ak ak∗ ) − αk (ak )αk (ak )∗ }ξ  + ξ, {αk (ak uk ) + αk (u∗k ak∗ )}ξ  λ + λ ξ, {1 − αk (uk )∗ αk (uk )}ξ .

Taking λk < 0, k ∈ N, with λk → −∞ and λk ξ, {1 − αk (uk )∗ αk (uk )}ξ  → 0, one has

ξ, (au + u∗ a ∗ )ξ  ≥ lim sup ξ, {αk (ak uk ) + αk (u∗k ak∗ )}ξ .

(7.18)

k

From (7.17) and (7.18) it follows that αk (ak uk ) + αk (u∗k ak∗ ) −→ au + u∗ a ∗ weakly.

(7.19)

Next, replace uk with iuk in (7.15) to obtain λi{αk (ak )αk (uk ) − αk (uk )∗ αk (ak )∗ } ≤ αk (ak ak∗ ) − αk (ak )αk (ak )∗ + λi{αk (ak uk ) − αk (u∗k ak∗ )} + λ2 {αk (u∗k uk ) − αk (uk )∗ αk (uk )}. From this, the same argument as above gives αk (ak uk ) − αk (u∗k ak∗ ) −→ au − u∗ a ∗ weakly.

(7.20)

Combining (7.19) and (7.20) yields that αk (ak uk ) → au weakly. By replacing uk , ak with u∗k , ak∗ in the above argument we have αk (ak∗ u∗k ) → a ∗ u∗ weakly so that αk (uk ak ) → ua weakly. Lemma 7.6 Let A ⊂ B(H) be a unital C ∗ -algebra. Let βk : A → B(H) be a sequence of unital Schwarz maps and ∈ H be such that , βk (a)  = , a  for all k and a ∈ A. If u, v ∈ A are unitaries and βk (u) → u and βk (v) → v in the strong* topology, then lim v , βk (a)u  = v , au , k

a ∈ A.

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

109

Proof It suffices to show that v , au  is a unique limit point of { v , βk (a)u }∞ k=1 . For this we may and do assume that H is a separable Hilbert space. Indeed, for each a ∈ A, let A0 be the C ∗ -subalgebra of A generated by 1, u, v, a, βk (a) (k ∈ N), and B0 be the C ∗ -subalgebra of B(H) generated by A0 , βk (A0 ) (k ∈ N). Then H0 := B0 is a separable subspace of H and we may consider βk (x)|H0 (x ∈ A0 ). Since the weak topology is metrizable on bounded subsets of B(H), by taking a subsequence we can assume that βk (a) → b ∈ B(H) in the weak topology. The repeated use of Lemma 7.4 implies that βk (v ∗ au) → v ∗ bu in the weak topology. Therefore, lim v , βk (a)u  = , v ∗ bu  = lim , βk (v ∗ au)  = v , au , k

k

as asserted. Proof (Lemma 7.2) For each k let Vk : Hk → H be a contraction as given in Step 1 of the proof of Lemma 7.1. Let ϕk , ψk and , k be as in the proof of Lemma 7.1. Set ut := [Dψ : Dϕ]t and uk,t := [Dψk : Dϕk ]t (t ∈ R). We divide the proof into several steps to make it readable. 1/4

1/4

Step 1. For each k we show that  ϕ αk (y)  ≤  ϕk y k  for all y ∈ Mk . For every y ∈ Mk we have  ϕ1/2 αk (y) 2 = αk (y)∗ 2 = ϕ(αk (y)αk (y)∗ ) ≤ ϕ(αk (yy ∗)) = y ∗ k 2 =  ϕ1/2 y k 2 , k 1/2

1/2

1/2

1/2

which implies that Vk maps D( ϕk ) to D( ϕ ) and  ϕ Vk ξ  ≤  ϕk ξ  for 1/2 all ξ ∈ D( ϕk ). Hence the result follows from Proposition B.11 of Appendix B. 1/4 Step 2. We show that limk  ϕ (αk (uk,t ) − ut )  = 0 for all t ∈ R. Let xk := 1/2 ∗ αk (uk,t ) − ut ∈ M. By (7.2) one has xk  ∞→ 0 and  ϕ xk  = xk  → 0. With the spectral decomposition ϕ = 0 s des one has   ϕ1/4 xk 2 =

∞

s 1/2 des (xk )2

0



1/2  

∞

=

∞

2

des (xk )

1/2 2

s des (xk )

0

0

= xk   ϕ1/2 xk  −→ 0. 1/4

1/4

Step 3. We show that limk  ϕk uk,t k  =  ϕ ut  for all t ∈ R. Since λ−1 ϕ ≤ ψ ≤ λϕ, it follows from Lemma A.58 of Sect. A.7 that the function s ∈ R → [Dψ : Dϕ]s (resp., [Dϕ : Dψ]s ) extends to a strongly continuous (Mvalued) function [Dψ : Dϕ]z (resp., [Dϕ : Dψ]z ) on the strip −1/2 ≤ Im z ≤ 0 which is analytic in the interior. Since [Dψ : Dϕ]s = [Dϕ : Dψ]∗s , we see

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7 Reversibility and Measurements

that us = [Dψ : Dϕ]s has the (strongly) analytic continuation uz to the strip −1/2 < Im z < 1/2 given by uz :=

 [Dψ : Dϕ]z , [Dϕ :

Dψ]∗z

,

−1/2 < Im z ≤ 0, 0 ≤ Im z < 1/2.

Since σs (ut ) = u∗s us+t by Theorem A.52 (ii), we further see that the function ϕ s ∈ R → σs (ut ), with t ∈ R fixed, has the analytic continuation to the strip −1/2 < Im z < 1/2 as ϕ

σzϕ (ut ) = u∗z uz+t ,

−1/2 < Im z < 1/2.

Similarly, for each k the function s ∈ R → uk,s has the analytic continuation ϕ uk,z to −1/2 < Im z < 1/2 so that s ∈ R → σs k (uk,t ), with t fixed, has the ϕk analytic continuation σz (uk,t ) = u∗k,z uk,z+t to −1/2 < Im z < 1/2. Since ϕk (σsϕk (u∗k,t )σ−sk (uk,t )) = ϕ(αk (u∗k,s+t uk,s u∗k,−s uk,−s+t )) ϕ

and the repeated use of Lemma 7.4 based on (7.2) gives αk (u∗k,s+t uk,s u∗k,−s uk,−s+t ) −→ u∗s+t us u∗−s us+t

weakly,

it follows that lim ϕk (σsϕk (uk,t )∗ σ−sk (uk,t )) = ϕ(u∗s+t us u∗−s us+t ) = ϕ(σsϕ (ut )∗ σ−s (ut )). ϕ

ϕ

k

Vitali’s theorem (see, e.g., [115, Sec. 7.3, p. 156]) implies that lim ϕk (σz k (uk,t )∗ σ−zk (uk,t )) = ϕ(σz (ut )∗ σ−s (ut )), ϕ

ϕ

ϕ

ϕ

k

−1/2 < Im z < 1/2.

Putting z = i/4 yields k k (uk,t )∗ σ−i/4 (uk,t )) = ϕ(σ−i/4 (ut )∗ σ−i/4 (ut )). lim ϕk (σ−i/4

ϕ

ϕ

ϕ

ϕ

k

ϕ

1/4

ϕ

(7.21)

k Since analytic continuation gives σ−i/4 (ut ) = ϕ ut and σ−i/4 (uk,t ) k =

1/4

ϕk uk,t k , one can rewrite (7.21) as lim  ϕ1/4 uk,t k 2 =  ϕ1/4 ut 2 k k

for all t ∈ R, as required.

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

111 1/4

Step 4. Let H1/4 be the Hilbert space introduced by completing D( ϕ ) with 1/4 1/4 1/4 respect to the inner product ξ, η1/4 :=

ϕ ξ, ϕ η for ξ, η ∈ D( ϕ ). 1/4 For each k let Hk,1/4 be the Hilbert space by completing D( ϕk ) similarly. Step 1/4 1/4 1 says that Vk maps D( ϕk ) to D( ϕ ) contractively with respect to the 1/4inner product, so Vk extends to a contraction from Hk,1/4 to H1/4 (denoted by the  same Vk ). Let Vk : H1/4 → Hk,1/4 denote the adjoint of Vk : Hk,1/4 → H1/4 . Then we show that ∗ (x) k = Vk x , αk,ϕ 

x ∈ M.

For every x ∈ M and y ∈ Mk we have

Vk y k , x 1/4 =

ϕ1/4 Vk y k , ϕ1/4 x  = αk (y) , ϕ1/2x  = αk (y) , J x ∗  = αk (y) , J x ∗J  = αk (y)J xJ ,  = J x , αk (y ∗ )  and 





y k , ϕ1/4 Vk x  =

ϕ1/2 y k , Vk x 

y k , Vk x 1/4 =

ϕ1/4 k k k = Jk y ∗ k , Vk x  = Jk Vk x , y ∗ k . 



Therefore,

Jk Vk x , y ∗ k  = J x , αk (y ∗ ) , 

x ∈ M, y ∈ Mk .

Comparing this with condition (6.1) yields the assertion. 1/4 ∗ (u ) − u )  = 0 for all t ∈ R. For Step 5. We show that limk  ϕ (αk ◦ αk,ϕ t t any t ∈ R set ξk := uk,t k ∈ Hk,1/4 and η := ut ∈ H1/4 . Then, by Step 3, ξk 1/4



 1/2 := ξk , ξk 1/4 =  ϕ1/4 uk,t k  −→  ϕ1/4 ut  = η1/4 , k 

and Vk ξk − η1/4 → 0 by Step 2. Hence Lemma 7.3 implies that Vk η − ξk 1/4 → 0. By Step 4 this means that ∗ (αk,ϕ (ut ) − uk,t ) k  −→ 0.  ϕ1/4 k

Therefore, by Step 1 we have ∗ (ut ) − uk,t )  −→ 0,  ϕ1/4 αk (αk,ϕ

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7 Reversibility and Measurements

and consequently, ∗  ϕ1/4 (αk ◦ αk,ϕ (ut ) − ut )  ∗ (ut ) − uk,t )  +  ϕ1/4 (αk (uk,t ) − ut )  −→ 0 ≤  ϕ1/4 αk (αk,ϕ

thanks to Step 2. Step 6. We prove (7.13). Note that ∗ ∗ (ut ) 2 ≤ ϕ(αk ◦ αk,ϕ (u∗t ut )) = ϕ(1) = ut 2 , αk ◦ αk,ϕ

t ∈ R. (7.22)

Step 5 implies that ∗

ϕ1/4 x , (αk ◦ αk,ϕ (ut ) − ut )  −→ 0,

x ∈ M.

∗ (u ) } is bounded), it follows that Since ϕ M is dense in H (and {αk ◦ αk,ϕ t ∗ αk ◦ αk,ϕ (ut ) → ut weakly as k → ∞ for all t ∈ R. This and (7.22) imply ∗ (u ) − u  → 0, showing that α ◦ α ∗ (u ) → u strongly. that αk ◦ αk,ϕ t t k t k,ϕ t Furthermore, it is immediate to see that Steps 2, 3 and 5 are also valid when ∗ (u∗ ) → ut , uk,t are replaced with u∗t , u∗k,t , respectively. Hence we have αk ◦ αk,ϕ t ∗ ut strongly as well. ∗ : M → M. By Proposition 6.3 (1) Step 7. We prove (7.14). Let βk := αk ◦ αk,ϕ ∗ note that , βk (x)  = ϕ ◦ αk ◦ αk,ϕ (x) = ϕ(x) = , x  for all x ∈ M. From this and (7.13) one can use Lemma 7.6 to have 1/4

lim us , βk (x)u−s  = us , xu−s , k

x ∈ M, s ∈ R.

From the proof of Step 3 the function s ∈ R → us = [Dψ, Dϕ]s extends to a strongly continuous function uz on −1/2 ≤ Im z ≤ 1/2 which is analytic in the interior. Hence Vitali’s theorem implies that lim uz , βk (x)u−z  = uz , xu−z , k

−1/2 < Im z < 1/2.

For z = ip (0 < p < 1/2) one has lim u−ip , βk (x)u−ip  = u−ip , xu−ip , k

0 < p < 1/2.

(7.23)

7.1 Approximation of Connes’ Cocycle Derivatives and Approximate. . .

113

Note (see Lemma A.58) that u−ip − u−i/2  → 0 as 0 < p  1/2 and u−i/2 = . Since | u−i/p , βk (x)u−ip  − , βk (x)| ≤ | u−ip − , βk (x)u−ip | + | , βk (x)(u−ip − )| ≤ x u−ip  u−ip −  + x  u−ip −  −→ 0 as 0 < p  1/2 uniformly for k. Hence limk , βk (x) exists and the LHS of (7.23) converges as p  1/2 to limk , βk (x), while the RHS converges to

, x. Therefore, limk , βk (x) = , x, i.e., limk ψ(βk (x)) = ψ(x). Problem 7.7 The assumption λ−1 ϕ ≤ ψ ≤ λϕ in Lemma 7.2 (also Theorem 7.8) may be rather too strong in the von Neumann algebra setting, so it is desirable to remove that. But the problem does not seem easy because the assumption has been used in an essential way in Steps 3 and 7 of the proof of Lemma 7.2 to appeal to Vitali’s theorem. We now present the approximate reversibility theorem, refining [108, Theorem 3.7] (also [101, Theorem 9.12]). The proof based on Lemmas 7.1 and 7.2 is similar to that of Theorem 6.19 based on Lemmas 6.14 and 6.15. Theorem 7.8 Let αk : Mk → M (k ∈ N) be a sequence of unital 2-positive maps with von Neumann algebras Mk . Let ψ, ϕ ∈ M∗+ and assume that λ−1 ϕ ≤ ψ ≤ λϕ for some λ > 0. Then the following conditions are equivalent: ∗ = ψ in σ (M , M) (ϕ ◦α ◦α ∗ = ϕ is automatic), where α ∗ (i) limk ψ ◦αk ◦αk,ϕ ∗ k.ϕ k,ϕ is Petz’ recovery map of αk with respect to ϕ in the sense of Proposition 6.6. (ii) Approximate reversibility: There exist unital normal 2-positive maps βk : M → Mk (k ∈ N) such that

ψ ◦ αk ◦ βk −→ ψ,

ϕ ◦ αk ◦ βk −→ ϕ

in σ (M∗ , M).

(iii) limk Sf (ψ ◦ αk ϕ ◦ αk ) = Sf (ψϕ) for every operator convex function f on (0, ∞). (iv) limk Sf (ψ ◦ αk ϕ ◦ αk ) = Sf (ψϕ) for some operator convex function f on (0, ∞) such that supp μf has a limit point in (0, ∞). Proof Write ψk := ψ ◦ αk , ϕk := ϕ ◦ αk , e := s(ϕ) = s(ψ) (∈ M) and ek := s(ϕk ) = s(ψk ) (∈ Mk ). ∗ (i) ⇒ (ii). For each k let βk := αk,ϕ : M → ek Mk ek be given as in  Proposition 6.6 and define βk (x) := βk (x) + ρ(x)(1 − ek ) for x ∈ M, where ρ is a normal state on M. Then βk : M → Mk is a unital normal 2-positive map. From (i) we have for every x ∈ M, k (x) = ψ ◦ αk ◦ βk (x) → ψ(x), ψ ◦ αk ◦ β

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7 Reversibility and Measurements

k (x) = ϕ ◦ α ◦ βk (x) = ϕ(x) by Proposition 6.6 (1). Hence (ii) as well as ϕ ◦ αk ◦ β holds. (ii) ⇒ (iii). Assume (ii). For every f as stated in (iii) we have Sf (ψϕ) ≤ lim inf Sf (ψ ◦ αk ◦ βk ϕ ◦ αk ◦ βk ) k

≤ lim inf Sf (ψ ◦ αk ϕ ◦ αk ) k

≤ lim sup Sf (ψ ◦ αk ϕ ◦ αk ) k

≤ Sf (ψϕ) by the lower semicontinuity in σ (M∗ , M) and the monotonicity of Sf , see Theorem 2.7 (i) and (iv). (iii) ⇒ (iv) is obvious. (iv) ⇒ (i). For each k let αˆ k := eαk (·)e|ek Mk ek : ek Mk ek → eMe, which is a unital normal 2-positive map (see Remark 6.7). Then as in the proof of (iv) ⇒ (i) of Theorem 6.19 the following hold: ψk |ek Mk ek = (ψ|eMe ) ◦ αˆ k ,

ϕk |ek Mk ek = (ϕ|eMe ) ◦ αˆ k ,

and for any operator convex function f on (0, ∞), Sf (ψk ϕk ) = Sf (ψ|eMe ◦ αˆ k (ϕ|eMe ) ◦ αˆ k ), as well as Sf (ψϕ) = Sf (ψ|eMe ϕ|eMe ). Here, note that Sf (ψϕ) < +∞ always holds by Corollary 4.19. Assume (iv) with f as stated. Then it follows from Lemmas 7.1 and 7.2 that (ψ|eMe ) ◦ αˆ k ◦ βˆk −→ ψ|eMe

in σ (M∗ , M),

where βˆk : eMe → ek Mk ek is Petz’ recovery map of αˆ k . For every x ∈ M, since ∗ (x) = βˆ (exe) (see Remark 6.7), we have αk,ϕ k ∗ (x) = (ψ|eMe ) ◦ αˆ k ◦ βˆk (exe) −→ ψ(exe) = ψ(x). ψk ◦ αk,ϕ

Hence (i) follows.

7.2 Reversibility via Measurements In this section we will study the reversibility via measurement procedure, i.e., quantum-classical channels given by unital normal positive maps α : A → M with commutative A and its relation with equalities between Sf ,  Sf and Sfmeas ,

7.2 Reversibility via Measurements

115

the standard, the maximal and the measured f -divergences discussed in Chaps. 2, 4 and 5. Recall that Sfmeas ≤ Sf ≤  Sf , see (5.7). Let f be an operator convex function on (0, ∞) as before, and for ψ, ϕ ∈ M∗+ we consider the following conditions: (i) ψ, ϕ commute (see Lemma 4.20 and Definition A.57). (ii) Existence of a sufficient commutative subalgebra: There exist a commutative von Neumann subalgebra A of M and a normal conditional expectation E : M → A such that ψ ◦ E = ψ and ϕ ◦ E = ϕ. (iii) Reversibility via a measurement channel: There exist unital normal positive maps α : A → M and β : M → A with a commutative von Neumann algebra A such that ψ = ψ ◦ α ◦ β,

ϕ = ϕ ◦ α ◦ β.

(iv) Approximate reversibility via measurement channels: There exist unital normal positive maps αk : Ak → M and βk : M → Ak (k ∈ N) with commutative von Neumann algebras Ak such that ψ ◦ αk ◦ βk −→ ψ,

ϕ ◦ αk ◦ βk −→ ϕ

in σ (M∗ , M).

(v) There exists a unital normal positive map α : A → M with a commutative von Neumann algebra A such that Sf (ψ ◦ αϕ ◦ α) = Sf (ψϕ). pr

(vi) Sf (ψϕ) = Sf (ψϕ) (see Definition 5.1). (vii) Sfmeas (ψϕ) = Sf (ψϕ), that is, there exists a sequence of unital normal positive maps αk : Ak → M with commutative von Neumann algebras Ak such that lim Sf (ψ ◦ αk ϕ ◦ αk ) = Sf (ψϕ). k

(viii) Sfmeas (ψϕ) =  Sf (ψϕ).  (ix) Sf (ψϕ) = Sf (ψϕ). We first summarize in the next lemma an easy part of the proof of Theorem 7.10 below. Lemma 7.9 For the above conditions we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (vii), (iii) ⇒ (v) ⇒ (vii), (ii) ⇒ (vi) ⇒ (vii), (iii) ⇒ (viii) ⇒ (ix).

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7 Reversibility and Measurements

Proof (i) ⇒ (ii). Assume (i) and let e := s(ψ + ϕ), ψ0 := ψ|eMe and ϕ0 := ϕ|eMe . From the proof of [55, Proposition 6.7], there exists a commutative von Neumann subalgebra A0 of eMe and a normal conditional expectation E0 : eMe → A0 such that ψ0 = ψ0 ◦ E0 and ϕ0 = ϕ0 ◦ E0 . Hence (ii) holds with A = A0 + C(1 − e) and E : M → A given by E(x) := E0 (exe) + ρ((1 − e)x(1 − e))(1 − e),

x ∈ M,

where ρ is a normal state on (1 − e)M(1 − e). (ii) ⇒ (iii) is clear with the injection α : A → M and β = E, and (iii) ⇒ (iv) is obvious. (iv) ⇒ (vii) is a special case of (ii) ⇒ (iii) of Theorem 7.8, which holds for arbitrary ψ, ϕ ∈ M∗+ . (iii) ⇒ (v) follows from Theorem 2.7 (iv), and (v) ⇒ (vii) is obvious (see also Proposition 5.2 for (vii)). (ii) ⇒ (vi) is seen as Sf (ψϕ) = Sf (ψ|A ϕ|A )

= sup Sf (ψ|A0 ϕ|A0 ) : A0 a finite-dimensional subalgebra of A pr

≤ Sf (ψϕ), where the second equality is due to Theorem 2.7 (vii). (vi) ⇒ (vii) is obvious from (5.7). (iii) ⇒ (viii) is seen since (iii) implies that  Sf (ψ ◦ α ◦ βϕ ◦ α ◦ β) ≤ Sf (ψ ◦ αϕ ◦ α) ≤ Sfmeas (ψϕ), Sf (ψϕ) =  where the first inequality above is due to Theorem 4.4 (i) and Proposition 4.21. Finally, (viii) ⇒ (ix) is obvious. The main aim of the present section is to prove the following theorem. In [108, Theorem 3.3] (also [101, Theorem 9.10]) Petz proved the implication (iv) ⇒ (i), so the theorem below supplements Petz’ result in [108] with some other equivalent conditions. Theorem 7.10 Assume that supp μf , the support of the representing measure of f (see (2.5)), has a limit point in (0, ∞). If s(ψ) = s(ϕ) and Sf (ψϕ) < +∞, then the above conditions (i)–(ix) are all equivalent. Once we have shown the implications in Lemma 7.9, it only remains to prove (vii) ⇒ (i) and (ix) ⇒ (v). Proof (Theorem 7.10) In view of Lemma 7.9 the proof will be completed when we show the following two implications. (ix) ⇒ (v). By Theorem 4.17 there exist a unital normal positive map β : M → A with a commutative von Neumann algebra A and p, q ∈ A+ ∗ such that ψ = p ◦ β, ϕ = q ◦ β and Sf (pq) =  Sf (ψϕ). From (ix) this implies that Sf (pq) = Sf (ψϕ) < +∞. Hence, from Theorem 6.19 it follows that β is

7.2 Reversibility via Measurements

117

reversible for {p, q}, so there exists a unital normal positive map α : A → M such that p = ψ ◦ α and q = ϕ ◦ α. Since Sf (ψ ◦ αϕ ◦ α) = Sf (pq) = Sf (ψϕ), condition (v) holds. (Note that the proof of this part works when s(ψ) ≤ s(ϕ).) (vii) ⇒ (i). Condition (vii) means that there exists a sequence of unital normal positive maps αk : Ak → M with commutative Ak such that (7.1) holds. Put e := s(ψ) = s(ϕ). For each k, let ek (∈ Ak ) be the support projection of the normal positive map eαk (·)e : Ak → eMe, and define αˆ k := eαk (·)e|ek Ak ek : ek Ak ek → eMe, which is obviously unital (i.e., αˆ k (ek ) = e) and faithful. Note that Sf (ψϕ) = Sf (eψeeϕe). Moreover, (ψ ◦ αk )(a) = (eψe) ◦ αˆ k (ek aek ),

a ∈ Ak ,

and similarly for ϕ, so that Sf (ψ ◦ αk ϕ ◦ αk ) = Sf ((eψe) ◦ αˆ k (eϕe) ◦ αˆ k ). Hence, replacing ψ, ϕ, αk with eψe, eϕe, αˆ k , we may assume that s(ψ) = s(ϕ) = 1 and all αk are faithful. Now, let uk,t := [Dψ ◦ αk : Dϕ ◦ αk ]t and ut := [Dψ : Dϕ]t for t ∈ R. Then Lemma 7.1 implies that αk (uk,t ) → ut strongly* for all t ∈ R. By Lemma 7.4 we have αk (uk,s uk,t ) −→ us ut weakly,

s, t ∈ R.

Since Ak is commutative and so {uk,t }t ∈R is a one-parameter unitary group by Proposition A.56, we have us ut = ut us for all s, t ∈ R, which means by Proposition A.56 again that ψ and ϕ are commuting. Theorem 7.11 Assume that supp μf has a limit point in (0, ∞). Let ψ, ϕ ∈ M∗+ , and assume that s(ψ) ≤ s(ϕ) and Sf (ψϕ) < +∞. Let (i) –(ix) denote the respective conditions corresponding to (i)–(ix), where ψ is replaced with ψ + ϕ; for example, (vii) Sfmeas (ψ + ϕϕ) = Sf (ψ + ϕϕ). Then the conditions (i)–(v), (viii), (ix) and all (i) –(ix) are equivalent. Proof Each condition of (i)–(iv) is unchanged when ψ is replaced with ψ + ϕ. Note that s(ψ + ϕ) = s(ϕ) from s(ψ) ≤ s(ϕ) and Sf (ψ + ϕϕ) < +∞ by Proposition 2.9. Hence we can apply Theorem 7.10 to ψ + ϕ and ϕ to see that the conditions (i)–(iv) and (i) –(ix) are all equivalent. Furthermore, we have (iii) ⇒ (v) and (iii) ⇒ (viii) ⇒ (ix) by Lemma 7.9, (ix) ⇒ (v) by the above proof of Theorem 7.10 (this part has been done with s(ψ) ≤ s(ϕ)), and (v) ⇒ (iii) from

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7 Reversibility and Measurements

Theorem 6.19. Hence (v), (viii) and (ix) are also equivalent to (iii), so the result follows. Problem 7.12 Conditions (vi) and (vii) are missing in Theorem 7.11. But it is unknown to us how to prove the implication (vii) ⇒ (i) under s(ψ) ≤ s(ϕ). Here, note that although [Dψ : Dϕ]t is defined even when s(ψ) ≤ s(ϕ) = 1 (in this case, [Dψ : Dϕ]0 = s(ψ)), Lemma 7.1 does not hold in general. Indeed, one can easily construct an example of ψ, ϕ and von Neumann subalgebras Mk ⊂ M such that ψ|Mk and ϕ|Mk are faithful and Sf (ψ ◦ αk ϕ ◦ αk ) → Sf (ψϕ) < +∞, for example, in the case where M is commutative and f (x) = x α , 1 < α < 2. But in this case, αk ([Dψk : Dϕk ]0 ) = αk (1) = 1 = s(ψ) = [Dψ : Dϕ]0 , so the strong convergence (7.2) is impossible. This phenomenon is similar to convergence of positive self-adjoint operators. Let Ak , A be positive self-adjoint operators on a Hilbert space such that Ak → A in the sense that (sI + Ak )−1 → (sI + A)−1 strongly for all s > 0 (see Proposition B.8). When all Ak are non-singular but A is singular, note that Aitk cannot converge to s(A)Ait for t ∈ R; in fact, A0k = I → s(A) for t = 0. Therefore, we need to find another route to prove (vii) ⇒ (i) for the case s(ψ) ≤ s(ϕ). Problem 7.13 A significant problem is the attainability question for Sfmeas , that is, whether there exists a unital normal positive map α : A → M with commutative A such that Sf (ψ ◦ αϕ ◦ α) = Sfmeas (ψϕ). Theorem 7.10 tells us that the attainability holds for f satisfying the assumption in the theorem if s(ψ) = s(ϕ) and Sfmeas (ψϕ) = Sf (ψϕ) < +∞. When M = B(H) with dim H < ∞, the attainability is true for arbitrary ψ, ϕ ∈ B(H)+ ∗ [58, Proposition 4.17]. If the question were affirmative in general for ψ, ϕ ∈ M∗+ with s(ψ) ≤ s(ϕ), then conditions (v) and (vii) would be equivalent, so that Problem 7.12 becomes meaningless.

Chapter 8

Preservation of Maximal f -Divergences

8.1 Maximal f -Divergences and Operator Connections In this chapter we will characterize the preservation of  Sf under a unital normal positive map γ , i.e., the equality case in the monotonicity inequality  Sf (ψ ◦ γ ϕ ◦ ϕ) ≤  Sf (ψϕ). To do this, we work in the standard form (M, L2 (M), J = ∗ , L2 (M)+ ), where Lp (M) is Haagerup’s Lp -space (see Sect. A.6). Recall that M∗ is identified with L1 (M) by the linear bijection ψ ↔ hψ and the tr-functional on L1 (M) is given by tr(hψ ) = ψ(1). To present the main result, we introduce the notion of operator connections of ψ, ϕ ∈ M∗+ , which is a type of extension of the Kubo–Ando operator connections [85]. This is an interesting topic on its own, so we develop the theory in some detail in Appendix D separately. When A, B ∈ B(H)+ are invertible, the operator perspective Pφ (A, B), introduced in [35, 36], is Pφ (A, B) := B 1/2 φ(B −1/2 AB −1/2 )B 1/2 for any continuous function φ on (0, ∞). The operator connection Aσ B in the Kubo–Ando sense [85] is defined by Aσ B = A1/2 k(A−1/2BA−1/2 )A1/2 (= Pk (B, A)) for invertible A, B ∈ B(H)+ , corresponding to a non-negative operator monotone function k = kσ on [0, ∞) (called the representing function of σ ). The σ is extended

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9_8

119

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8 Preservation of Maximal f -Divergences

to general A, B ∈ B(H)+ as Aσ B = lim (A + εI )σ (B + εI ) ε 0

in the strong operator topology (decreasingly). See the first paragraph of Appendix D for a more intrinsic (axiomatic) definition of operator connections [85]. In the following we modify the operator perspectives and connections given above into those for functionals ψ, ϕ ∈ M∗+ . Definition 8.1 Let ψ, ϕ ∈ M∗+ and assume that (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ , see the beginning of Sect. 4.1. By Lemma A.24 we have a unique A ∈ s(ϕ)Ms(ϕ) such that 1/2 1/2 1/2 hψ = Ahϕ (note that hϕ ∈ L2 (M)+ is the vector representative of ϕ). Define Tψ/ϕ := A∗ A. For every continuous function φ on [0, ∞), since hϕ φ(Tψ/ϕ )hϕ L1 (M), we define the M∗ -valued perspective Pφ (ψ, ϕ) by 1/2

hPφ (ψ,ϕ) = hϕ1/2 φ(Tψ/ϕ )hϕ1/2 .

1/2

∈

(8.1)

Definition 8.2 Let σ be an operator connection in the Kubo–Ando sense with the representing function kσ (a non-negative operator monotone function on [0, ∞)). When (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ , we define the M∗ -valued connection ϕσ ψ as ϕσ ψ = Pkσ (ψ, ϕ),

i.e.,

hϕσ ψ = hϕ1/2 kσ (Tψ/ϕ )hϕ1/2 .

(8.2)

For general ψ, ϕ ∈ M∗+ , choosing an ω ∈ M∗+ with ω ∼ ϕ + ψ, we define ϕσ ψ := lim (ϕ + εω)σ (ψ + εω) ∈ M∗+ ε 0

in the norm.

(8.3)

In fact, from Lemma D.4 and Theorem D.6 of Appendix D, the limit in (8.3) as ε 0 exists independently of the choice of ω ∼ ϕ + ψ, and it coincides with definition (8.2) when (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ . Indeed, a slightly more intrinsic definition of ϕσ ψ is provided in Appendix D. It is instructive to note here that the connection ϕσ ψ for ψ, ϕ ∈ M∗+ is explicitly related to the maximal f -divergence  Sf (ψϕ) as follows: (ϕσ ψ)(1) = − S−kσ (ψϕ),

(8.4)

where kσ is the representing function of σ (so −kσ is an operator convex function on (0, ∞)). In fact, identity (8.4) is shown in Proposition D.10, and it will be useful in Sect. 8.2. Now let γ : N → M be a unital normal positive linear map between von Neumann algebras, and let γ∗ : L1 (M) → L1 (N) be the predual map of γ via M∗ ∼ = L1 (M) and N∗ ∼ = L1 (N), i.e., γ∗ (hψ ) = hψ◦γ for ψ ∈ M∗ and the

8.1 Maximal f -Divergences and Operator Connections

121

corresponding hψ ∈ L1 (M). For every ϕ ∈ M∗+ let e0 ∈ N be the support projection of ϕ ◦ γ , and γϕ∗ : M → e0 Ne0 be the (extended) Petz’ recovery map of γ with respect to ϕ given in Proposition 6.6. The next lemma is the description of γϕ∗ in terms of Haagerup’s L1 -elements hϕ and hϕ◦γ . Note that (8.5) has a strong resemblance to (6.4) in the finite-dimensional case. Lemma 8.3 For every ϕ ∈ M∗+ let γϕ∗ be as stated above. Then for every x ∈ M, γϕ∗ (x) ∈ e0 Ne0 is determined by   1/2 ∗ 1/2 γϕ (x)hϕ◦γ = γ∗ hϕ1/2 xhϕ1/2 . hϕ◦γ

(8.5)

Proof Write β = γϕ∗ . For every x ∈ M the defining condition of β(x) is (6.1) for all y ∈ e0 Ne0 , and it can be rewritten in terms of hϕ , hϕ◦γ as 1/2 1/2 tr β(x)hϕ◦γ yhϕ◦γ = tr xhϕ1/2 γ (y)hϕ1/2,

y ∈ e0 Ne0 ,

  1/2 1/2 tr hϕ◦γ β(x)hϕ◦γ y = tr γ∗ hϕ1/2 xhϕ1/2 y,

y ∈ e0 Ne0 .

that is,

 1/2 1/2  1/2 1/2  1/2 1/2 Since e0 hϕ◦γ e0 = hϕ◦γ and e0 γ∗ hϕ xhϕ e0 = γ∗ hϕ xhϕ as easily verified, the above holds for all y ∈ N, which is equivalent to (8.5). In this section we present the next theorem on the preservation of the maximal f -divergence, i.e., the equality case in the monotonicity inequality of  Sf (ψϕ), which is the extension of [58, Theorem 3.34] to the von Neumann algebra setting. It is remarkable that we treat general ψ, ϕ ∈ M∗+ with no condition on their support projections, differently from the reversibility theorems in Chaps. 6 and 7, so that even in the finite-dimensional case, the next theorem improves [58, Theorem 3.34]. Theorem 8.4 Let γ be as above. Let ψ, ϕ ∈ M∗+ and set ω := ψ + ϕ. As in 1/2 1/2 Definition 8.1 let Tψ/ω := A∗ A with A ∈ s(ω)Ms(ω) satisfying hψ = Ahω , and Tψ◦γ /ω◦γ := A∗0 A0 with A0 ∈ s(ω ◦ γ )Ns(ω ◦ γ ) satisfying hψ◦γ = A0 hω◦γ . Then the following conditions are equivalent: 1/2

1/2

(i)  Sf (ψ ◦ γ ϕ ◦ γ ) =  Sf (ψϕ) for any operator convex function f on (0, ∞); (ii)  Sf (ψ ◦ γ ϕ ◦ γ ) =  Sf (ψϕ) < +∞ for some nonlinear operator convex function f on (0, ∞) with limt →0+ tf (t) = 0 (in particular, this is the case if f (0+ ) < +∞) and limt →∞ f (t)/t 2 = 0 (in particular, this is the case if f (∞) < +∞); (iii)  Sf (ψ ◦ γ ω ◦ γ ) =  Sf (ψω) for any operator convex function f on [0, ∞); (iv)  Sf (ψ ◦ γ ω ◦ γ ) =  Sf (ψω) for some nonlinear operator convex function f on [0, ∞); (v) St 2 (ψ ◦ γ ω ◦ γ ) = St 2 (ψω) (equivalently D2 (ψ ◦ γ ω ◦ γ ) = D2 (ψω) as long as ψ = 0);

122

(vi) (vii) (viii) (ix) (x) (xi) (xii)

8 Preservation of Maximal f -Divergences

(ϕ ◦ γ )σ (ψ ◦ γ ) = (ϕσ ψ) ◦ γ for any operator connection σ ; (ϕ ◦ γ )σ (ψ ◦ γ ) = (ϕσ ψ) ◦ γ for some nonlinear operator connection σ ; (ω ◦ γ )σ (ψ ◦ γ ) = (ωσ ψ) ◦ γ for any operator connection σ ; (ω ◦ γ )σ (ψ ◦ γ ) = (ωσ ψ) ◦ γ for some nonlinear operator connection σ ; Pφ (ψ ◦ γ , ω ◦ γ ) = Pφ (ψ, ω) ◦ γ for any continuous function φ on [0, ∞);  1/2 1/2 1/2 1/2  hψ◦γ A0 A∗0 hψ◦γ = γ∗ hψ AA∗ hψ ; γω∗ ((Tψ/ω )2 ) = (γω∗ (Tψ/ω ))2 .

Moreover, if γ is 2-positive and hence so is γω∗ , then the above conditions are also equivalent to (xiii) Tψ/ω ∈ Mγω∗ (the multiplicative domain of γω∗ ). For example, the function f (t) = t log t on (0, ∞) satisfies f (0+ ) < +∞ and limt →∞ f (t)/t 2 = 0. so that a typical realization of condition (ii) is that DBS (ψ ◦ γ ϕ ◦ γ ) = DBS (ψϕ) < +∞, see Example 4.22. In the rest of this chapter let ψ0 := ψ ◦ γ , ϕ0 := ϕ ◦ γ , ω0 := ω ◦ γ = ψ0 + ϕ0 , e := s(ω) ∈ M and e0 := s(ω0 ) ∈ N. The following two simple lemmas are included in this section and the proof of Theorem 8.4 will be given in the next section. Lemma 8.5 We have γω∗ (Tψ/ω ) = Tψ0 /ω0 . Proof By (8.5) we have hω1/2 γ ∗ (Tψ/ω )hω1/2 = γ∗ (hω1/2 Tψ/ω hω1/2 ) = γ∗ (hψ hψ ) 0 ω 0 1/2 1/2

= γ∗ (hψ ) = hψ0 = hω1/2 Tψ0 /ω0 hω1/2 . 0 0 Hence, for every y, z ∈ N,   ! hω1/2 y, γω∗ (Tψ/ω ) − Tψ0 /ω0 hω1/2 z = 0. 0 0 Since hω0 N = e0 L2 (H0 ) and γω∗ (Tψ/ω ), Tψ0 /ω0 ∈ e0 Ne0 , the assertion follows. 1/2

Lemma 8.6 For every operator convex function f on [0, ∞), Pf (ψ ◦ γ , ω ◦ γ ) ≤ Pf (ψ, ω) ◦ γ . Proof By definition (8.1) and Lemma 8.5, as well as (8.5) applied to ω, we have f (Tψ0 /ω0 )hω1/2 = hω1/2 f (γω∗ (Tψ/ω ))hω1/2 hPf (ψ0 ,ω0 ) = hϕ1/2 0 0 0 0  1/2  1/2 ∗ 1/2 ≤ hω0 γω (f (Tψ/ω ))hω0 = γ∗ hω f (Tψ/ω )hω1/2 = γ∗ (hPf (ψ,ω) ) = hPf (ψ,ω)◦γ ,

8.2 Proof of the Theorem

123

where the inequality above is due to the Jensen inequality for operator convex functions (see [25, Theorem 2.1]). Hence we have the asserted inequality.

8.2 Proof of the Theorem We divide the proof of Theorem 8.4 into two parts. The integral expression of an operator convex function on (0, 1) given in Appendix C will be useful in Part 2. Proof (Part 1) We prove that (iii)–(v) and (viii)–(xii) are equivalent, and they are equivalent to (xiii) when γ is 2-positive. In view of (4.5), it is obvious that (iii) ⇒ (v) ⇒ (iv). That (x) ⇒ (viii) ⇒ (ix) is trivial. By (8.4) (Proposition D.10) we have (x) ⇒ (iii) and (ix) ⇒ (iv). (xi) ⇐⇒ (v). Note that (A∗0 A0 )2 hω1/2 = hPt 2 (ψ0 ,ω0 ) hψ0 A0 A∗0 hψ0 = hω1/2 0 0 1/2

1/2

≤ hPt 2 (ψ,ω)◦γ (by Lemma 8.6)    1/2 1/2  = γ∗ hω1/2 (A∗ A)2 hω1/2 = γ∗ hψ AA∗ hψ . Hence (xi) is equivalent to tr hψ0 A0 A∗0 hψ0 = tr hψ AA∗ hψ , 1/2

1/2

1/2

1/2

or equivalently, tr hω0 (A∗0 A0 )2 hω0 = tr hω (A∗ A)2 hω , which is (v) by (4.5) and Proposition 4.5 (6). (xi) ⇐⇒ (xii). It follows from (8.5) that 1/2

1/2

1/2

1/2

   1/2 1/2  γ ∗ ((A∗ A)2 )hω1/2 = γ∗ hω1/2 (A∗ A)2 hω1/2 = γ∗ hψ AA∗ hψ . hω1/2 0 ω 0 It also follows from Lemma 8.5 that (γω∗ (A∗ A))2 hω1/2 = hω1/2 (A∗0 A0 )2 hω1/2 = hψ0 A0 A∗0 hψ0 . hω1/2 0 0 0 0 1/2

1/2

Hence (xi) is equivalent to hω0 γω∗ ((A∗ A)2 )hω0 = hω0 (γω∗ (A∗ A))2 hω0 , which implies (xii) as in the proof of Lemma 8.5. The converse is obvious. ∗ ∗ (xii) ⇒ (x). Let γ' ω be the restriction of γω to the commutative von Neumann subalgebra of M generated by Tψ/ω and 1. Then condition (xii) implies that Tψ/ω ∗ is in the multiplicative domain Mγ'ω∗ of γ' ω (see Definition 6.9). Then, since the ∗ restriction of γ' ∗ is a ∗-homomorphism by Proposition 6.10, we have, for ω to Mγ' ω any continuous function φ on [0, ∞), 1/2

1/2

1/2

1/2

∗ ∗ '∗ φ(γω∗ (Tψ/ω )) = φ(γ' ω (Tψ/ω )) = γω (φ(Tψ/ω )) = γω (φ(Tψ/ω )),

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8 Preservation of Maximal f -Divergences

which implies by Lemma 8.5 that φ(Tψ0 /ω0 ) = γω∗ (φ(Tψ/ω )). Multiplying hω0 from both sides gives

1/2

φ(Tψ0 /ω0 )hω1/2 = hω1/2 γ ∗ (φ(Tψ/ω ))hω1/2 = γ∗ (hω1/2 φ(Tψ/ω )hω1/2 ) hω1/2 0 0 0 ω 0 by Lemma 8.3. This equality is (x). (iv) ⇒ (v). Recall [63, Theorem 8.1] that an operator convex function f on [0, ∞) has the integral expression f (t) = f (0+ ) + at + bt 2 +



 (0,∞)

t t − 1+s t +s

 dμ(s),

t ∈ (0, ∞),

where a ∈ R, b ≥ 0 and μ is a positive measure on (0, ∞) with (0,∞) (1 + s)−2 dμ(s) < +∞. With fs (t) := −t/(t + s) (t ∈ [0, ∞)) for s > 0, one has  Sf (ψω) = f (0+ )ω(1) + aψ(1) + bSt 2 (ψω)    ψ(1)  + Sfs (ψω) dμ(s), + (0,∞) 1 + s

(8.6)

 Sf (ψ0 ω0 ) = f (0+ )ω0 (1) + aψ0 (1) + bSt 2 (ψ0 ω0 )    ψ0 (1)  + Sfs (ψ0 ω0 ) dμ(s) + (0,∞) 1 + s = f (0+ )ω(1) + aψ(1) + bSt 2 (ψ0 ω0 )    ψ(1)  + Sfs (ψ0 ω0 ) dμ(s). + (0,∞) 1 + s

(8.7)

By comparing (8.6) and (8.7), in view of the monotonicity in Theorem 4.4 (i), one has St 2 (ψ0 ω0 ) = St 2 (ψω)

if b > 0,

 Sfs (ψ0 ω0 ) =  Sfs (ψω)

for μ-a.e. s > 0.

Since f is nonlinear, b > 0 or μ = 0. So it suffices to show that (v) holds if  Sfs (ψ0 ω0 ) =  Sfs (ψω) for some s ∈ (0, ∞). Since fs (A∗ A) = −1 + s(A∗ A + −1 s1) and similarly for fs (A∗0 A0 ), the assumption means that for some s ∈ (0, ∞), (A∗0 A0 + se0 )−1 hω1/2 = tr hω1/2 (A∗ A + se)−1 hω1/2 , tr hω1/2 0 0

(8.8)

8.2 Proof of the Theorem

125

where (A∗ A + se)−1 is the inverse in eMe and (A∗0 A0 + se0 )−1 is in e0 Ne0 . We 1/2 1/2 1/2 1/2 have hω = Bhψ+sω with an invertible B ∈ eMe and hω0 = B0 hψ0 +sω0 with an invertible B0 ∈ e0 Ne0 . Note that hψ+sω = B −1 hω1/2 = hω1/2 B ∗−1 , 1/2

hψ0 +sω0 = B0−1 hω1/2 = hω1/2 B0∗−1 . 0 0 1/2

(8.9)

Since hω1/2 (A∗ A + se)hω1/2 = hψ + shω = hψ+sω = hω1/2 (BB ∗ )−1 hω1/2 , we have A∗ A + se = (BB ∗ )−1 . Similarly, A∗0 A0 + se0 = (B0 B0∗ )−1 . Hence (8.8) is rewritten as tr hω1/2 B0 B0∗ hω1/2 = tr hω1/2 BB ∗ hω1/2 , 0 0 that is, tr hψ0 +sω0 (B0∗ B0 )2 hψ0 +sω0 = tr hψ+sω (B ∗ B)2 hψ+sω , 1/2

1/2

1/2

1/2

which means that (v) holds for ω, ψ + sω in place of ψ, ω. Here note that the above proofs of (v) ⇐⇒ (xi) ⇐⇒ (xii) ⇒ (x) ⇒ (iii) have been carried out in the setting of arbitrary (ψ, ω) ∈ (M∗+ × M∗+ )≤ . So we find that (iii) holds for ω, ψ + sω in place of ψ, ω. Applying this to f (t) = (t + ε)−1 for any ε > 0 gives tr hψ0 +sω0 (B0∗ B0 + εe0 )−1 hψ0 +sω0 = tr hψ+sω (B ∗ B + εe)−1 hψ+sω . 1/2

1/2

1/2

1/2

Letting ε 0 gives tr hψ0 +sω0 (B0∗ B0 )−1 hψ0 +sω0 = tr hψ+sω (B ∗ B)−1 hψ+sω . 1/2

1/2

1/2

1/2

Therefore, from (8.9) we have (B0 B0∗ )−2 hω1/2 = tr hω1/2 (BB ∗ )−2 hω1/2 , tr hω1/2 0 0 i.e., (A∗0 A0 + se0 )2 hω1/2 = tr hω1/2 (A∗ A + se)2 hω1/2 , tr hω1/2 0 0 which means by (4.5) that St 2 (ψ0 ω0 ) + 2sψ0 (1) + s 2 ω0 (1) = St 2 (ψω) + 2sψ(1) + s 2 ω(1), so that (v) holds.

126

8 Preservation of Maximal f -Divergences

Finally, assume that γ is 2-positive. Then so is γω∗ by Proposition 6.6 (2), and (xii) ⇐⇒ (xiii) is obvious by Definition 6.9. Proof (Part 2) We prove that (i), (ii), (vi) and (vii) are equivalent to the conditions treated in Part 1. (x) ⇒ on (0, ∞). Set g(t) :=  t (i).  Let f be any operator convex function (1 − t)f 1−t for t ∈ [0, 1], where g(0) := f (0+ ) and g(1) := f (∞) are in (−∞, +∞]. For each n ∈ N let gn (t) := g(t) ∧ n, t ∈ [0, 1]. Then by (x) we have hω1/2 gn (Tψ0 /ω0 )hω1/2 = γ∗ (hω1/2 gn (Tψ/ω )hω1/2 ). 0 0 Taking tr of both sides above yields that   1 2 gn (t) dEψ0 /ω0 (t)hω1/2  = 0 0

1 0

gn (t) dEψ/ω (t)hω1/2 2 ,

1

1 where Tψ/ω = 0 t dEψ/ω (t) and Tψ0 /ω0 = 0 t dEψ0 /ω0 (t) are the spectral decompositions. By the monotone convergence theorem letting n → ∞ gives 

1 0

 g(t) dEψ0 /ω0 (t)hω1/2 2 = 0

1 0

g(t) dEψ/ω (t)hω1/2 2 ,

which means that  Sf (ψ0 ϕ0 ) =  Sf (ψϕ) by Theorem 4.8. (i) ⇒ (vi). Let σ be any operator connection with the representing operator monotone function kσ ≥ 0 on [0, ∞). Then by (i) for f = −kσ and by (8.4) (Proposition D.10) we have S−kσ (ψ0 ϕ0 ) = − S−kσ (ψϕ) (ϕ0 σ ψ0 )(1N ) = − = (ϕσ ψ)(1M ) = ((ϕσ ψ) ◦ γ )(1N ). Since ϕ0 σ ψ0 ≥ (ϕσ ψ) ◦ γ by Proposition D.9, (vi) follows. (vi) ⇒ (vii) is trivial. (vii) ⇒ (ii). Assume that (vii) holds for a nonlinear operator connection σ with the representation function of kσ ; then f := −kσ is a nonlinear operator convex function on [0, ∞) and f (∞) < +∞. Similarly to the proof of (i) ⇒ (vi) we have Sf (ψϕ). − Sf (ψ0 ϕ0 ) = (ϕ0 σ ψ0 )(1N ) = (ϕσ ψ)(1M ) = − (ii) f as stated, and let g(t) := (1 −  (v). Assume that (ii) holds with  t ⇒ for t ∈ [0, 1], where g(0) := f (0+ ) and g(1) := f (∞) ∈ (−∞, +∞]. t)f 1−t Then by Theorem 4.8 we have 

1 0

 g(t) dEψ0 /ω0 (t)hω1/2 2 = 0

1 0

g(t) dEψ/ω (t)hω1/2 2

(8.10)

8.2 Proof of the Theorem

127

with the spectral measures Eϕ/ω of Tϕ/ω and Eϕ0 /ω0 of Tϕ0 /ω0 . Since lim tg(t) = lim tf (t) = 0,

t →0+

lim (1 − t)g(t) = lim

t →0+

t →1−

t →∞

f (t) = 0, t2

it follows from Corollary C.3 that g has the integral expression  g(t) = α + βt + (−∞,1)

(2t − 1)2 dν(s), 1 − st

t ∈ [0, 1],

(8.11)

where α, β ∈ R and ν is a positive measure on (−∞, 1) with s)−1 dν(s) < +∞. We set φs (t) :=

(2t − 1)2 , 1 − st



(−∞,1) (2

−

t ∈ [0, 1], s ∈ (−∞, 1).

By Fubini’s theorem the LHS of (8.10) is 



1

αω0 (1) + βψ0 (1) + (−∞,1)



0

= αω(1) + βψ(1) + (−∞,1)

φs (t) dEψ0 /ω0 (t)hω1/2 2 0

 dν(s)

tr hω1/2 φs (Tψ0 /ω0 )hω1/2 dν(s) 0 0

and the RHS of (8.10) is  αω(1) + βψ(1) + (−∞,1)

tr hω1/2 φs (Tψ/ ω )hω1/2 dν(s).

Therefore, (8.10) implies that 

 (−∞,1)

tr hω1/2 φs (Tψ0 /ω0 )hω1/2 dν(s) = 0 0

(−∞,1)

tr hω1/2 φs (Tψ/ ω )hω1/2 dν(s). (8.12)

For every s ∈ (−∞, 1), since φs is operator convex on [0, 1], note by Lemma 8.5 that hω1/2 φs (Tψ0 /ω0 )hω1/2 = hω1/2 φs (γω∗ (Tψ/ω ))hω1/2 0 0 0 0 ≤ hω1/2 γ ∗ (φs (Tψ/ω ))hω1/2 = γ∗ (hω1/2 φs (Tψ/ω )hω1/2 ), 0 ω 0 where we have used [25, Theorem 2.1] for the inequality above and the last equality is due to (8.5). Therefore, φs (Tψ0 /ω0 )hω1/2 ) ≤ tr hω1/2 φs (Tψ/ω )hω1/2 ), tr hω1/2 0 0

s ∈ (−∞, 1).

(8.13)

128

8 Preservation of Maximal f -Divergences

Since f is nonlinear and so is g, note that ν((−∞, 1)) > 0 in expression (8.11). Hence by (8.12) and (8.13) we find that there exists an s ∈ (−∞, 1) such that tr hω1/2 φs (Tψ0 /ω0 )hω1/2 = tr hω1/2 φs (Tψ/ω )hω1/2 . 0 0

(8.14)

When s = 0, since φ0 (t) = (2t − 1)2 , equality (8.14) implies that 1/2 1/2 1/2 2 1/2 tr hω0 Tψ20 /ω0 hω0 = tr hω Tψ/ω hω so that  St 2 (ψ0 ω0 ) =  St 2 (ψω), i.e., (v) holds. Next, when s = 0, note that  φs (t) =

1−

2 s

+ 2s (1 − st) 1 − st

2

  2 2 1 4(1 − s) 4 = 1− − − t. s 1 − st s2 s

Hence, (8.14) implies that tr hω1/2 (e0 − sA∗0 A0 )−1 hω1/2 = tr hω1/2 (e − sA∗ A)−1 hω1/2 . 0 0

(8.15)

Now, since ω ∼ (1 − s)ψ + ϕ, by Lemma A.24 choose invertible B ∈ eMe and B0 ∈ e0 Ne0 such that 1/2

1/2

hω1/2 = Bh(1−s)ψ+ϕ ,

hω1/2 = B0 h(1−s)ψ0+ϕ0 . 0

We then have hω1/2 (e − sA∗ A)hω1/2 = hω − shψ = h(1−s)ψ+ϕ = hω1/2 (BB ∗ )−1 hω1/2 thanks to h(1−s)ψ+ϕ = B −1 hω = hω B ∗−1 . Therefore, we have e − sA∗ A = (BB ∗ )−1 so that (e − sA∗ A)−1 = BB ∗ , and similarly e0 − sA∗0 A0 = (B0 B0∗ )−1 so that (e0 − sA∗0 A0 )−1 = B0 B0∗ . Thus, (8.15) is rewritten as 1/2

1/2

1/2

tr hω1/2 B0 B0∗ hω1/2 = tr hω1/2 BB ∗ hω1/2 , 0 0 that is, tr h(1−s)ψ0 +ϕ0 (B0∗ B0 )2 h(1−s)ψ0 +ϕ0 = tr h(1−s)ψ+ϕ (B ∗ B)2 h(1−s)ψ+ϕ , 1/2

1/2

1/2

1/2

which means that (v) holds for ω, (1 − s)ψ + ϕ in place of ψ, ω. Furthermore, similarly to an argument in the proof of (iv) ⇒ (v) of Part 1 based on (v) ⇒ (iii), we find that (B0 B0∗ )−2 hω1/2 = tr hω1/2 (BB ∗ )−2 hω1/2 . tr hω1/2 0 0

8.2 Proof of the Theorem

129

Since (BB ∗ )−2 = (e − sA∗ A)2 and (B0 B0∗ )−2 = (e0 − sA∗0 A0 )2 , we have ω0 (1) − 2sψ0 (1) + s 2 St 2 (ψ0 ω0 ) = ω(1) − 2sψ(1) + s 2 St 2 (ψω), so that (v) holds. Remarks 8.7 (1) When (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ , we can replace ω with ϕ in conditions (v) and (x)–(xiii) of Theorem 8.4. Indeed, all the proofs of Part 1 are valid for ψ, ϕ themselves instead of replacing ϕ with ω = ψ + ϕ. (2) As remarked in Remark 6.21, even in the finite-dimensional case, the preservation of St 2 (=  St 2 ) under a CP map γ does not imply the reversibility of γ . Therefore, by Theorem 8.4 (and (1) above) we see that a CP map γ is not necessarily reversible for (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ even if  Sf (ψ ◦ γ ϕ ◦ γ ) =  Sf (ψϕ) holds for all operator convex functions f on (0, ∞). Thus, the preservation of  Sf is strictly weaker than that of Sf . (3) Assume that the domain N of γ is abelian (i.e., γ is a quantum-classical channel). By Theorem 4.18 and Proposition 4.21, condition (iii) implies that Sf (ψ ◦ γ (ψ + ϕ) ◦ γ ) = Sf (ψψ + ϕ) for any operator convex function f on [0, ∞), which implies by Theorem 6.19 that γ is reversible for {ψ, ψ + ϕ}. Hence from Theorem 7.11 it follows that ψ, ψ + ϕ commute and so ψ, ϕ commute. Hence, in this case, the conditions of Theorem 8.4 are also equivalent to the commutativity of ψ, ϕ. Here note that ψ, ϕ are arbitrary without the assumption s(ψ) ≤ s(ϕ), while it is assumed in Theorem 7.11. Problem 8.8 It is desirable to remove the assumption limt →0+ tf (t) = 0 and limt →∞ f (t)/t 2 = 0 on f in condition (ii) of Theorem 8.4. To do this, we need to take care of only the case where f (t) = t 2 and f (t) = t −1 (see the proof of (ii) ⇒ (v) of Part 2). Since t −1 is the transpose of t 2 , the question is whether or not St 2 (ψ ◦ γ ϕ ◦ γ ) = St 2 (ψϕ) < +∞ implies (v). In view of Remark 8.7 (1), this holds true when (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ . By Proposition 4.5 (4) note that St 2 (ψϕ) < +∞ implies s(ψ) ≤ s(ϕ). Therefore, the question is affirmative in the finite-dimensional case, because s(ψ) ≤ s(ϕ) ⇐⇒ (ψ, ϕ) ∈ (M∗+ × M∗+ )≤ in that case.

Appendix A

Preliminaries on von Neumann Algebras

A.1 Introduction of von Neumann Algebras Appendix A1 consists of eight sections, in which we give concise accounts of selected topics of von Neumann algebras used in the main body of this monograph. The first section is a very brief introduction of von Neumann algebras. The set B(H) of all bounded linear operators on a Hilbert space H with the inner product

·, · is a vector space with the operator sum a + b and the scalar multiplication λa (a, b ∈ B(H), λ ∈ C) and is a Banach space with the operator norm a := sup{aξ  : ξ ∈ H, ξ  ≤ 1}. Moreover, B(H) becomes a Banach ∗-algebra with the operator product ab and the adjoint operation a → a ∗ . A subspace of B(H) is called an algebra if it is closed under the product, and a ∗-subalgebra if it is further closed under the ∗-operation. In general, an operator algebra means a ∗-subalgebra of B(H). The (operator) norm topology, the strong operator topology and the weak operator topology are defined on B(H), which are weaker in this order. For a net {aα } and a in B(H), we say that {aα } converges to a in the strong operator topology (or aα → a strongly) if (aα − a)ξ  → 0 for all ξ ∈ H, and {aα } converges to a in the weak operator topology (or aα → a weakly) if ξ, (aα − a)η → 0 for all ξ, η ∈ H. Moreover, the convergence aα → a in the strong* operator topology (or strongly*) means that (aα − a)ξ  → 0 and (aα − a)∗ ξ  → 0 for all ξ ∈ H. Since B(H) is the dual Banach space of the Banach space C1 (H) of trace-class operators with the trace-norm, the weak topology σ (B(H), C1 (H)) is also defined 1 The contents of Appendix A are extracted from the author’s lecture notes [56], a corrected and enlarged version of the manuscript for an intensive course (2019) at Budapest University of Technology and Economics. The proofs and more details omitted in this appendix are found in [56].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9

131

132

A

Preliminaries on von Neumann Algebras

on B(H), which is called the σ -weak topology and is particularly important in the study of von Neumann algebras. In fact, a net {aα } in B(H)  converges to a ∈ B(H) in the σ -weak topology (we write  aα → a σ -weakly)  if ∞ n=1 ξn (aα − a)ηn  → 0 for all ξn , ηn ∈ H (n ∈ N) with n ξn 2 < +∞, n ηn 2 < +∞. We write B(H)sa for the set of all self-adjoint (i.e., a = a ∗ ) a ∈ B(H). The order a ≤ b for a, b ∈ B(H)sa means that ξ, aξ  ≤ ξ, bξ  for all ξ ∈ H, which is a partial order on B(H)sa . When a net {aα } in B(H)sa is increasing and bounded above, it has the supremum a ∈ B(H)sa and aα → a strongly; in this case, we write aα  a. A ∗-subalgebra of B(H) is called a von Neumann algebra (also W ∗ -algebra) if it contains the identity operator 1 and is closed in the weak operator topology. For S ⊂ B(H), define the commutant S of S by S := {a ∈ B(H ) : ab = ba for all b ∈ S}, and also S := (S ) . For a ∗-subalgebra M of B(H) with 1 ∈ M, the double commutation theorem or von Neumann’s density theorem says that the following three conditions are equivalent: • M is a von Neumann algebra (i.e., closed in the weak operator topology); • M is closed in the strong (equivalently strong*) operator topology; • M = M. From this, the polar decompositions and the spectral decompositions of operators in a von Neumann algebra M are taken inside M itself, so M contains plenty of projections. Starting from the above theorem (1929), J. von Neumann developed basics, including the classification, as we will explain shortly, of the von Neumann algebra theory in a series of joint papers with F.J. Murray. C ∗ -algebras are another major subject of operator algebras, which are faithfully represented as norm-closed subalgebras of B(H). Sakai [116] gave the abstract characterization of von Neumann algebras in such a way that a C ∗ -algebra is isomorphic to a von Neumann algebra if and only if it is the dual Banach space of some Banach space. In this case, the predual space is unique in a strong sense. The term W ∗ -algebra is often used to stress this abstract (or Hilbert space-free) situation. Although von Neumann algebras are special C ∗ -algebras, both categories of operator algebras are quite different theoretically and methodologically. Let M∗ be the set of all σ -weakly continuous linear functionals on a von Neumann algebra M, which is a Banach space as a closed subspace of the dual Banach space M ∗ . Then the dual Banach space of M∗ is isometric to M, so M∗ is identified with the predual of M. A positive linear functional ϕ on M is in M∗ if and only if ϕ is normal (i.e., aα  a ⇒ ϕ(aα )  ϕ(a)). Let M, N be von Neumann algebras. A ∗-homomorphism π : M → N is normal (i.e., aα  a in M implies π(aα )  π(a)) if and only if it is continuous with respect to the σ -weak topologies on M, N. A ∗-homomorphism π : M → B(K) (K is a Hilbert space) is called a ∗-representation (or simply representation) of M. When π is a normal representation of M, the range π(M) is a von Neumann algebra and the kernel of π is represented as M(1 − e) for some central projection e (i.e., a projection in the

A.1 Introduction of von Neumann Algebras

133

center Z(M) := M ∩ M ), so π induces a ∗-isomorphism between Me and π(M). Note that a faithful representation is normal automatically. The notion of Murray–von Neumann equivalence on the set Proj(M) of all projections in a von Neumann algebra M is important: e, f ∈ Proj(M) is said to be equivalent (e ∼ f ) if there is a v ∈ M such that v ∗ v = e and vv ∗ = f . A projection e ∈ Proj(M) is called an abelian projection if eMe is an abelian algebra, and a finite projection if, for f ∈ Proj(M), e ∼ f ≤ e ⇒ f = e. The von Neumann algebra M is said to be finite if 1 is a finite projection, and semifinite if, for every central projection e = 0, there is a finite projection f ∈ Proj(M) such that 0 = f ≤ e. If M has no finite central projection (= 0), then M is said to be properly infinite. If M has no finite projection (= 0), then M is said to be purely infinite or of type III. A von Neumann algebra M is properly infinite if and only if M is isomorphic to the tensor product M ⊗ B(H) for separable Hilbert spaces H, and M is semifinite if and only if M has a faithful normal and semifinite trace. A von Neumann algebra M is called a factor if the center is trivial (i.e., Z(M) = C1). The factors are classified into one of the following types: • type In (n ∈ N) if M is isomorphic to the n × n matrix algebra Mn (type In for some n ∈ N if M is finite and has a finite abelian projection = 0), • type I∞ if M is isomorphic to B(H) (dim H = ∞), or equivalently, M is properly infinite and has an abelian projection = 0, • type II1 if M is finite and has no abelian projection = 0, • type II∞ if M is semifinite and properly infinite, and has no abelian projection = 0, • type III if M has no finite projection = 0. A finite factor has a faithful finite normal trace (unique up to positive constants). A factor of type II∞ is represented as (a factor of type II1 ) ⊗ B(H). Corresponding to the types of factors, the quotient set Proj(M)/ ∼ is identified with one of the following: {0, 1, . . . , n} for type In , {0, 1, . . . , ∞} for type I∞ , [0, 1] for type II1 , [0, ∞] for type II∞ , and {0, ∞} for type III. Von Neumann algebras of type I (i.e., a direct sum of factors of type I) are said to be discrete or atomic. Type III factors were further classified by Connes [27] as follows: type III1 , type IIIλ (0 < λ < 1), and type III0 . Assume that a von Neumann algebra M is on a separable Hilbert space, or equivalently, M has the separable predual M∗ . If M is generated by an increasing sequence of finite-dimensional ∗-subalgebras, then M is said to be hyperfinite or AFD (approximately finite dimensional). The uniqueness of type II1 AFD factors is an old result of Murray and von Neumann. A von Neumann algebra M ⊂ B(H) is said to be injective if there exists a (not necessarily normal) norm one projection from B(H) onto M. In 1976, Connes [29] proved that a von Neumann algebra is AFD if and only if it is injective, and at the same time, he proved that injective factors of types II1 , II∞ , IIIλ (0 < λ < 1) are unique for each type. In this fundamental article on the subject, it was also proved that any injective factor of type III0 is a Krieger factor (i.e., the crossed product of an abelian von Neumann

134

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Preliminaries on von Neumann Algebras

algebra by an ergodic automorphism). Thus, the complete classification of injective factors of type III0 was built upon Krieger’s work [84] in 1976. Later in 1985, Haagerup [47] (also [48]) proved the uniqueness of injective factors of type III1 , thus completing the classification of AFD (= injective) factors. In other words, the conjugacy class of the flow of weights due to Connes and Takesaki [30] is a complete invariant for injective factors of type III. We end the overview with a list [72, 102, 116, 123, 127, 128] of standard textbooks on von Neumann algebras.

A.2 Tomita–Takesaki Modular Theory Let M be a von Neumann algebra. Throughout the rest of Appendix A, we assume2 that there is a faithful ω ∈ M∗+ , or equivalently, M is σ -finite, i.e., mutually orthogonal projections in M are at most countable. By making the GNS (Gelfand– Naimark–Segal) cyclic representation {πω , Hω , } of M with respect to ω, we have a ∗-isomorphism πω : M → B(Hω ) with a cyclic and separating vector  ∈ Hω for πω (M) for which ω(x) = , πω (x),

x ∈ M.

Thus, by identifying M with πω (M), we may assume that M itself is a von Neumann algebra on H with a cyclic and separating vector  for M. Here,  is cyclic for M if M = H (where M := {x : x ∈ M}), and  is separating for M if, for x ∈ M, x = 0 ⇒ x = 0, equivalently M  = H. We begin to define two conjugate-linear operators S0 and F0 with the dense domains D(S0 ) = M and D(F0 ) = M  by S0 x := x ∗ ,

∗

F0 x  := x ,

x ∈ M, x ∈ M .

For any x ∈ M and x ∈ M note that

x , S0 x = x , x ∗  = x, x ∗  = x, F0 x , which implies that S0 and F0 are closable, F0 ⊂ S0∗ and S0 ⊂ F0∗ . So we set S := S0 and F := S ∗ = S0∗ , and take the polar decomposition of S as S = J 1/2,

:= S ∗ S = F S.

2 When M is not σ -finite, the construction of the modular theory is essentially similar to the σ -finite case (but technically more complicated) with a faithful normal semifinite weight on M based on the left Hilbert algebra theory.

A.2 Tomita–Takesaki Modular Theory

135

Since the ranges of S and S ∗ are dense, it follows that J is a conjugate-linear unitary and is a non-singular self-adjoint operator. We then have the following: (1) (2) (3) (4) (5)

J = J ∗ and J 2 = 1.

= F S and −1 = SF . S = J 1/2 = −1/2 J and F = J −1/2 = 1/2 J .

−1 = J J and J it = −it J for all t ∈ R. J  =  and  = . The next theorem is Tomita’s fundamental theorem.

Theorem A.1 (Tomita) With J and given above, we have J MJ = M ,

it M −it = M,

(A.1) t ∈ R.

(A.2)

Definition A.2 The operator is called the modular operator with respect to  (or ω), and J is called the modular conjugation with respect to  (or ω). The oneparameter automorphism group σt = σtω of M defined by σtω (x) := it x −it (x ∈ M, t ∈ R) is called the modular automorphism group with respect to  (or ω). Proposition A.3 If x ∈ M ∩ M (the center of M), then J xJ = x ∗ ,

σt (x) = it x −it = x,

t ∈ R.

Example A.4 Consider the simple case where ω = τ is a faithful normal finite trace of M, so M is a finite von Neumann algebra of type II1 (or M is finite dimensional). Since x2 = τ (x ∗ x) = τ (xx ∗ ) = x ∗ 2 for all x ∈ M, S is a conjugate-linear unitary, which means that S = J and = 1. Hence (A.2) trivially holds. For every x, y, y1 ∈ M note that J xJyy1 = J xy1∗y ∗  = yy1 x ∗  = yJ xy1∗ = yJ xJy1 so that J xJy = yJ xJ . Hence J MJ ⊂ M . Moreover, for every x ∈ M and x ∈ M note that

x, J x  = x , J x = x , x ∗  = x, x ∗  so that J x  = x ∗ . Hence, similarly to the above, J M J ⊂ M, so M ⊂ J MJ . Therefore, (A.1) holds. In this way, Tomita’s theorem in this case is quite easy. Let αt (t ∈ R) be a one-parameter weakly continuous (hence automatically strongly* continuous) automorphism group of M. For β ∈ R with β = 0, if β < 0, Dβ := {z ∈ C : 0 < Im z < −β},

D β := {z ∈ C : 0 ≤ Im z ≤ −β},

and if β > 0, Dβ := {z ∈ C : −β < Im z < 0} and D β is similar.

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Preliminaries on von Neumann Algebras

Definition A.5 A functional ϕ ∈ M∗+ is said to satisfy the KMS (Kubo–Martin– Schwinger) condition with respect to αt at β, or the (αt , β)-KMS condition, if, for every x, y ∈ M, there is a bounded continuous function fx,y (z) on D β , analytic in Dβ , such that fx,y (t) = ϕ(αt (x)y),

fx,y (t − iβ) = ϕ(yαt (x)),

t ∈ R.

This condition proposed by Haag, Hugenholtz and Winnink serves as a mathematical formulation of equilibrium states in the quantum statistical mechanics, which is also defined and more useful in C ∗ -algebraic dynamical systems, see [23]. To illustrate that, given a Hamiltonian H ∈ B(H)sa in a finite-dimensional H, consider the Gibbs state ϕ(x) = Tr e−βH x/Tr e−βH and the corresponding dynamics αt (x) = eit H xe−it H , t ∈ R. (The physical meaning of β is the inverse temperature.) For any x, y ∈ B(H) the entire function fx,y (z) := Tr(e−βH eizH xe−izH y)/Tr e−βH satisfies fx,y (t) = Tr(e−βH eit H xe−it H y)/Tr e−βH = ϕ(αt (x)y), fx,y (t − iβ) = Tr(eit H xe−it H e−βH y)/Tr e−βH = ϕ(yαt (x)),

t ∈ R,

so that ϕ satisfies the (αt , β)-KMS condition. Moreover, the Gibbs state ϕ is a unique state satisfying the (αt , β)-KMS condition. The following proposition is another justification for the KMS condition to describe equilibrium states. Proposition A.6 If ϕ ∈ M∗+ satisfies the (αt , β)-KMS condition, then ϕ ◦ αt = ϕ for all t ∈ R. The following is Takesaki’s theorem in [124], an important ingredient of the Tomita–Takesaki theory in addition to Tomita’s theorem. Theorem A.7 (Takesaki) In the same situation as in Theorem A.1, the ω satisfies the (σtω , −1)-KMS condition. Furthermore, σtω is uniquely determined as a weakly continuous one-parameter automorphism group of M for which ω satisfies the KMS condition at β = −1. Definition A.8 The centralizer of ω is defined as Mω := {x ∈ M : ω(xy) = ω(yx), y ∈ M}. It is obvious that Mω is a subalgebra of M including the center M ∩ M . Proposition A.9 The centralizer Mω coincides with the fixed-point algebra of σtω , i.e., Mω = {x ∈ M : σtω (x) = x, t ∈ R}.

A.3 Standard Forms

137

Proposition A.9 contains the second assertion of Proposition A.3. Also, it follows that ω is a trace if and only if σtω = id for all t ∈ R. The notion of conditional expectations in the von Neumann algebra setting was first introduced by Umegaki [132]. If M0 is a von Neumann subalgebra of M and E : M → M0 is a norm one projection onto M0 , then we call E a conditional expectation onto M0 . Tomiyama [131] showed that even when M is a C ∗ -algebra and M0 is a C ∗ -subalgebra of M, a norm one projection E : M → M0 satisfies (1) E is positive, (2) E(y1 xy2 ) = y1 E(x)y2 for all x ∈ M and y1 , y2 ∈ N, and (3) E(x)∗ E(x) ≤ E(x ∗ x) for all x ∈ M. In fact, it is well-known that E is completely positive, see the first paragraph of Sect. 6.1. Takesaki’s theorem [125] characterizes von Neumann subalgebras onto which the normal conditional expectation exists for a given faithful ω ∈ M∗+ . Theorem A.10 ([125]) For a von Neumann subalgebra M0 of M and a faithful ω ∈ M∗+ , there exists a (faithful normal) conditional expectation E : M → M0 satisfying ω = ω ◦ E on M if and only if M0 is globally invariant under σtω , i.e., σtω (M0 ) = M0 for all t ∈ R. ω|M0

Furthermore, in this case, such an E as above is unique and σt all t ∈ R.

= σtω |M0 for

A.3 Standard Forms The theory of standard forms of von Neumann algebras was developed, independently, by Araki [9] and Connes [28] in the case of σ -finite von Neumann algebras, and by Haagerup [42, 43] in the general case. Let M be a σ -finite von Neumann algebra, thus represented on a Hilbert space H with a cyclic and separating vector , for which we have the modular operator

and the modular conjugation J , as in Sect. A.2. Let j : M → M be the conjugatelinear ∗-isomorphism defined by j (x) := J xJ , x ∈ M. Definition A.11 The natural positive cone P = P in H associated with (M, ) is defined by P := {xj (x) : x ∈ M} = {xJ x : x ∈ M}. Theorem A.12 We have: (1) (2) (3) (4) (5)

. In particular, P is a closed cone. P = 1/4 M+  = −1/4 M+ J ξ = ξ for all ξ ∈ P.

it P = P for all t ∈ R. xj (x)P ⊂ P for all x ∈ M. P is self-dual, i.e., P = {η ∈ H : ξ, η ≥ 0, ξ ∈ P}.

(A.3)

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We thus conclude that any (σ -finite) von Neumann algebra is faithfully represented on a Hilbert space H with a conjugate-linear involution J and a self-dual cone P such that: (a) (b) (c) (d)

J MJ = M (Theorem A.1), J xJ = x ∗ for all x ∈ M ∩ M (Proposition A.3), J ξ = ξ for all ξ ∈ P, xj (x)P ⊂ P for all x ∈ M, where j (x) := J xJ .

Definition A.13 A quadruple (M, H, J, P) satisfying the above conditions (a)–(d) is called a standard form of a von Neumann algebra M. This is the abstract (or axiomatic) definition, and we have shown the existence of a standard form for any σ -finite von Neumann algebra. Examples A.14 (1) Let (X, X, μ) be a σ -finite (or more generally, localizable) measure space. The commutative von Neumann algebra M = L∞ (X, μ) is faithfully represented on the Hilbert space L2 (X, μ) as multiplication operators π(f )ξ := f ξ for f ∈ L∞ (X, μ) and ξ ∈ L2 (X, μ). The standard form of M is (L∞ (X, μ), L2 (X, μ), J ξ = ξ , L2 (X, μ)+ ), where L2 (X, μ)+ is the cone of non-negative functions ξ ∈ L2 (X, μ). (2) Let M = B(H), a factor of type I. Let C2 (H) be the space of Hilbert–Schmidt operators (i.e., a ∈ B(H) with Tr a ∗ a < +∞), which is a Hilbert space with the Hilbert–Schmidt inner product a, b := Tr a ∗ b for a, b ∈ C2 (H). Then M = B(H) is faithfully represented on C2 (H) as left multiplication operators π(x)a := xa for x ∈ B(H) and a ∈ C2 (H). The standard form of M is (B(H), C2 (H), J = ∗ , C2 (H)+ ), where J = ∗ is the adjoint operation and C2 (H)+ is the cone of positive a ∈ C2 (H). In this case, note that for any x ∈ B(H), J xJ is the right multiplication of x ∗ on C2 (H) and xj (x)C2 (H)+ = xC2 (H)+ x ∗ ⊂ C2 (H)+ . The following gives geometric properties of the cone P. Proposition A.15 Let (M, H, J, P) be a standard form. Then: (1) P is a pointed cone, i.e., P ∩ (−P) = {0}. (2) If ξ ∈ H and J ξ = ξ , then ξ has a unique decomposition ξ = ξ1 − ξ2 with ξ1 , ξ2 ∈ P and ξ1 ⊥ ξ2 . (3) H is linearly spanned by P. The next proposition is the description of the standard form of a reduced von Neumann algebra eMe, where e is a projection in M.

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139

Proposition A.16 Let (M, H, J, P) be a standard form. Let e ∈ M be a projection and set q := ej (e). Then: (1) exe → qxq (x ∈ M) is a ∗-isomorphism of eMe onto qMq. In particular, e = 0 ⇐⇒ q = 0. (2) qJ = J q and (qMq, qH, qJ q, qP) is a standard form of qMq ∼ = eMe. The next proposition shows the universality of (J, P) for the choice of a cyclic and separating vector ξ in P. Proposition A.17 Let (M, H, J, P) be a standard form. If ξ ∈ P is cyclic and separating for M, then Jξ = J,

Pξ = P,

where Jξ is the modular conjugation and Pξ is the natural positive cone associated with (M, ξ ). For each ξ ∈ H we denote by ωξ (∈ M∗+ ) the vector functional x → ξ, xξ  on M. The following is the most important property of the standard form, establishing the relation between P and M∗+ . Theorem A.18 Let (M, H, J, P) be a standard form. For every ϕ ∈ M∗+ there exists a ξ ∈ P such that ϕ(x) = ξ, xξ  for all x ∈ M. Furthermore, the following estimates hold: ξ − η2 ≤ ωξ − ωη  ≤ ξ − η ξ + η,

ξ, η ∈ P.

Consequently, the map ξ → ωξ is a homeomorphism from P onto M∗+ when P and M∗+ are equipped with the norm topology. The vector ξ ∈ P such that ωξ = ϕ is called the vector representative of ϕ ∈ M∗+ . Another essential property of the standard form is the universality (uniqueness) in the following strict sense. Theorem A.19 Let (M, H, J, P) and (M1 , H1 , J1 , P1 ) be standard forms of von Neumann algebras of M and M1 , respectively. If γ : M → M1 is a ∗-isomorphism, then there exists a unique unitary U : H → H1 such that: (1) γ (x) = U xU ∗ for all x ∈ M, (2) J1 = U J U ∗ , (3) P1 = U P. Example A.20 Let (M, H, J, P) be a standard form. We describe here the standard form of M (2) := M ⊗ M2 , the tensor product of M with the 2 × 2 matrix algebra M2 (C). To do this, choose a cyclic and  separating  vector  ∈ P and consider a x x 11 12 (2) faithful ω(2) ∈ (M (2))+ ; = ω (x11) + ω (x22 ). Then ∗ defined by ω x21 x22 the GNS cyclic representation {π (2) , H (2), (2) } of M (2) with respect to ω(2) is

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given as follows: H (2) = H ⊕ H ⊕ H ⊕ H = 

(2)

   0 = , 0 

$ %  ξ11 ξ12 : ξij ∈ H , ξ21 ξ22

(A.4)

     x11 x12 x x ξ11 ξ12 acts like 2 × 2 matrix product as 11 12 , whose x21 x22 x21 x22 ξ21 ξ22 4 × 4 representation is

and π (2)

⎡

x11 ⎢ 0 ⎢ ⎣x21 0

0 x11 0 x21

x12 0 x22 0

⎤⎡ ⎤ 0 ξ11 ⎢ξ12 ⎥ x12 ⎥ ⎥⎢ ⎥ 0 ⎦ ⎣ξ21 ⎦ x22 ξ22

⎡ for

⎤ ξ11 ⎢ξ12 ⎥ ⎢ ⎥ ∈ H (2). ⎣ξ21 ⎦ ξ22

(A.5)

Let S (2) and (2) be the operators for (M (2), (2) ) corresponding to S and for (M, ) (see Sect. A.2). Since S

(2)

    ∗  ∗  x11 x12 x11  x21 (2)  = ∗ ∗  , x21 x22 x12  x22

one can write

S (2)

⎡ S ⎢0 =⎢ ⎣0 0

0 0 S 0

0 S 0 0

⎤ 0 0⎥ ⎥ 0⎦ S

⎡

and (2)

⎢0 =⎢ ⎣0 0

0 0

0 0

0 0

⎤ 0 0⎥ ⎥. 0⎦

(A.6)

From this with S = J 1/2 one has the polar decomposition S (2) = J (2)( (2) )1/2 , where ⎡

J (2)

J ⎢0 =⎢ ⎣0 0

0 0 J 0

0 J 0 0

⎤ 0 0⎥ ⎥. 0⎦ J

(A.7)

Therefore, the standard form of M (2) is given as (M (2) , H (2), J (2) , P(2) ) with identifications (A.4), (A.5) and (A.7). Moreover, by (A.3), P(2) =

x 0 one has xJ (2)x(2) : x ∈ M (2) . In particular, restricting x ∈ M (2) to 1 0 x2 $  % ξ 0 P(2) ⊃ : ξ, η ∈ P . 0η

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A.4 Relative Modular Operators The notion of relative modular operators was introduced by Araki [11] to extend the relative entropy to the general von Neumann algebra setting. Let (M, H, J, P) be a standard form. Let ψ, ϕ ∈ M∗+ , whose vector representatives are , ∈ P, 0 0 respectively. Define the operators Sψ,ϕ and Fψ,ϕ by 0 Sψ,ϕ (x + η) := sM (ϕ)x ∗ ,

x ∈ M, η ∈ (1 − sM (ϕ))H,

0 Fψ,ϕ (x + ζ ) := sM (ϕ)x ∗ ,

x ∈ M , ζ ∈ (1 − sM (ϕ))H,

where sM (ϕ) is the orthogonal projection onto M (the M-support) and sM (ϕ) is that onto M (the M -support). Note that J sM (ϕ)J = sM (ϕ). In fact, since J = and J M J = M (Theorem A.1), J sM (ϕ)J H = J sM (ϕ)H = J M = J M J = M = sM (ϕ)H. 0 0 Note that Sψ,ϕ and Fψ,ϕ are well defined. In fact, assume that x1 + η1 = x2 + η2 for xi ∈ M and ηi ∈ (1 − sM (ϕ))H. Then (x1 − x2) = η2 − η1 = 0, so η1 = η2 and x1 sM (ϕ) = x2 sM (ϕ). Hence sM (ϕ)x1∗  = sM (ϕ)x2∗ , showing that 0 0 . For every x ∈ M, η ∈ (1 − s (ϕ))H Sψ,ϕ is well defined and similarly for Fψ,ϕ M and x ∈ M , ζ ∈ (1 − sM (ϕ))H, one has 0

Sψ,ϕ (x + η), x + ζ  = sM (ϕ)x ∗ , x + ζ  = x ∗ , x 

= x ∗ , x  = sM (ϕ)x ∗ , x + η 0 = Fψ,ϕ (x + ζ ), x + η. 0 and F 0 have the dense domains, we have their closures S Since Sψ,ϕ ψ,ϕ and Fψ,ϕ ψ,ϕ satisfying ∗ Sψ,ϕ = Fψ,ϕ ,

∗ Fψ,ϕ = Sψ,ϕ .

(A.8)

∗ S Definition A.21 For every ψ, ϕ ∈ M∗+ define ψ,ϕ := Sψ,ϕ ψ,ϕ , called the relative modular operator with respect to ψ, ϕ.

The notion of the relative modular operator plays a crucial role in this monograph, so we give the next proposition with proof. Proposition A.22 Let ψ, ϕ, ψi , ϕi ∈ M∗+ (i = 1, 2). ∗ S (1) The support projection of ψ,ϕ := Sψ,ϕ ψ,ϕ is sM (ψ)sM (ϕ). 1/2

(2) Sψ,ϕ = J ψ,ϕ (the polar decomposition).

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−1 (3) −1 ϕ,ψ = J ψ,ϕ J , where ϕ,ψ is defined with restriction to the support of ϕ,ψ , i.e., in the sense of the generalized inverse. (4) If ψ1 ≤ ψ2 , then ψ1 ,ϕ ≤ ψ2 ,ϕ . If s(ϕ1 ) = s(ϕ2 ) and ϕ1 ≥ ϕ2 , then ψ,ϕ1 ≤

ψ,ϕ2 . ˙ ψ2 ,ϕ (the form sum, see around (B.5) of Appendix B). (5) ψ1 +ψ2 ,ϕ = ψ1 ,ϕ +

Proof ∗ S (1) By (A.8) the support projection of Sψ,ϕ ψ,ϕ is the orthogonal projection onto the closure of the range of Fψ,ϕ , which is sM (ϕ)M  = sM (ϕ)sM (ψ)H. Hence the assertion follows. (2) First, prove the case ψ = ϕ. We write Sϕ := Sϕ,ϕ and ϕ := ϕ,ϕ . Let e := sM (ϕ). e := sM (ϕ) and q := ee = eJ eJ . By Proposition A.16, (qMq, qH, qJ q, qP) is a standard form of qMq ∼ = eMe. Define ϕ ∈ (qMq)+ ∗ by ϕ(qxq) := ϕ(exe) for x ∈ M, whose vector representative is = q . Note that is cyclic and separating for qMq on qH. Let ϕ and Jϕ be the modular operator and the modular conjugation with respect to ϕ. Since Sϕ ((1 − e)x ) = ex ∗ (1 − e) = 0 for all x ∈ M, Sϕ |(1−e)e H = 0 as well as Sϕ |(1−e )H = 0. Since (1 − e ) + (1 − e)e = 1 − q, we have Sϕ |(1−q)H = 0. Moreover, for every x ∈ M, note that ex = exq = qxq and Sϕ (ex ) = ex ∗ = (qxq)∗ , so Sϕ |qH coincides with Sϕ . Therefore,

Sϕ = Sϕ ⊕ 0,

so

ϕ = ϕ ⊕ 0

on the decomposition H = qH ⊕ (1 − q)H. Since Jϕ = qJ q by Proposition A.17, one has 1/2

1/2

Sϕ = Jϕ ϕ ⊕ 0 = (J q ⊕ J (1 − q))( ϕ ⊕ 0) = J ϕ1/2. Next, prove the case of general ψ, ϕ. Let M (2) := M ⊗ M2 , whose standard (2) (2) (2) form (M (2), H   , J , P ) was described in Example A.20. Define θ ∈ x11 x12 (M (2) )+ := ϕ(x11) + ψ(x22 ), whose vector representative ∗ by θ x21 x22     0 s (ϕ) 0 in P(2) is = or in the . It is clear that sM (2) (θ ) = M 0  0 sM (ψ) 4 × 4 form, ⎤ ⎡ 0 0 sM (ϕ) 0 ⎢ 0 sM (ϕ) 0 0 ⎥ ⎥, sM (2) (θ ) = ⎢ ⎣ 0 0 sM (ψ) 0 ⎦ 0 0 0 sM (ψ)

(A.9)

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143

and also by (A.7),

s(M (2) ) (θ ) = J (2)sM (2) (θ )J (2)

⎤ ⎡ 0 0 0 sM (ϕ) ⎢ 0 sM (ψ) 0 0 ⎥ ⎥. =⎢ ⎣ 0 0 ⎦ 0 sM (ϕ) 0 0 0 sM (ψ) (A.10)

Furthermore, Sθ is defined as Sθ

  x11 x12 x21 x22

+

 ∗ ∗   x x21 η11 η12 = sM (2) (θ ) 11 ∗ x∗ η21 η22 x12 22

(A.11)

      η η x11 x12 ∈ M (2) and 11 12 ∈ 1 − s(M (2) ) (θ ) H (2). Rewrite (A.11) x21 x22 η21 η22 in the 4 × 4 form and combine with (A.9) and (A.10). Then extending (A.6) we find that ⎤ ⎡ 0 0 Sϕ 0 ⎢ 0 0 Sψ,ϕ 0 ⎥ ⎥ Sθ = ⎢ (A.12) ⎣ 0 Sϕ,ψ 0 0 ⎦ 0 0 0 Sψ for

and so

⎤ ⎡ Sϕ∗ Sϕ 0 0 0

ϕ 0 0 ⎢ 0 S∗ S 0 0 ⎥ 0 0

⎥ ⎢ ⎢ ϕ,ψ ϕ,ψ ϕ,ψ ∗

θ = Sθ Sθ = ⎢ ⎥=⎢ ∗ S 0 ⎦ ⎣ 0 0 Sψ,ϕ 0 ψ,ϕ ⎣ 0 ψ,ϕ ∗ 0 0 0 0 0 0 Sψ Sψ ⎡

⎤ 0 0 ⎥ ⎥. 0 ⎦

ψ (A.13)

From the first case (applied to θ ∈ (M (2) )+ ∗ ) and (A.7) it follows that

1/2

Sθ = J (2) θ

⎡ ⎤ 1/2 0 0 0 J ϕ ⎢ ⎥ 1/2 ⎢ 0 0 J ψ,ϕ 0 ⎥ ⎢ ⎥. =⎢ 1/2 0 0 ⎥ ⎣ 0 J ϕ,ψ ⎦ 1/2 0 0 0 J ψ

(A.14)

1/2

Comparing the (2,3)-entries of (A.12) and (A.14) implies that Sψ,ϕ = J ψ,ϕ . (3) For the case ψ = ϕ, from the above proof of (2) and the property (4) of Sect. A.2, we have −1

−1 ϕ = ϕ ⊕ 0 = (Jϕ ϕ Jϕ ) ⊕ 0 = J ϕ J.

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For general ψ, ϕ apply the above case to θ in the proof of (2); then by (A.13) and (A.7) we have ⎤ ⎡ ⎤

−1 0 0 0 J ϕ J 0 0 0 ϕ ⎥ ⎢ 0 −1 0 0 ⎥ ⎢ 0 0 ⎥ 0 J ψ,ϕ J ⎢ ϕ,ψ ⎥, ⎥=⎢ ⎢ −1 ⎣ 0 ψ,ϕ 0 ⎦ 0 ⎦ 0 0 J ϕ,ψ J ⎣ 0 0 0 0 J ψ J 0 0 0 −1 ψ ⎡

showing that −1 ψ,ϕ = J ϕ,ψ J . (4) Let i be the vector representative of ψi . If ψ1 ≤ ψ2 , then for every x ∈ M and η ∈ (1 − sM (ϕ))H one has  ψ1 ,ϕ (x + η)2 = Sψ1 ,ϕ (x + η)2 = x ∗ 1 2 1/2

= ψ1 (xx ∗ ) ≤ ψ2 (xx ∗ ) =  ψ2 ,ϕ (x + η)2 . 1/2

1/2

Since M + (1 − sM (ϕ))H is a core of ψi ,ϕ , it follows from Proposition B.4 of Appendix B that ψ1 ,ϕ ≤ ψ2 ,ϕ . Hence the first assertion holds. For the second, assume that s(ϕ1 ) = s(ϕ2 ) and ϕ1 ≥ ϕ2 . Then s( ϕ1 ,ψ ) = s( ϕ2 ,ψ ) by (1) above. Hence from the first assertion and Proposition B.6 one has −1 ϕ1 ,ψ ≤

−1

−1 ϕ2 ,ψ . Since ψ,ϕi = J ϕi ,ψ J by (3) above, the second assertion holds. (5) Let 3 be the vector representative of ψ1 + ψ2 as well as i of ψi . Note that 1/2 1/2 M + (1 − sM (ϕ))H is the common core of ψi ,ϕ (i = 1, 2) and ψ1 +ψ2 ,ϕ . For every x ∈ M and η ∈ (1 − sM (ϕ))H one has 1/2

1/2

 ψ1 ,ϕ (x + η)2 +  ψ2 ,ϕ (x + η)2 = sM (ϕ)x ∗ 1 2 + sM (ϕ)x ∗ 2 2 = ψ1 (xsM (ϕ)x ∗ ) + ψ2 (xsM (ϕ)x ∗ ) = (ψ1 + ψ2 )(xsM (ϕ)x ∗ ) = sM (ϕ)x ∗ 3 2 =  ψ1 +ψ2 ,ϕ (x + η)2 . 1/2

This immediately implies that 1/2

1/2

1/2

 ψ1 +ψ2 ξ 2 =  ψ1 ,ϕ ξ 2 +  ψ2 ,ϕ ξ 2 ,

1/2

1/2

ξ ∈ D( ψ1 ,ϕ ) ∩ D( ψ2 ,ϕ ),

˙ ψ2 ,ϕ . which means that ψ1 +ψ2 ,ϕ = ψ1 ,ϕ + Examples A.23 (1) Let M = L∞ (X, μ) as in Example A.14 (1). For every ψ, ϕ ∈ L1 (X, μ)+ ∼ = M∗+ , it is easy to verify that ψ,ϕ is the multiplication of 1{ϕ>0} (ψ/ϕ), which is the Radon–Nikodym derivative of ψ dμ with respect to ϕ dμ (restricted to the support of ϕ) in the classical sense.

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(2) Let M = B(H) as in Example A.14 (2). For every ψ, ϕ ∈ B(H)+ ∗ we have the density operators (positive trace-class operators) Dψ , D such that ψ(X) = ϕ  Tr D X for X ∈ B(H) and similarly for D . Let D = aP and Dϕ = ψ ϕ ψ a a>0  bQ be the spectral decompositions of D , D , where P and Q b ψ ϕ a b are b>0 finite-dimensional orthogonal projections. Then the relative modular operator

ψ,ϕ on C2 (H) is given as

ψ,ϕ = LDψ RDϕ−1 =



ab−1LPa RQb ,

(A.15)

a>0, b>0

where L[−] and R[−] denote the left and the right multiplications and Dϕ−1 is the generalized inverse of Dϕ . The relative modular operator ψ,ϕ is a type of division ψ/ϕ regarded as the “non-commutative Radon–Nikodym derivative” of ψ with respect to ϕ. When ψ is dominated by ϕ, i.e., ψ ≤ λϕ, one can consider another type (a slightly more primitive) of division ψ/ϕ as defined in the next lemma, which is a variant of the factorization technique due to [34]. The lemma is often used in the main body of this monograph, so we give a proof for the convenience of the reader. Lemma A.24 Assume that ψ ≤ λϕ for some λ > 0. Let ,  ∈ P be the vector representatives of ϕ, ψ, respectively. Then there exists a unique A ∈ s(ϕ)Ms(ϕ) such that  = A . The A satisfies A ≤ λ1/2 and s(AA∗ ) = s(ψ). Moreover, if νϕ ≤ ψ ≤ λϕ for some λ, ν > 0, then the above A satisfies νs(ϕ) ≤ A∗ A ≤ λs(ϕ). Proof Since x 2 = J x J 2 = ψ((J x J )∗ (J x J )) ≤ λϕ((J x J )∗ (J x J )) = λx 2 ,

x ∈ M ,

there is a unique A ∈ B(H) such that A(1 − s(ϕ)) = 0 and A(x ) = x  for all x ∈ M. Moreover, A ≤ λ1/2 , and for every unitary u in M s(ϕ), u ∗ Au (x ) = u ∗ u x  = x  = A(x ),

x ∈ M,

which implies that u ∗ Au = A, so A ∈ (M s(ϕ)) = s(ϕ)Ms(ϕ). Since the closure of the range of A is M , we have s(AA∗ ) = s(ψ). Next, assume that νϕ ≤ ψ in addition to ψ ≤ λϕ. Then s(ψ) = s(ϕ), and there is a unique B ∈ s(ϕ)Ms(ϕ) such that = B. It is easy to see that AB = BA = s(ϕ), hence B = A−1 in s(ϕ)Ms(ϕ). Since BB ∗ ≤ ν −1 s(ϕ), we have νs(ϕ) ≤ A∗ A ≤ λs(ϕ). Consider Example A.23 (2) with dim H < ∞. When ψ ≤ λϕ the operator A 1/2 −1/2 given in Lemma A.24 may be written as A = Dψ Dϕ , so we have A∗ A =

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−1/2

Dϕ Dψ Dϕ . Both of the relative modular operator ψ,ϕ and the operator −1/2 −1/2 play an equally important role in quantum information, but in Dϕ Dψ Dϕ separate situations (see, e.g., [58, Sec. 3.1]).

A.5 τ -Measurable Operators The non-commutative integration theory was initiated by Segal [118] and further developed in, e.g., [32, 121, 137], where measurable operators affiliated with a von Neumann algebra with a trace were discussed. Later in [99], Nelson introduced the notion of τ -measurable operators in a stricter connection with a given trace τ . This section is a brief survey of the theory of τ -measurable operators. Assume that M is a semifinite von Neumann algebra on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let a : D(a) → H be a linear operator, where D(a) is a linear subspace of H. We say that a is affiliated with M, denoted by a ηM, if x a ⊂ ax for all x ∈ M , or equivalently, if u au ∗ = a for all unitaries u ∈ M . The following facts are easy to verify: (a) If a, b are linear operators affiliated with M, then a + b with D(a + b) = D(a) ∩ D(b) and ab with D(ab) = {ξ ∈ D(b) : bξ ∈ D(a)} are affiliated with M. (b) If a is densely defined and a ηM, then a ∗ ηM. (c) If a is closable and a ηM, then a ηM. (d) Assume that a is densely defined and closed, so we have ∞the polar decomposition a = w|a| and the spectral decomposition |a| = 0 λ deλ . Then a ηM if and only if w, eλ ∈ M for all λ ≥ 0. Definition A.25 Let a be a densely defined closed operator such that a ηM. We say that a is τ -measurable if, for any δ > 0, there exists an e ∈ Proj(M) such that eH ⊂ D(a) and τ (e⊥ ) ≤ δ (note that eH ⊂ D(a) ⇐⇒ ae < ∞ due to the  the set of such τ -measurable operators. closed graph theorem). We denote by M Proposition A.26 Let a be a densely defined closed operator affiliated with M ∞ with a = w|a| and |a| = 0 λ deλ as above. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

 a ∈ M;  |a| ∈ M; τ (eλ⊥ ) → 0 as λ → ∞; τ (eλ⊥ ) < +∞ for some λ > 0.

For each ε, δ > 0 define  : ae ≤ ε and τ (e⊥ ) ≤ δ for some e ∈ Proj(M)}. N(ε, δ) := {a ∈ M

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Then the following assertions hold:  Moreover, a ∈ N(ε, δ) ⇐⇒ a ∗ ∈ N(ε, δ).  then a ∗ ∈ M. (1) If a ∈ M,  then a + b and ab are densely defined and closable, and a + b, (2) If a, b ∈ M,  Moreover, if a ∈ N(ε1 , δ1 ) and b ∈ N(ε2 , δ2 ), then a + b ∈ N(ε1 + ab ∈ M. ε2 , δ1 + δ2 ) and ab ∈ N(ε1 ε2 , δ1 + δ2 ).  we may use the convention In view of the above assertions, for every a, b ∈ M that a +b and ab mean the strong sum a + b and the strong product ab, respectively. A big advantage of τ -measurable operators is that we can freely take adjoint, sum  The following is the main result on τ -measurable operators. and product in M.  is a complete metrizable Hausdorff topological ∗-algebra with Theorem A.27 M the strong sum, the strong product, and {N(ε, δ) : ε, δ > 0} as a neighborhood  basis of 0. Moreover, M is dense in M. + be the set of positive self-adjoint a ∈ M  (denoted by a ≥ 0). Note that Let M    M+ is a closed convex cone in M and M is an ordered topological space with the  The trace τ on M+ order a ≥ b defined as a − b ≥ 0 for self-adjoint a, b ∈ M. ∞ ∞  extends to all a ∈ M+ as τ (a) := 0 λ dτ (eλ ), where a = 0 λ deλ is the spectral decomposition.  given in Theorem A.27 is called the measure topology. This The topology on M topology is not necessarily locally convex. For instance, if M is a finite non-atomic von Neumann algebra with a faithful normal finite trace τ , then a non-empty open  is only the whole M  and there is no non-zero continuous linear convex set in M  functional on M. Examples A.28  = B(H) and the measure (1) When M = B(H) and τ is the usual trace Tr, M topology is the operator norm topology.  is the set of all (2) Let M be finite with a faithful normal finite trace τ . Then M densely defined closed operators x ηM. (3) Let (X, X, μ) be a localizable measure space, where (X, X, μ) is localizable if, for every A ∈ X, there is a B ∈ X such that B ⊂ A and μ(B) < +∞. For algebra A = L∞ (X, μ) = L1 (X, μ)∗ with τ (f ) := an abelian von Neumann ∞  X f dμ for f ∈ L (X, μ)+ , A is the space of measurable functions f on X such that there is an A ∈ X such that μ(A) < +∞ and f is bounded on X \ A,  ⇐⇒ f (x) = g(x) μ-a.e. where f = g in A We end the section with the following two definitions.  and t > 0 the (tth) generalized s-number μt (a) is Definition A.29 For a ∈ M defined to be μt (a) := inf{s ≥ 0 : τ (e(s,∞) (|a|) ≤ t},

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where e(s,∞)(|a|) denotes the spectral projection of |a| corresponding to the interval (s, ∞). Note that μt (a) < +∞ for all t > 0 since τ (e(s,∞)(|a|)) → 0 as s → ∞ by Proposition A.26.  define Definition A.30 For each a ∈ M ap := τ (|a|p )1/p ∈ [0, +∞],

0 < p < ∞.

Furthermore, a∞ := a ∈ [0, +∞], the operator norm with the convention that a = +∞ unless a ∈ M. The non-commutative Lp -space on (M, τ ) is defined as  : ap < +∞}, Lp (M, τ ) := {a ∈ M

0 < p ≤ ∞.

(In particular, L∞ (M, τ ) = M.) The trace τ uniquely extends to a positive linear functional on L1 (M, τ ) and it satisfies |τ (a)| ≤ a1 for all a ∈ L1 (M, τ ). When 1 ≤ p ≤ ∞ and 1/p+1/q = 1, for every a ∈ Lp (M, τ ) and b ∈ Lq (M, τ ) we have ab ∈ L1 (M, τ ) and |τ (ab)| ≤ ab1 ≤ ap bq

(Hölder’s inequality).

More generally, when p, q, r > 0 and 1/p + 1/q = 1/r, Hölder’s inequality holds as abr ≤ ap bq ,

 a, b ∈ M.

If 1 ≤ p < ∞ and 1/p + 1/q = 1, then Lp (M, τ ) is a Banach space with respect to the norm  · p and its dual Banach space is Lq (M, τ ) under the duality pairing (a, b) ∈ Lp (M, τ ) × Lq (M, τ ) −→ τ (ab) ∈ C. In particular, we have M∗ = L1 (M, τ ) via the correspondence ψ ∈ M∗ ↔ a ∈ L1 (M, τ ) given by ψ(x) = τ (ax), x ∈ M, and ψ ∈ M∗+ ⇐⇒ a ∈ L1 (M, τ )+ + ). This a is often called the Radon–Nikodym derivative of ψ with (= L1 (M, τ )∩ M respect to τ and denoted by dψ/dτ . More details on the generalized s-numbers and the non-commutative Lp (M, τ ) are found in [37], which is the best literature on those. More general Haagerup’s Lp -spaces will be explained in the next section.

A.6 Haagerup’s Lp -Spaces We begin with the crossed product of M by the modular automorphism group, based on which Haagerup’s Lp -spaces are constructed. Let σt = σtω (t ∈ R) be the modular automorphism group with respect to a faithful ω ∈ M∗+ (see

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Definition A.2). (We may take a faithful normal semifinite weight ω when M is not σ -finite.) Define a faithful normal representation πσ of M and a one-parameter strongly continuous unitary representation λ on L2 (R, H) ∼ = H ⊗ L2 (R, dt) by (πσ (x)ξ )(s) := σ−s (x)ξ(s),

(λ(t)ξ )(s) := ξ(s − t),

s ∈ R, ξ ∈ L2 (R, H),

satisfying the covariance property πσ (σt (x)) = λ(t)πσ (x)λ(t)∗ for all x ∈ M and t ∈ R. Then the crossed product N := M σ R = {πσ (M) ∪ λ(R)} is the von Neumann algebra generated by πσ (x) (x ∈ M) and λ(t) (t ∈ R). The crossed product construction is more generally performed for a continuous (locally compact abelian) group action on M, for which Takesaki’s duality theorem [126] was established, where the dual action plays a role. In the case of σ ω the dual action θ = σ'ω is determined by θs (πσ (x)) = πσ (x) and θs (λ(t)) = eist λ(t) for x ∈ M and s, t ∈ R. Then N θ = πσ (M) holds, where N θ is the fixed-point algebra of θ . We may assume that M ⊂ N under identifying x ∈ M with πσ (x). An important fact is that N is a semifinite von Neumann algebra with a faithful normal semifinite trace τ (called the canonical trace on N) satisfying the trace scaling property τ ◦ θs = e−s τ,

s ∈ R.

This fact is shown by making use of the dual weight  ω on N, while we omit its  be the space of details here (see [44, 45] for theory of dual weights). Thus, let N τ -measurable operators affiliated with N = M σ R. It is easy to see that θs (s ∈  as a one-parameter group of homeomorphic ∗R) on N uniquely extends to N isomorphisms with respect to the measure topology. It is also worth noting that the triplet (N := M σ ω R, θ := σ'ω , τ ) is canonical in the sense that for another faithful ω1 ∈ M∗+ there is a unitary U on L2 (R, H) such that U NU ∗ = N1 ,

Ad(U ) ◦ θs = θ1s ◦ Ad(U ) (s ∈ R),

τ = τ1 ◦ Ad(U ),

where (N1 , θ1 , τ1 ) is the triplet associated with ω1 . When M is a factor of type III, the flow of weights of M mentioned at the end of Sect. A.1 is defined by (X, FtM ) := (Z(N), θt |Z(N) ), which is an ergodic flow and can be used for the type IIIλ (0 ≤ λ ≤ 1) classification, see [30].

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Definition A.31 ([46]) For 0 < p ≤ ∞, Haagerup’s Lp -space Lp (M) is defined by  : θs (x) = e−s/p x, s ∈ R}. Lp (M) := {x ∈ N + where N + is the positive part of N.  Let Lp (M)+ = Lp (M) ∩ N  which are M-bimodules and Clearly, Lp (M)’s are closed linear subspaces of N, closed under a → a ∗ and a → |a|. Moreover, they are linearly spanned by their positive part Lp (M)+ . It follows that every a ∈ L∞ (M) is bounded, so we have L∞ (M) = M since N θ = M. The following is the starting point of the theory of Haagerup’s Lp -spaces developed in [129]. Lemma A.32 There is a bijection ψ ∈ M∗+ → hψ ∈ L1 (M)+ such that for every ψ, ϕ ∈ M∗+ and x ∈ M, hψ+ϕ = hψ + hϕ ,

hxψx ∗ = xhψ x ∗ ,

where (xψx ∗ )(y) := ψ(x ∗ yx), y ∈ M. Although we omit the details here, note that the bijection ψ ∈ M∗+ → hψ ∈  on N is equal to τ (hψ ·), + is determined in such a way that the dual weight ψ and that s(ψ) = s(hψ ) for the support projections. By linearly extending the above bijection we have the next theorem. L1 (M)

Theorem A.33 There is a linear bijection ψ ∈ M∗ → hψ ∈ L1 (M) such that for every ψ ∈ M∗ and x, y ∈ M, hxψy ∗ = xhψ y ∗ ,

hψ ∗ = h∗ψ .

Moreover, if ψ = u|ψ| is the polar decomposition of ψ ∈ M∗ (see [127, Sec. III.4]), then hψ = uh|ψ| is the polar decomposition of hψ . Hence |hψ | = h|ψ| and the partial isometry part of hψ is u ∈ M. Due to the linear bijection in Theorem A.33, define a linear functional tr on L1 (M) by tr(hψ ) := ψ(1),

ψ ∈ M∗ .

Then we have tr(|hψ |) = tr(h|ψ| ) = |ψ|(1) = ψ,

ψ ∈ M∗ .

(A.16)

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 with the polar decomposition a = u|a|. Then for every Lemma A.34 Let a ∈ N p ∈ [1, ∞), a ∈ Lp (M) ⇐⇒ u ∈ M and |a|p ∈ L1 (M). Definition A.35 In view of Lemma A.34, for every a ∈ Lp (M) define ap ∈ [0, +∞) by ap := tr(|a|p )1/p

if 0 < p < ∞,

a∞ := a

if p = ∞.

In the case p = 1, by (A.16), a1 := tr(|a|) for a ∈ L1 (M) is the norm on L1 (M) copied from the norm on M∗ by the linear bijection ψ → hψ . In this way, (L1 (M),  · 1 ) becomes a Banach space identified with M∗ . Example A.36 Assume that M is semifinite with a faithful normal semifinite trace τ0 . Then (N, θs , τ ) is identified with    M ⊗ L∞ (R), id ⊗ (f → f (· + s)), τ0 ⊗ · et dt . R

In this case, for each ψ ∈ M∗+ we have hψ = (dψ/dτ0 ) ⊗ e−t , where dψ/dτ0 is the Radon–Nikodym derivative of ψ with respect to τ0 (see the end of Sect. A.5). Hence, for each p ∈ (0, ∞], Lp (M) = Lp (M, τ0 ) ⊗ e−t /p and a ⊗ e−t /p Lp (M) = aLp (M,τ0 ) for all a ∈ Lp (M, τ0 ). With neglecting the superfluous tensor factor e−t /p , we may identify Lp (M) with Lp (M, τ0 ). In the rest of the section we present properties of Haagerup’s Lp -spaces. See [129] for their proofs and more details. Proposition A.37 Let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1. If a ∈ Lp (M) and b ∈ Lq (M), then ab, ba ∈ L1 (M) and tr(ab) = tr(ba). Theorem A.38 (Hölder’s Inequality) Let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1. If a ∈ Lp (M) and b ∈ Lq (M), then |tr(ab)| ≤ ab1 ≤ ap bq .

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Moreover, for every p, q, r ∈ (0, ∞] with 1/p + 1/q = 1/r, if a ∈ Lp (M) and b ∈ Lq (M), then ab ∈ Lr (M) and abr ≤ ap bq . The reference [129] contains only the first assertion of the above theorem, while the second assertion and other inequalities for Haagerup’s  · p are found in [37]. Proposition A.39 Let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1. Then for every a ∈ Lp (M), ap = sup{|tr(ab)| : b ∈ Lq (M), bq ≤ 1}.

(A.17)

Theorem A.40 (1) For every p ∈ [1, ∞], (Lp (M),  · p ) is a Banach space. (2) In particular, L2 (M) is a Hilbert space with respect to the inner product

a, b := tr(a ∗ b) (= tr(ba ∗)) for a, b ∈ L2 (M). (3) For any p ∈ [1, ∞), the norm topology on Lp (M) coincides with the relative . More precisely, the uniform topology induced from the measure topology on N p . structure on L (M) by  · p coincides with that induced from N Theorem A.41 Let 1 ≤ p < ∞ and 1/p + 1/q = 1. Then the dual Banach space of Lp (M) is Lq (M) under the duality pairing (a, b) ∈ Lp (M) × Lq (M) −→ tr(ab) ∈ C. Proposition A.42 Let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1. Let a ∈ Lq (M). Then a ≥ 0 ⇐⇒ tr(ab) ≥ 0 for all b ∈ Lp (M)+ . Theorem A.43 For each x ∈ M we define the left action λ(x) and the right action ρ(x) on the Hilbert space L2 (M) by λ(x)a := xa,

ρ(x)a := ax,

a ∈ L2 (M),

and the involution J on L2 (M) by J a := a ∗ . Then: (1) λ (resp., ρ) is a normal faithful representation (resp., anti-representation) of M on L2 (M). (2) The von Neumann algebras λ(M) and ρ(M) are the commutants of each other and ρ(M) = J λ(M)J. (3) (λ(M), L2 (M), J, L2 (M)+ ) is a standard form of M.

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Remark A.44 For any projection e ∈ M, Haagerup’s Lp -space Lp (eMe) is identified with eLp (M)e. Furthermore, since ej (e)L2 (M) = eL2 (M)e, we see from Proposition A.16 that (eMe, eL2(M)e, J = ∗ , eL2 (M)+ e) is a standard form of eMe, where eMe acts on eL2 (M)e as the left multiplication. Finally in this section we give a lemma (due to Kosaki [78]), which will be useful in the next section. Lemma A.45 For every ϕ ∈ M∗+ let s(ϕ) be the M-support of ϕ and σt be the modular automorphism group with respect to ϕ|s(ϕ)Ms(ϕ). Then ϕ

σt (x) = hitϕ xh−it ϕ , ϕ

x ∈ s(ϕ)Ms(ϕ), t ∈ R.

Proof First, assume that ϕ is faithful. Then αt (x) := hitϕ xh−it ϕ (x ∈ M, t ∈ R) defines a strongly continuous one-parameter automorphism group. Let x, y ∈ M and assume that x is entire α-analytic with the analytic extension αz (x) (z ∈ C), see, is e.g., [22, Sec. 2.5.3]. By analytic continuation it follows that his ϕ αz (x) = αz+s (x)hϕ for all s ∈ R and z ∈ C. For every ξ ∈ D(hϕ ) (⊂ K := L2 (R, H), the representing Hilbert space for N), by Theorem B.1 of Appendix B there exists a K-valued bounded strongly continuous function f (ζ ) on −1 ≤ Im ζ ≤ 0, analytic in −1 < Im ζ < 0, such that f (s) = his ϕ ξ (s ∈ R). Then, for each z ∈ C, αz+ζ (x)f (ζ ) is a bounded strongly continuous function on −1 ≤ Im ζ ≤ 0, analytic in −1 < Im ζ < 0, such that αz+s (x)f (s) = his ϕ αz (x)ξ (s ∈ R). Hence by Theorem B.1 again, αz (x)ξ ∈ D(hϕ ) and hϕ αz (x)ξ = αz−i (x)hϕ ξ , from which we find that hϕ αz (x) = αz−i (x)hϕ (z ∈ C), see [129, Chap. I, Proposition 12]. Since ϕ(αt (x)y) = tr(hϕ αt (x)y) = tr(αt −i (x)hϕ y) = ϕ(yαt −i (x)),

t ∈ R,

it follows that ϕ satisfies the (αt , −1)-KMS condition, see [23, Definition 5.3.1 and ϕ Proposition 5.3.7]. Hence Theorem A.7 implies that σt = αt . For general ϕ ∈ M∗+ 1 let e := s(ϕ). Since hϕ ∈ eL (M)e corresponds to ϕ|eMe (see Remark A.44), the result follows from the above case. The above lemma says that itϕ (xhϕ ) = hitϕ (xhϕ )h−it ϕ for all x ∈ M and t ∈ it it −it 2 R. This shows that ϕ ξ = hϕ ξ hϕ for all ξ ∈ L (M) and t ∈ R. Also, from the 1/2

1/2

p/2

1/2

p/2

(1−p)/2

uniqueness of analytic continuation it follows that ϕ (xhϕ ) = hϕ xhϕ for 0 ≤ p ≤ 1. Furthermore, in a way similar to the proof of Proposition A.22 (2), we have for any ψ, ϕ ∈ M∗+ ,

itψ,ϕ ξ = hitψ ξ h−it ϕ , p/2

p/2

ψ,ϕ (xhϕ1/2) = hψ xh(1−p)/2 , ϕ

ξ ∈ L2 (M), t ∈ R,

(A.18)

x ∈ M, 0 ≤ p ≤ 1,

(A.19)

with the convention that h0ψ = s(ψ), h0ϕ = s(ϕ) and 0ψ,ϕ = s(ψ)J s(ϕ)J .

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A.7 Connes’ Cocycle Derivatives Consider the tensor product M (2) := M ⊗ M2 of M with the 2 × 2 matrix algebra M2 (C). For each ϕ, ψ ∈ M∗+ the balanced functional θ = θ (ϕ, ψ) on M (2) is defined by θ



2

 xij ⊗ eij

:= ϕ(x11 ) + ψ(x22 ),

xij ∈ M,

i,j =1

where eij (i, j = 1, 2) are the matrix units of M2 . The functional θ was already used in the proof of Proposition A.22 (2). The support projection of θ is s(θ ) = s(ϕ) ⊗ e11 +s(ψ)⊗e22 . Concerning the modular automorphism group σtθ on s(θ )M (2) s(θ ), the following hold:   (1) s(ϕ) ⊗ e11 , s(ψ) ⊗ e22 ∈ s(θ )M (2) s(θ ) θ (the centralizer of θ |s(θ)M (2)s(θ)), ϕ (2) σtθ (x ⊗ e11 ) = σt (x) ⊗ e11 for all t ∈ R and x ∈ s(ϕ)Ms(ϕ), ψ (3) σtθ (x ⊗ e22 ) = σt (x) ⊗ e22 for all t ∈ R and x ∈ s(ψ)Ms(ψ), (4) σtθ (s(ψ)Ms(ϕ) ⊗ e21 ) ⊂ s(ψ)Ms(ϕ) ⊗ e21 . Definition A.46 ([26]) Let ϕ, ψ ∈ M∗+ and θ = θ (ϕ, ψ). By the above (4) there exists a strongly* continuous map t ∈ R → ut ∈ s(ψ)Ms(ϕ) such that σtθ (s(ψ)s(ϕ) ⊗ e21 ) = ut ⊗ e21 ,

t ∈ R.

The map t → ut is called Connes’ cocycle (Radon–Nikodym) derivative of ψ with respect to ϕ, and denoted by ut = [Dψ : Dϕ]t , t ∈ R. The next proposition specifies the relation between Connes’ cocycle derivative [Dψ : Dϕ]t and Araki’s relative modular operator ψ,ϕ . Proposition A.47 For every ϕ, ψ ∈ M∗+ we have [Dψ : Dϕ]t J s(ϕ)J = itψ,ϕ −it ϕ ,

[Dψ : Dϕ]t J s(ψ)J = itψ −it ϕ,ψ ,

t ∈ R. (A.20)

Remark A.48 For any projection e ∈ M, recall that x ∈ eMe →  xe ∈ eMee (e := J eJ ) is a ∗-isomorphism (see Proposition A.16 (1)). This may justify writing (A.20) in a simpler way as follows: [Dψ : Dϕ]t = itψ,ϕ −it ϕ

if s(ψ) ≤ s(ϕ),

[Dψ : Dϕ]t = itψ −it ϕ,ψ

if s(ϕ) ≤ s(ψ).

The expression of [Dψ : Dϕ]t in terms of Haagerup’s L1 -elements is quite convenient to derive properties of Connes’ cocycle derivative [Dψ : Dϕ]t . To prove

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this, we first briefly examine Haagerup’s Lp -spaces for M (2) = M ⊗ M2 . Take the tensor product ω ⊗ Tr of a faithful ω ∈ M∗+ and the trace functional Tr on M2 . Then σtω⊗Tr = σtω ⊗ id2 , where id2 is the identity map on M2 . From the construction of the crossed products N := M σ ω R and N (2) := M (2) σ ω ⊗id2 R (see the first paragraph of Sect. A.6), it is easy to see that: (a) N (2) = N ⊗ M2 , (b) the canonical trace on N (2) is τ ⊗ Tr, where τ is the canonical trace on N, (c) the dual action on N (2) is θs ⊗ id2 (s ∈ R), where θs is the dual action on N.  ⊗ M2 , where N  and N (2) are the (2) = N Based on these facts we have N spaces of τ -measurable and τ ⊗ Tr-measurable operators affiliated with N and N (2) , respectively. Therefore, for 0 < p ≤ ∞, Haagerup’s Lp -space (2) = N  ⊗ M2 : (θs ⊗ id2 )(a) = e−s/p a, s ∈ R} Lp (M (2) ) := {a ∈ N is given as  $  % a a Lp (M (2)) = Lp (M) ⊗ M2 = a = 11 12 : aij ∈ Lp (M), i, j = 1, 2 , a21 a22  (2)   and its positive part is (Lp (M) ⊗ M2 ) ∩ N . Moreover, by closely looking the + construction of the functional tr, we notice that (d) the tr-functional on L1 (M (2)) = L1 (M) ⊗ M2 is tr ⊗ Tr, where tr is the trfunctional on L1 (M). In this way, the standard form of M (2) = M ⊗ M2 in terms of Haagerup’s L2 space is canonically given as (M ⊗ M2 , L2 (M) ⊗ M2 , J = ∗ , (L2 (M) ⊗ M2 )+ ), where [xij ]2i,j =1 ∈ M ⊗ M2 acts on L2 (M) ⊗ M2 as the 2 × 2 matrix left multi    ∗ x11 x12 ξ11 ξ12 ξ11 ξ12 ∗ plication = , and J = is the matrix ∗-operation x x ξ21 ξ22 ξ21 ξ22  ∗ ∗  21 22 ξ11 ξ21 2 2 ∗ ξ ∗ for [ξij ]i,j =1 ∈ L (M) ⊗ M2 . ξ12 22 Lemma A.49 For every ϕ, ψ ∈ M∗+ we have [Dψ : Dϕ]t = hitψ h−it ϕ ,

t ∈ R,

(A.21)

where hϕ , hψ are the elements of Haagerup’s L1 -space L1 (M) corresponding to ϕ, ψ.

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Proof In view of the above description of Lp (M ⊗M2 ), in the fact in (d),   particular, h 0 ϕ . Hence it follows the element of L1 (N) corresponding to θ (ϕ, ψ) is hθ = 0 hψ from Lemma A.45 that for [xij ] ∈ s(θ )Ns(θ ), σtθ

   it   −it x11 x12 h 0 x11 x12 hϕ 0 = ϕ 0 hψ 0 hψ x21 x22 x21 x22 / . −it it hitϕ x11h−it ϕ hϕ x12 hψ t ∈ R. = it −it , it hψ x21 h−it ϕ hψ x22 hψ

(A.22)

Therefore, /    . 0 0 0 0 0 0 θ = σt = it −it hψ hϕ 0 s(ψ)s(ϕ) 0 [Dψ : Dϕ]t 0



so that (A.21) follows. Remark A.50 In fact, the assertions stated just before Definition A.46 are immediately seen from expression (A.22) and Lemma A.45. Example A.51 Assume that M is a semifinite von Neumann algebra with a faithful normal semifinite trace τ . As explained in Example A.36, Haagerup’s L1 -space L1 (M) in this case is identified with the conventional L1 -space L1 (M, τ ) with respect to τ . More precisely, for each ψ ∈ M∗ , hψ in L1 (M) and the Radon– Nikodym derivative dψ/dτ ∈ L1 (M, τ ) are in the relation that hψ = (dψ/dτ ) ⊗ e−t . Hence, for every ϕ, ψ ∈ M∗+ we have  [Dψ : Dϕ]t =

dψ dτ

it 

dϕ dτ

−it .

In particular, when B = B(H) with the usual trace Tr, we have [Dψ : Dϕ]t = Dψit Dϕ−it , where Dϕ , Dψ are the density (trace-class) operators representing ϕ, ψ ∈ B(H)+ ∗. Below we present important properties of Connes’ cocycle derivative [Dψ : Dϕ]t , which were first given by Connes [26, 27] for the case of faithful normal semifinite weights. Their proofs and more details are found in [122, 128]. Theorem A.52 Let ϕ, ψ ∈ M∗+ and assume that s(ψ) ≤ s(ϕ). Then ut := [Dψ : Dϕ]t satisfies the following properties: (i) ut u∗t = s(ψ) = u0 and u∗t ut = σt (s(ψ)) for all t ∈ R. In particular, ut ’s are partial isometries with the final projection s(ψ). ϕ (ii) us+t = us σs (ut ) for all s, t ∈ R (the cocycle identity). ϕ (iii) u−t = σ−t (u∗t ) for all t ∈ R. ϕ

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(iv) σt (x) = ut σt (x)u∗t for all t ∈ R and x ∈ s(ψ)Ms(ψ). (v) For every x ∈ s(ψ)Ms(ϕ) and y ∈ s(ϕ)Ms(ψ), there exists a continuous bounded function F on 0 ≤ Im z ≤ 1, analytic in 0 < Im z < 1, such that ψ

ϕ

ϕ

ϕ

F (t + i) = ϕ(yut σt (x)),

F (t) = ψ(ut σt (x)y),

t ∈ R.

Furthermore, ut (t ∈ R) is uniquely determined by a strongly* continuous map t ∈ R → ut ∈ M satisfying the above (i), (ii), (iv) and (v). (Note that (iii) follows from (i) and (ii).) Proposition A.53 Let ϕ, ψ, ω ∈ M∗+ . (1) [Dψ : Dϕ]∗t = [Dϕ : Dψ]t for all t ∈ R. (2) If either s(ψ) ≤ s(ω) or s(ϕ) ≤ s(ω), then [Dψ : Dω]t [Dω : Dϕ]t = [Dψ : Dϕ]t for all t ∈ R (the chain rule). (3) If s(ψ) ≤ s(ω) and s(ϕ) ≤ s(ω), then [Dψ; Dω]t = [Dϕ : Dω]t for all t ∈ R if and only if ψ = ϕ. (4) For every α ∈ Aut(M) (the automorphisms of M), [D(ψ ◦ α) : D(ϕ ◦ α)]t = α −1 ([Dψ : Dϕ]t ) for all t ∈ R. The following is Connes’ inverse theorem in [27, Theorem 1.2.4]. Theorem A.54 ([27]) Let ϕ is a faithful normal semifinite weight on M. Let t ∈ R → ut ∈ M is a strongly* continuous map satisfying us+t = us σsϕ (ut ), u−t =

ϕ σ−t (u∗t ),

t ∈ R, t ∈ R.

(A.23)

Then there exists a unique normal semifinite weight ψ on M such that ut = [Dψ : Dϕ]t for all t ∈ R. In [27] ut ’s are assumed to be unitaries in M. In this case, (A.23) is redundant and ψ given in the theorem is faithful as well. The above version without ut ’s being unitaries is taken from [122, Theorem 51]. Note that even when ϕ ∈ M∗+ , ψ in the theorem cannot be in M∗+ in general. This fact suggests that the von Neumann algebra theory cannot be self-completed when we stick to functionals in M∗ , so the normal semifinite weight theory is unavoidable. We state the next proposition in terms of weights, which is used in Sect. 6.4 together with Theorem A.54. The proposition was originally given in [27, Lemma 1.4.4] for faithful normal semifinite weights ψ, ϕ, but it is not difficult to remove the assumption of ψ being faithful.3 Proposition A.55 Let M0 be a von Neumann subalgebra of M and E : M → M0 be a faithful normal conditional expectation onto M0 . Let ϕ, ψ be normal semifinite

3

The author is indebted to A. Jenˇcová for this observation.

158

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Preliminaries on von Neumann Algebras

weights on M0 and assume that ϕ is faithful. Then [D(ψ ◦ E) : D(ϕ ◦ E)]t = [Dψ : Dϕ]t ,

t ∈ R.

The next proposition supplements Lemma 4.20 with additional characterizations in terms of [Dψ : Dϕ]t . Proposition A.56 Let ϕ, ψ ∈ M∗+ with s(ψ) ≤ s(ϕ). The following conditions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

ϕ

ψ ◦ σt = ψ for all t ∈ R; [Dψ : Dϕ]t ∈ (s(ϕ)Ms(ϕ))ϕ (the centralizer of ϕ|s(ϕ)Ms(ϕ)) for all t ∈ R; [Dψ : Dϕ]t ∈ (s(ψ)Ms(ψ))ψ for all t ∈ R; t ∈ R → [Dψ : Dϕ]t is a one-parameter group of unitaries in s(ψ)Ms(ψ); it it is his ψ hϕ = hϕ hψ for all s, t ∈ R;  (the τ -measurable operators affiliated with hψ hϕ = hϕ hψ as elements of N N).

Definition A.57 We say that ψ commutes with ϕ if the equivalent conditions of Proposition A.56 hold. Condition (i) is often used to define the commutativity for normal functionals (also normal semifinite weights), but (v) and (vi) are quite natural definitions of commutativity that are available for any ϕ, ψ ∈ M∗+ without s(ψ) ≤ s(ϕ), as stated in Lemma 4.20. For use in Sects. 3.3 and 5.3 we state the following lemmas. The first lemma was also given in [54, Lemma A.1], which generalizes [26, Theorem 3] and [28, Lemma 3.13]. In fact, the lemma can be shown by (A.21) and [123, Sec. 9.24] together with a common argument in analytic function theory. See also [81]. In the notation of [30, Definition 4.1] condition (ii) can be written as ψ ≤ λ1/δ ϕ (δ/2). Lemma A.58 For every ψ, ϕ ∈ M∗+ and δ > 0, the following conditions are equivalent: (i) hδψ ≤ λhδϕ , i.e., λhδϕ − hδψ ∈ L1/δ (M)+ for some λ > 0; (ii) s(ψ) ≤ s(ϕ) and [Dψ : Dϕ]t extends to a σ -weakly continuous (M-valued) function [Dψ : Dϕ]z on −δ/2 ≤ Im z ≤ 0 which is analytic in the interior. If the above conditions hold, then [Dψ : Dϕ]z  ≤ λ1/2 and [Dψ : Dϕ]z is strongly continuous on −δ/2 ≤ Im z ≤ 0, and p/2

hψ

= [Dψ : Dϕ]−ip/2 hp/2 ϕ ,

0 < p ≤ δ.

(A.24)

The assumption of Lemma A.24 is the δ = 1 case of the above lemma, so A in Lemma A.24 is equal to [Dψ : Dϕ]−i/2 . Kosaki [81] proved the following, which contains Sakai’s quadratic Radon– Nikodym theorem [116, 1.24.3] as the special case when δ = p = 1.

A.8 Kosaki’s Lp -Spaces

159

Lemma A.59 If ψ, ϕ ∈ M∗+ and hδψ ≤ λhδϕ for some δ, λ > 0, then for every p > 0 there exists a unique kp ∈ M+ with 0 ≤ kp ≤ λp/2δ such that s(kp ) ≤ s(ϕ) and p p p p hψ = kp hϕ kp , where s(kp ) is the support projection of kp . Moreover, hψ = kp hϕ kp p/2 p p/2

p/2

p/2

implies that (hϕ hψ hϕ )1/2 = hϕ kp hϕ . The construction of kp in [81] is as follows. By virtue of Furuta’s inequality [41] (extended to τ -measurable operators) the above assumption implies that p/2 p p/2 p (hϕ hψ hϕ )1/2 ≤ λp/2δ hϕ for any p > 0. Hence there exists a unique bp ∈ M with bp  ≤ λp/4δ such that s(bp∗ bp ) ≤ s(ϕ) and (hϕ hψ hϕ )1/4 = p/2 p p/2

bp hϕ . In fact, bp = [Dχ(p) : Dϕ]−ip/2 , where χ(p) ∈ M∗+ is given by p/2 p p/2 p/2 p p/2 hχ(p) = (hϕ hψ hϕ )1/2p (see (A.24)). Let kp := bp∗ bp . Then hϕ hψ hϕ = p/2

p/2

p

p/2

hϕ kp hϕ kp hϕ

p

p

p/2 p p/2

p/2

p/2

and so hψ = kp hϕ kp and (hϕ hψ hϕ )1/2 = hϕ kp hϕ .

A.8 Kosaki’s Lp -Spaces In this section, in connection with Haagerup’s Lp -spaces in Sect. A.6, we give a brief survey of Kosaki’s Lp -spaces [78]. A merit of Kosaki’s Lp -spaces is that the construction itself allows us to apply complex interpolation techniques of the Riesz– Thorin type to them. Let ϕ0 ∈ M∗+ be a distinguished faithful state and h0 := hϕ0 ∈ L1 (M). For each η 1−η η ∈ [0, 1], M is embedded into L1 (M) by x ∈ M → h0 xh0 ∈ L1 (M). Define η 1−η η 1−η η 1−η the norm h0 xh0  := x on h0 Mh0 (⊂ L1 (M)), so that h0 Mh0 ∼ = M. η 1−η 1 Then (h0 Mh0 , L (M)) becomes a pair of compatible Banach spaces. Definition A.60 Let 1 < p < ∞ and 0 ≤ η ≤ 1. Kosaki’s Lp -space Lp (M, ϕ0 )η with respect to ϕ0 is defined to be the complex interpolation space η

1−η

C1/p (h0 Mh0

, L1 (M))

equipped with the complex interpolation norm  · p,η (=  · C1/p ), see [18] for general theory on the complex interpolation method. Moreover, we write η 1−η L1 (M, ϕ0 )η := L1 (M) with  · 1,η =  · 1 and L∞ (M, ϕ0 )η := h0 Mh0 η 1−η (identified with M) with h0 xh0 ∞,η = x. p In particular, L (M, ϕ0 )η for η = 0, 1 and 1/2 are respectively called the left, the right and the symmetric Lp -spaces. We may write the norm  · p,η as  · p,ϕ0 ,η if we need to specify its dependence on ϕ0 . η

1−η

Note that a more general embedding x ∈ M → h0 xk0 ∈ L1 (M) was treated in [78], where k0 := hψ0 for another faithful state ψ0 ∈ M∗+ .

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Preliminaries on von Neumann Algebras

Since Hölder’s inequality (Theorem A.38) gives 1−η

η

h0 xh0

1−η

η

1 ≤ h0 1 x h0 1

= x,

x ∈ M,

we find from general properties of the complex interpolation method (see [18]) that for 1 < p < p < ∞, (M =) L∞ (M, ϕ0 )η ⊂ Lp (M, ϕ0 )η

⊂ Lp (M, ϕ0 )η ⊂ L1 (M, ϕ0 )η (= L1 (M)), η

1−η

(x =) h0 xh0

η

1−η

p,η

η

1−η

p ,η ≥ h0 xh0

∞,η ≥ h0 xh0 ≥ h0 xh0

η

1−η

1 ,

x ∈ M.

1−η

η

Note that h0 Mh0 is dense in Lp (M, ϕ0 )η for every p ∈ [1, ∞). The following is a main theorem [78, Theorem 9.1]. Theorem A.61 Let 1 ≤ p ≤ ∞ and 1/p + 1/q = 1. Then η/q

(1−η)/q

Lp (M, ϕ0 )η = h0 Lp (M)h0 η/q

(1−η)/q

h0 ah0

p,η = ap ,

(⊂ L1 (M)), a ∈ Lp (M).

η/q (1−η)/q That is, Lp (M) ∼ . = Lp (M, ϕ0 )η by the isometry a → h0 ah0

In particular, when η = 0, 1 and 1/2, we have Kosaki’s left, right and symmetric Lp -spaces as 1/q

1/q

Lp (M, ϕ0 )L := Lp (M)h0 ,

Lp (M, ψ0 )R := h0 Lp (M), 1/2q

Lp (M, ϕ0 ) := Lp (M, ϕ0 )η=1/2 = h0

1/2q

Lp (M)h0

with the norms 1/q

1/q

1/2q

ah0 p,0 = h0 ap,1 = h0

1/2q

ah0

p,1/2 = ap ,

a ∈ Lp (M).

When 1 ≤ p < ∞ and 1/p + 1/q = 1, the Lp -Lq -duality of Kosaki’s can be given by transforming that of Haagerup’s Lp -spaces in view of Theorem A.61. For instance, for η, η ∈ [0, 1] the duality pairing between Lp (M, ϕ0 )η and Lq (M, ϕ0 )η is written as Lp -spaces

η/q

(1−η)/q

h0 ah0

η /p

, h0

(1−η )/p !

bh0

p,q

= tr(ab),

a ∈ Lp (M), b ∈ Lq (M).

A.8 Kosaki’s Lp -Spaces

161

When η = 1 − η, this duality is somewhat convenient in the sense that for every x, y ∈ M, 1−η

η

h0 xh0

1−η

, h0

η/q

η!

yh0

η/p

p,q

(1−η)/p

= h0 (h0 xh0 =

(1−η)/q

)h0

(1−η)/p

, h0

η/p (1−η)/p (1−η)/q η/q tr(h0 xh0 h0 yh0 )

=

(1−η)/q

(h0

η/q

η/p !

yh0 )h0

p,q

η 1−η tr(h0 xh0 y),

whose last expression is independent of p ∈ [1, ∞). Remark A.62 When ϕ ∈ M∗+ is not necessarily faithful with the support e := s(ϕ) ∈ M, Kosaki’s Lp -space with respect to ϕ is defined with restriction to eMe. More specifically, for 0 ≤ η ≤ 1 and 1 < p < ∞, with the embedding x ∈ eMe → η 1−η hϕ xhϕ ∈ eL1 (M)e (= L1 (eMe)) (see Remark A.44), Theorem A.61 says that p (1−η)/q p (1−η)/q Lp (M, ϕ)η = hη/q = hη/q (⊂ eL1 (M)e) ϕ eL (M)ehϕ ϕ L (M)hϕ η/q

(1−η)/q

with the norm hϕ ahϕ p,η = ap for a ∈ eLp (M)e, where 1/p+1/q = 1. In particular, the symmetric case Lp (M, ϕ) := Lp (M, ϕ)η=1/2 = hϕ1/2q Lp (M)hϕ1/2q plays a role in Sect. 3.3.

Appendix B

Preliminaries on Positive Self-Adjoint Operators

Let A be a positive self-adjoint operator on a Hilbert space H having the spectral decomposition 

∞

A=

t dEt 0

with the spectral resolution Et (0 ≤ t < ∞). ∞ It is well-known that ξ ∈ H belongs to D(A), the domain of A, if and only if 0 t 2 dEt ξ 2 < +∞, and in this case, ∞ Aξ 2 = 0 t 2 dEt ξ 2 . For a complex-valued Borel function f on [0, ∞), the Borel functional calculus f (A) is defined by 

∞

f (A) :=

f (t) dEt , 0

  ∞ ξ ∈ H : 0 |f (t)|2 dEt ξ 2 < +∞ . If ξ ∈ ∞ D(f (A)), then we have f (A)ξ 2 = 0 |f (t)|2 dEt ξ 2 . The operator f (A) is self-adjoint, positive self-adjoint, and bounded if f is real-valued, non-negative, and bounded, respectively. For instance, whose domain is D(f (A)) =



∞

A = p

(s1 + A)−1 =



t p dEt

(p > 0),

0 ∞ 0

1 dEt , s+t

A(1 + sA)−1 =



∞ 0

t dEt 1 + st

(s > 0).

Moreover, when A is non-singular (i.e., the support s(A) = 1), A

−1



∞

=

t 0

−1

 dEt ,

∞

A = is

t is dEt

(s ∈ R).

0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9

163

164

B Preliminaries on Positive Self-Adjoint Operators

In fact, A−1 and Ais are defined for singular A as well with restriction to s(A)H (or in the sense of the generalized inverse), that is, A−1 = (0,∞) t −1 dEt and similarly for Ais . (Note that A0 is used to mean either A0 = 1 or A0 = s(A) according to the situation.) A fundamental property of positive self-adjoint operators is stated in the next theorem, whose details are found in [123, Chap. 9] (also [56, Appendix A]). Theorem B.1 Let A be a self-adjoint operator on H with the spectral decomposi∞ tion A = 0 t dEt . Let α > 0. Then, for every ξ ∈ H the following conditions are equivalent: (i) ξ ∈ D(Aα ); ∞ (ii) 0 t 2α dEt ξ 2 < +∞; (iii) there exists an H-valued bounded weakly (equivalently, strongly) continuous function f on −α ≤ Im z ≤ 0, weakly (equivalently, strongly) analytic in −α < Im z < 0, such that f (s) = Ais ξ for all s ∈ R. Furthermore, the function f (z) in (iii) is unique and Aα ξ = f (−iα) holds. The above theorem may be a reformulation of the famous Stone’s representation theorem from the viewpoint of the analytic generator. For a strongly continuous oneparameter unitary group Us = eisH on H, where H is a self-adjoint generator, the H is usually obtained by the real analytic method as H = limt →0 (Ut − 1)/it, while the above theorem says that the analytic generator A = eH can be obtained by a complex analytic method. From the analytic continuation characterization in (iii) above, one can show the following: Proposition B.2 Let A be a positive self-adjoint operator on H. Let α, β ≥ 0. Then we have Aα+β = Aα Aβ ; more precisely, for ξ ∈ H, ξ ∈ D(Aα+β ) if and only if ξ ∈ D(Aβ ) and Aβ ξ ∈ D(Aα ), and in this case, Aα+β ξ = Aα (Aβ ξ ). (Either convention A0 = 1 or A0 = s(A) is available here.) Moreover, we have (A−1 )α = (Aα )−1 (simply denoted by A−α ) and A−(α+β) = −α A A−β . For Borel functional calculus we have general formulas like (f + g)(A) = f (A) + g(A),

(fg)(A) = f (A)g(A).

A point of the above proposition is that Aα+β = Aα B β holds without closure for α, β ∈ R with αβ ≥ 0. The equality further extends to all α, β ∈ C with Re α · Re β ≥ 0. Another fundamental aspect of positive self-adjoint operators on H is their correspondence to closed (densely defined) positive quadratic forms on H. A function

B Preliminaries on Positive Self-Adjoint Operators

165

q : D(q) → [0, +∞), where D(q) is a linear subspace of H, is called a positive quadratic form if: • q(λξ ) = |λ|2 q(ξ ) for all ξ ∈ D(q), λ ∈ C, • q(ξ + η) + q(ξ − η) = 2q(ξ ) + 2q(η) for all ξ, η ∈ D(q). Such a q is said to be closed if {ξn } ⊂ D(q), ξ ∈ H, ξn −ξ  → 0 and q(ξn −ξm ) → 0 as n, m → ∞, then ξ ∈ D(q) and q(ξn − ξ ) → 0. Now we state the next important theorem. For details see [74, Chap. 6, Sec. 2.6], [114, Theorem VIII.15], [117, Chap. 10], [119, Theorem 2] or [120, Sec. 7.5] (also [56, Appendix A]). Theorem B.3 Let q be a densely defined (i.e., D(q) is dense in H) positive quadratic form, and define q(ξ ) :=

 q(ξ ) +∞

if ξ ∈ D(q), if ξ ∈ H \ D(q).

(B.1)

Then the following conditions are equivalent: (a) q is closed; (b) q is lower semicontinuous on H; (c) there exists a (unique) positive self-adjoint operator A on H such that D(A1/2 ) = D(q) and q(ξ ) = A1/2 ξ 2 ,

ξ ∈ D(A1/2 ).

(B.2)

In this way, a positive self-adjoint operator A on H corresponds one-to-one to a closed densely defined positive quadratic form q given in (B.2). Below we will sometimes use the notation A1/2 ξ 2 for all ξ ∈ H to mean q(ξ ) in (B.1), that is, we set   ∞ A1/2 ξ 2 < +∞ if ξ ∈ D(A1/2 ), 1/2 2 2 (B.3) A ξ  := t dEt ξ  = +∞ otherwise. 0 The following fact is well-known, see, e.g., [122, Sec. A.4]. Proposition B.4 Let A, B be positive self-adjoint operators on H. Then the following conditions are equivalent: (i) D(B 1/2 ) ⊂ D(A1/2 ) and A1/2 ξ 2 ≤ B 1/2 ξ 2 for all ξ ∈ D(B 1/2 ); (ii) there exists a core D of B 1/2 such that D ⊂ D(A1/2 ) and A1/2ξ 2 ≤ B 1/2 ξ 2 for all ξ ∈ D; (iii) (s1 + B)−1 ≤ (s1 + A)−1 for some (equivalently, any) s > 0; (iv) A(1 + sA)−1 ≤ B(1 + sB)−1 for some (equivalently, any) s > 0.

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B Preliminaries on Positive Self-Adjoint Operators

For positive self-adjoint operators A, B we write A ≤ B (in the form sense) if the equivalent conditions of Proposition B.4 hold. From Proposition B.4 we have the next result. Proposition B.5 For densely defined closed operators A, B on H the following are equivalent: (i) A∗ A ≤ B ∗ B; (ii) there exists a core D of B such that D ⊂ D(A) and Aξ  ≤ Bξ  for all ξ ∈ D. Proposition B.6 Let A, B be positive self-adjoint operators with s(A) = s(B), where s(A) is the support projection of A. If A ≤ B, then A−1 ≥ B −1 , where A−1 is defined with restriction to s(A)H (i.e., in the sense of the generalized inverse). Proof Let e := s(A) = s(B). By Proposition B.4 it is clear that A ≤ B if and only if (e + A)−1 ≥ (e + B)−1 , which is equivalent to (e + A−1 )−1 ≤ (e + B −1 )−1 since (e + A−1 )−1 = e − (e + A)−1 . Hence the assertion holds. Lemma B.7 Let A, B be positive self-adjoint operators on H such that A ≤ B. Then, for every operator monotone function f ≥ 0 on [0, ∞) we have f (A) ≤ f (B), that is, f (A)1/2ξ 2 ≤ f (B)1/2 ξ 2 for all ξ ∈ H (in the sense of (B.3)). ∞ Proof Note that f (A)1/2ξ 2 = 0 f (t) dEt ξ 2 for every ξ ∈ H with the ∞ spectral decomposition A = 0 t dEt . Now we recall that f has the integral expression  f (t) = a + bt +

[0,∞)

t dμ(s), t +s

t ∈ [0, ∞),

where a, b ≥ 0 and μ is a positive measure on [0, ∞) with +∞. By Fubini’s theorem we have 

∞



[0,∞) (1 + s)

−1 dμ(s)


0; (ii) (i + An )−1 → (i + A)−1 strongly; (iii) An (1 + sAn )−1 → A(1 + sA)−1 strongly for some (equivalently, any) s > 0. Moreover, assume that all An and A are non-singular, so we write An = eHn and A = eH , where Hn := log An and H := log A. Then the above conditions are also equivalent to the following: (iv) (i + Hn )−1 → (i + H )−1 strongly; (v) Aitn → Ait strongly for all t ∈ R. If the equivalent conditions (i)–(iii) of Proposition B.8 hold, then we say that An converges to A in the strong resolvent sense. In particular, when A1 ≤ A2 ≤ · · · (in the form sense), it is also well-known ([74, Chap. 8, Sec. 3.4], [120, Sec. 7.5]) that An → A in the strong resolvent sense (or (1 + An )−1 (1 + A)−1 strongly) if 1/2 and only if A1/2 ξ 2 = supn An ξ 2 for all ξ ∈ H. On the other hand, when A1 ≥ A2 ≥ · · · , the convergence An → A in the strong resolvent sense (or 1/2 (1 + An )−1  (1 + A)−1 strongly) does not imply that A1/2ξ 2 = infn An ξ 2 , 1/2 2 because q0 (ξ ) = infn An ξ  is not necessarily lower semicontinuous on H, while A1/2 ξ 2 is lower semicontinuous on H (from condition (b) of Theorem B.3). For more in the latter case, see [120, Sec. 7.5], [82]. The next proposition is used in Sect. 3.3. Proposition B.9 Let An (n ∈ N) and A be positive self-adjoint operators on H such that (1 + An )−1 (1 + A)−1

strongly,

168

B Preliminaries on Positive Self-Adjoint Operators

that is, An increases to A in the strong resolvent sense. Then, for every operator monotone function f ≥ 0 on [0, ∞), we have (1 + f (An ))−1 (1 + f (A))−1

strongly,

that is, f (An ) increases to f (A) in the strong resolvent sense. Proof Since f (A1 ) ≤ f (A2 ) ≤ · · · by Lemma B.7, in view of the fact mentioned before the proposition, it suffices to show that f (An )1/2 ξ 2  f (A)1/2ξ 2 for all ξ ∈ H. Note that f (A)1/2ξ 2 is expressed as in (B.4) and similarly for f (An )1/2 ξ 2 . Since (s1 + An )−1 (s1 + A)−1 strongly for any s > 0 by Proposition B.8, we can apply the monotone convergence theorem to see that  [0,∞)

  ξ 2 − s ξ, (s1 + An )−1 ξ  dμ(s)



 [0,∞)

  ξ 2 − s ξ, (s1 + A)−1 ξ  dμ(s).

1/2

Together with An ξ 2  A1/2 ξ 2 this yields the desired conclusion. Here we recall the notion of form sums for two positive self-adjoint operators on H, which was introduced in [74]. Let A, B be positive self-adjoint operators on H. Then q(ξ ) := A1/2 ξ 2 + B 1/2 ξ 2 for ξ ∈ D(q) := D(A1/2 ) ∩ D(B 1/2 ) is a closed positive quadratic form on K := D(q), so there exists a unique self-adjoint operator C on K such that D(C 1/2 ) = D(q) and C 1/2 ξ 2 = A1/2 ξ 2 + B 1/2 ξ 2 ,

ξ ∈ D(A1/2 ) ∩ D(B 1/2 ).

(B.5)

˙ B. The form sum A + ˙B The C is called the form sum of A, B and denoted by A + is a positive self-adjoint operator on H whenever D(A1/2) ∩ D(B 1/2 ) is dense in H. For instance, this is the case of Proposition A.22 (5). The next proposition is given here to use it in Sect. 3.1. Proposition B.10 Let f be a positive operator monotone decreasing function on (0, ∞) (for example, f (t) = t α (t > 0) with −1 ≤ α ≤ 0). Assume that A, B are positive self-adjoint operators on H such that D(A1/2 ) ∩ D(B 1/2 ) is dense in H.4

˙ (1 − λ)B In fact, without this assumption let K := D(A1/2 ) ∩ D(B 1/2 ). Then C := λA + (0 < λ < 1) is a positive self-adjoint operator on K (considered ∞ on K ⊥ ). The conclusion remains true if f (C)1/2 ξ  is understood as  f (C)1/2 ξ 2 = f (t) dGt ξ 2 + f (∞)PK ⊥ ξ 2 ,

4

[0,∞)

∞

where C = 0 t dGt is the spectral decomposition of C on K, PK ⊥ is the projection onto K ⊥ , and f (∞) := limt→∞ f (t).

B Preliminaries on Positive Self-Adjoint Operators

169

Then for every ξ ∈ H we have ˙ (1 − λ)B)1/2 ξ  ≤ f (A)1/2 ξ λ f (B)1/2 ξ 1−λ , f (λA +

0 ≤ λ ≤ 1,

where f (A)1/2 ξ  etc. are understood in the sense of (B.3), i.e.,  f (A)1/2ξ 2 =

[0,∞)

f (t) dEt ξ 2

∞ with the spectral decomposition A = 0 t dEt and f (0) := f (0+ ) ∈ (0, +∞]. ∞ Proof Let B = 0 t dFt be the spectral decomposition as well as that of A above. ˙ (1 − λ)B. First, when A, B ∈ We may assume that 0 < λ < 1. Let C := λA + B(H)+ and they are invertible, the result is a consequence of [6, Theorem 3.1] saying that A → log ξ, f (A)ξ  is convex on {A ∈ B(H)+ : invertible} for every ξ ∈ H. Next, assume that A, B ≥ εI (hence C ≥ εI as well) for some ε > 0. For each n ∈ N set   An := t dEt + n(1 − En ), Bn := t dFt + n(1 − Fn ), [0,n]

[0,n]

Cn := λAn + (1 − λ)Bn . Note that An , Bn and Cn are all bounded and invertible. Then from the first case above one has f (Cn )1/2 ξ  ≤ f (An )1/2 ξ λ f (Bn )1/2 ξ 1−λ .

(B.6)

Since C −1 ≤ Cn−1 in B(H)+ and f (t −1 ) is operator monotone on (0, ∞), it follows that f (C) ≤ f (Cn ) in B(H)+ and hence f (C)1/2 ξ  ≤ f (Cn )1/2ξ .

(B.7)

Moreover, one sees that  f (An )1/2 ξ 2 =

[0,n]



−→

f (t) dEt ξ 2 + f (n)(1 − En )ξ 2

∞

f (t) dEt ξ 2 = f (A)1/2 2

0

as n → ∞, and similarly f (Bn )1/2 ξ 2 → f (B)1/2 ξ 2 . Combining these with (B.6) and (B.7) gives the result in this case.

170

B Preliminaries on Positive Self-Adjoint Operators

Finally, let A, B be general as in the proposition. For every ε > 0, since λ(A + ˙ (1 − λ)(B + ε1) = C + ε1, the above second case implies that ε1) + f (C + ε1)1/2 ξ  ≤ f (A + ε1)1/2ξ λ f (B + ε1)1/2 ξ 1−λ .

(B.8)

By the monotone convergence theorem we have  f (A + ε1)1/2 ξ 2 =

[0,∞)

f (t + ε) dEt ξ 2

 

[0,∞)

f (t) dEt ξ 2 = f (A)1/2ξ 2 ,

and similarly for f (B + ε1)1/2ξ 2 and f (C + ε1)1/2 ξ 2 . Therefore, we have the result by taking the limit of (B.8) as ε 0. Finally, we state an interpolation result related to positive self-adjoint operators. See [18] and [113, Appendix to IX.4] for general theory of interpolation Banach spaces. Let A be a positive self-adjoint operator on H. For each θ ∈ (0, 1] consider the inner product on D(Aθ/2 ) defined by

ξ, ηθ := Aθ/2 ξ, Aθ/2 η,

ξ, η ∈ D(Aθ/2 ),

and let Hθ be the Hilbert space by completing D(Aθ/2 ) with respect to ·, ·θ . (Alternatively, Hθ is also obtained by completing D(Aθ ) with respect to ξ, ηθ :=

ξ, Aθ η for ξ, η ∈ D(Aθ ).) It follows from [113, p. 35] that (H0 = H, H1 ) becomes a compatible pair of Hilbert spaces. Moreover, it is known [93] that Hθ (0 < θ < 1) becomes an exact interpolation space of exponent θ between H0 = H and H1 . In fact, the Hθ is an interpolation space in the complex interpolation method and is also called the geometric interpolation space of exponent θ . More specifically, we state the next proposition from [93, Theorem 1.1], whose statement is slightly different from (but equivalent to) that in [93] and is convenient to use in Sect. 7.1. Proposition B.11 Let A, B be positive self-adjoint operators on Hilbert spaces H, K, respectively. For each θ ∈ (0, 1] let Kθ be the interpolation Hilbert space induced from (K, B) similarly to Hθ from (H, A) stated as above. If V : H → K is a bounded linear operator with norm V 0 = V H→K such that V D(A1/2 ) ⊂ D(B 1/2 ) and B 1/2 V ξ  ≤ V 1 A1/2ξ ,

ξ ∈ D(A1/2 ),

with V 1 < +∞, then for every θ ∈ (0, 1] we have V D(Aθ/2 ) ⊂ D(B θ/2 ) and θ θ/2 B θ/2 V ξ  ≤ V 1−θ ξ , 0 V 1 A

ξ ∈ D(Aθ/2 ).

Appendix C

Operator Convex Functions on (0, 1)

As is well-known, e.g., [51, Theorem 2.7.6], a real function h on (−1, 1) is operator convex if and only if it has the integral expression  h(t) = a + bt +

[−1,1]

t2 dλ(u), 1 − ut

t ∈ (−1, 1),

(C.1)

where a, b ∈ R and a finite positive measure λ on [−1, 1] are uniquely determined. On the other hand, a real function f on (0, ∞) is operator convex if and only if it has the integral expression in (2.5) of Sect. 2.2, where a, b ∈ R, c ≥ 0 and a positive measure μ on [0, ∞) with [0,∞) (1 + s)−1 dμ(s) < +∞ are uniquely determined.  t  on (0, 1) induced In Theorem 4.8 of Sect. 4.2 the function g(t) := (1 − t)f 1−t from an operator convex function f on (0, ∞) plays a role. In this appendix we prove the next theorem, which says that the correspondence f ↔ g is bijective between the operator convex functions on (0, ∞) and those on (0, 1).  t  Theorem C.1 Let f be a real function on (0, ∞) and let g(t) := (1 − t)f 1−t for t ∈ (0, 1). Then the following conditions are equivalent: (a) (b) (c)

f is operator convex on (0, ∞). g is operator convex on (0, 1). There exist α, β−1∈ R, γ ≥ 0 and a positive measure ν on (−∞, 1] satisfying dν(w) < +∞ such that (−∞,1] (2 − w) g(t) = α + βt + γ

(2t − 1)2 + t

 (−∞,1]

(2t − 1)2 dν(w), 1 − wt

t ∈ (0, 1). (C.2)

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9

171

172

C Operator Convex Functions on (0, 1)

Moreover, in this case, g(0+ ) = f (0+ ),

g(1− ) = f (∞),

(C.3)

and α, β, γ and the measure ν in (c) are uniquely determined by f (hence by g). Proof  (b) ⇐⇒ (c). Obviously, g is operator convex on (0, 1) if and only tif+1so is g t +1 on (−1, 1). So we may show that expression (C.1) of h(t) = g 2 is 2 transformed into (C.2) of g and vice versa. From (C.1) we write for every t ∈ (0, 1), 

(2t − 1)2 dλ(u) [−1,1] 1 − u(2t − 1)  λ({−1}) (2t − 1)2 (2t − 1)2 dλ(u) + . = (a − b) + 2bt + 2u 2 t (−1,1] 1 − 1+u t 1 + u

g(t) = a + b(2t − 1) +

2u By transformation w = 1+u : (−1, 1] → (−∞, 1], define a measure ν on (−∞, 1] 1 −1 = 2−w by dν(w) = (1 + u) dλ(u). Since 1+u 2 , we have



(2 − w)−1 dν(w) =



(−∞,1]

dλ(u) < +∞. (−1,1]

Hence we have expression (C.2) of g. The converse transformation from (C.2) into (C.1) can similarly be shown, so (b) ⇐⇒ (c) follows. (a) ⇐⇒ (b). For this we may show that expression (2.5) of f is transformed to (C.2) of g and vice versa. From (2.5) we write for every t ∈ (0, 1), 

t g(t) = (1 − t)f 1−t



= a(1 − t) + b(2t − 1) + c = (a − b) + (2b − a)t + c  + (0,∞)

(2t − 1)2 + 1−t

 [0,∞)

(2t − 1)2 dμ(s) t + s(1 − t)

(2t − 1)2 (2t − 1)2 + μ({0}) 1−t t

(2t − 1)2 dμ(s) . s 1 − s−1 s t

By transformation w = s−1 s : (0, ∞) → (−∞, 1), define a measure ν on (−∞, 1) 1 s −1 = 1+s , we have by dν(w) = s dμ(s). Since 2−w  (0,∞)

(2 − w)−1 dν(w) =



(1 + s)−1 dμ(s) < +∞. (0,∞)

C Operator Convex Functions on (0, 1)

173

Hence, setting ν({1}) := c, we find that g has an expression in the form (C.2). Conversely, from expression (C.2) of g we have for every t ∈ (0, ∞),  f (t) = (t + 1)g

t t +1



(t − 1)2 + t



(t − 1)2 dν(w) (−∞,1] (t + 1) − wt  (t − 1)2 (t − 1)2 dν(w) + ν({1})(t − 1)2 + . = α + (α + β)t + γ 1 t (−∞,1) t + 1−w 1 − w

= α(t + 1) + βt + γ

1 By transformation s = 1−w : (−∞, 1) → (0, ∞), define a measure μ on (0, ∞) 1 −1 −1 dμ(s) = by dμ(s) = (1 − w) dν(w). Since 1+s = 1−w 2−w , we have (0,∞) (1 + s) −1 dν(w) < +∞. Hence, setting μ({0}) := γ , we find that f has an (−∞,1) (2 − w) expression in the form (2.5). So (a) ⇐⇒ (b) has been shown. Finally, the equalities in (C.3) are immediate and the uniqueness in expression (C.2) follows from that in (C.1) (or in (2.5)).

Remark C.2 The above proof indeed presents a proof of the integral expression in (2.5) of an operator convex function f on (0, ∞) based on the well-known integral expression in (C.1) of an operator convex function on (−1, 1). From γ = μ({0}) and ν({1}) = c given in the proof of Theorem C.1, one can easily find that γ = lim tf (t) = lim tg(t), t →0+

t →0+

ν({1}) = lim

t →∞

f (t) = lim (1 − t)g(t). t2 t →1−

Thus, we have the following corollary, which is used in the proof of Theorem 8.4 in Sect. 8.2. Corollary C.3 Let g be an operator convex function on (0, 1) such that lim tg(t) = lim (1 − t)g(t) = 0.

t →0+

t →1−

Then g has the integral expression  g(t) = α + βt + (−∞,1)

(2t − 1)2 dν(w), 1 − wt

t ∈ (0, 1),

where α, β ∈ R and ν is a positive measure on (−∞, 1) with w)−1 dν(w) < +∞.



(−∞,1) (2

−

174

C Operator Convex Functions on (0, 1)

Corollary C.4 Let g be a continuous real function on [0, 1]. Then the following conditions are equivalent: (i) g is operator convex on [0, 1]. (ii) There exist α, β ∈ R and an operator monotone function k on [0, ∞) such that 

 t g(t) = α + βt − (1 − t)k , 1−t

t ∈ [0, 1).

(iii) There exist α, β ∈ R and a positive measure μ on (0, ∞) satisfying −1 dμ(s) < +∞ such that (1 (0,∞) + s)  g(t) = α(1 − t) + βt − (0,∞)

t (1 − t) dμ(s), t + s(1 − t)

t ∈ [0, 1].

Proof By Theorem C.1, g is operator convex on [0, 1] if and only if there exists an operator convex f on (0, ∞) with f (0+ ), f (∞) < +∞ such that  tfunction  g(t) = (1 − t)f 1−t , t ∈ (0, 1). Hence the corollary is easily seen from [63, Proposition 8.4] and we omit the details.

Appendix D

Operator Connections of Normal Positive Functionals

Let H be an infinite-dimensional Hilbert space. An operator connection σ in the Kubo–Ando sense [85] is a binary operation (A, B) ∈ B(H)+ ×B(H)+ → Aσ B ∈ B(H)+ satisfying the following properties: (I) Joint monotonicity: A1 ≤ A2 and B1 ≤ B2 imply A1 σ B1 ≤ A2 σ B2 . (II) Transformer inequality: C(Aσ B)C ≤ (CAC)σ (CBC) for any C ∈ B(H)+ . (III) Downward continuity: An A and Bn B imply An σ Bn Aσ B, where An A means that A1 ≥ A2 ≥ · · · and An → A in the strong operator topology. It is well-known [85, Theorem 3.3] that there is a one-to-one correspondence between the operator connections σ and the operator monotone functions k ≥ 0 on [0, ∞), in such a way that k and σ are determined from each other by k(t)I = I σ (tI ),

t > 0,

Aσ B = A1/2k(A−1/2 BA−1/2 )A1/2 for every A, B ∈ B(H)+ with A invertible. In this case, write k = kσ and call it the representing function of σ . The transpose  σ of σ is the operator connection −1 corresponding to the operator monotone function k σ (t) := tkσ (t ), t > 0, the transpose of kσ . We have A σ B = Bσ A for A, B ∈ B(H)+ . The parallel sum A : B is the operator connection of A, B corresponding to the operator monotone function (1 + t −1 )−1 = t/(1 + t) on [0, ∞). The theory of operator connections has recently been extended in [57, 82, 83] to unbounded positive operators (or positive forms), including positive τ -measurable operators affiliated with a semifinite von Neumann algebra and positive elements of Haagerup’s Lp -spaces over a general von Neumann algebra.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9

175

176

D Operator Connections of Normal Positive Functionals

In this section we will discuss operator connections restricted to functionals in M∗+ via the identification with Haagerup’s L1 (M) under the correspondence ψ ∈ M∗ ↔ hψ ∈ L1 (M) (Theorem A.33 of Sect. A.6). We begin with the following definition, which is along the same lines as [111] and also makes sense in view of Lemma D.3 below. Definition D.1 For each ϕ, ψ ∈ M∗+ we have positive operators Tϕ/(ϕ+ψ) , Tψ/(ϕ+ψ) in s(ϕ + ψ)Ms(ϕ + ψ) (see Lemma A.24 and Definition 8.1). Let σ be an operator connection (in the Kubo–Ando sense). Note that Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) ∈ M+ , since u ∗ (Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) )u = (u ∗ Tϕ/(ϕ+ψ) u )σ (u ∗ Tψ/(ϕ+ψ) u ) = Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) for any unitary u ∈ M (the commutant of M). Hence we can define the connection ϕσ ψ ∈ M∗+ of ϕ, ψ by 1/2

1/2

hϕσ ψ = hϕ+ψ (Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) )hϕ+ψ . Proposition D.2 For every ϕ, ψ ∈ M∗+ , ϕ σ ψ = ψσ ϕ. Proof Since 1/2

1/2

1/2

1/2

σ Tψ/(ϕ+ψ) )hϕ+ψ hϕ σ ψ = hϕ+ψ (Tϕ/(ϕ+ψ) = hϕ+ψ (Tψ/(ϕ+ψ) σ Tϕ/(ϕ+ψ) )hϕ+ψ = hψσ ϕ , the assertion follows. As for the Kubo–Ando operator connections, the transformer inequality in the above (II) extends to C ∗ (Aσ B)C ≤ (C ∗ AC)σ (C ∗ BC) for any C ∈ B(H). Moreover, concerning the equality case of the transformer inequality, the next lemma was shown in [40, Theorem 3], whose proof is included here for the convenience of the reader. The lemma will be useful in showing the main properties of ϕσ ψ below. First recall the integral expression of the representing function kσ :  kσ (t) = α + βt + (0,∞)

t (1 + s) dμ(s), t +s

t ∈ [0, ∞),

where α, β ≥ 0 and μ is a finite positive measure on (0, ∞).

(D.1)

D Operator Connections of Normal Positive Functionals

177

Lemma D.3 Let A, B ∈ B(H)+ and C ∈ B(H) be such that s(A + B) ≤ s(CC ∗ ), where s(·) denotes the support projection. Then for any connection σ , C ∗ (Aσ B)C = (C ∗ AC)σ (C ∗ BC). Proof Recall the integral expression [85, Theorem 3.4] corresponding to (D.1):  Aσ B = αA + βB + (0,∞)

1+s {(sA) : B} dμ(s). s

(D.2)

Hence it suffices to prove that C ∗ (A : B)C = (C ∗ AC) : (C ∗ BC).

(D.3)

Assume C ∈ B(H) with s(A + B) ≤ E := s(CC ∗ ). Note that C = EC and CH = EH. Recall the well-known variational expression of A : B (see, e.g., [51, Lemma 3.1.5]): for every ξ ∈ H,

ξ, (A : B)ξ  = inf{ η, Aη + ζ, Bζ ) : η, ζ ∈ H, η + ζ = ξ }.

(D.4)

This gives

ξ, C(A : B)Cξ  = inf{ η, Aη + ζ, Bζ  : η, ζ ∈ H, η + ζ = Cξ }. Let η, ζ ∈ H be such that η + ζ = Cξ and hence Eη + Eζ = Cξ . Choose ηn ∈ H such that Cηn → Eη. Set ζn := ξ − ηn ; then Cηn + Cζn = Cξ and Cζn → Cξ − Eη = Eζ . By (D.4) we have

ηn , CACηn  + ζn , CBCζn  ≥ ξ, {(CAC) : (CBC)}ξ , whose limit as n → ∞ is

η, Aη + ζ, Bζ  ≥ ξ, {(CAC) : (CBC)}ξ  thanks to s(A + B) ≤ E. Therefore, C(A : B)C ≥ (CAC) : (CBC), and the reverse inequality is the transformer inequality. So the desired equality follows. Lemma D.4 Let ϕ, ψ, ω ∈ M∗+ be such that ϕ + ψ ≤ λω for some λ > 0. Then hϕσ ψ = hω1/2 (Tϕ/ω σ Tψ/ω )hω1/2 . 1/2

1/2

In particular, if ψ ≤ λϕ for some λ > 0, then hϕσ ψ = hϕ kσ (Tψ/ϕ )hϕ , where kσ is the representing function of σ .

178

D Operator Connections of Normal Positive Functionals

Proof By Lemma A.24 there are A, B ∈ s(ϕ + ψ)Ms(ϕ + ψ) and C ∈ s(ω)Ms(ω) such that 1/2

1/2

hϕ1/2 = Ahϕ+ψ , 1/2

Since hϕ

1/2

1/2

hψ = Bhϕ+ψ ,

1/2

1/2

hϕ+ψ = Chω1/2 .

1/2

= AChω and hψ = BChϕ+ψ , we have

Tϕ/ω = (AC)∗ (AC) = C ∗ Tϕ/(ϕ+ψ) C,

Tψ/ω = (BC)∗ (BC) = C ∗ Tψ/(ϕ+ψ) C.

Moreover, note that s(Tϕ/(ϕ+ψ) +Tψ/(ϕ+ψ) ) ≤ s(ϕ +ψ) = s(CC ∗ ) by Lemma A.24 (in fact, Tϕ/(ϕ+ψ) + Tψ/(ϕ+ψ) = s(ϕ + ψ) holds). Therefore, by Lemma D.3, hω1/2 (Tϕ/ω σ Tψ/ω )hω1/2 = hω1/2 ((C ∗ Tϕ/(ϕ+ψ) C)σ (C ∗ Tψ/(ϕ+ψ) C))hω1/2 = hω1/2 C ∗ (Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) )Chω1/2 1/2

1/2

= hϕ+ψ (Tϕ/(ϕ+ψ) σ Tψ/(ϕ+ψ) )hϕ+ψ = hϕσ ψ , showing the first result. The latter result follows from the first by taking ω = ϕ. Example D.5 Consider the case when M = B(H) on an arbitrary Hilbert space H, which is standardly represented by the left multiplication on the Hilbert–Schmidt class C2 (H), a Hilbert space with the Hilbert–Schmidt inner product. The traceclass C1 (H) is identified with B(H)∗ as usual. Let A, B ∈ C1 (H)+ , corresponding to ϕ = Tr(A ·), ψ = Tr(B ·) ∈ B(H)+ ∗ . We have ϕσ ψ in Definition D.1 as well as Aσ B in the Kubo–Ando sense [85]. We write A = (A + B)1/2 TA/(A+B) (A + B)1/2 and B = (A + B)1/2 TB/(A+B) (A + B)1/2 with positive operators TA/(A+B) , TB/(A+B) in s(A+B)B(H)s(A+B). Note that these are indeed Tϕ/(ϕ+ψ) , Tψ/(ϕ+ψ) , respectively. By Lemma D.3, Aσ B = (A + B)1/2 (TA/(A+B) σ TB/(A+B) )(A + B)1/2 , which means that ϕσ ψ corresponds to Aσ B, or ϕσ ψ coincides with Aσ B. The next theorem summarizes basic properties of the connection ϕσ ψ. Theorem D.6 Let ϕ, ψ, ϕi , ψi ∈ M∗+ . (i) Joint monotonicity: If ϕ1 ≤ ϕ2 and ψ1 ≤ ψ2 , then ϕ1 σ ψ1 ≤ ϕ2 σ ψ2 . (ii) Transformer inequality: For every a ∈ M, a ∗ (ϕσ ψ)a ≤ (a ∗ ϕa)σ (a ∗ψa),

(D.5)

where (a ∗ ϕa)(x) := ϕ(axa ∗), x ∈ M. Moreover, if s(ϕ + ψ) ≤ s(aa ∗ ) (in particular, if a is invertible), then equality holds in inequality (D.5). (iii) Downward continuity: If ϕn ϕ and ψn ψ in M∗+ , then ϕn σ ψn ϕσ ψ, where ϕn ϕ means that ϕ1 ≥ ϕ2 ≥ · · · and ϕn − ϕ → 0. In particular,

D Operator Connections of Normal Positive Functionals

179

for any ω ∈ M∗+ , ϕσ ψ = lim (ϕ + εω)σ (ψ + εω) ε 0

in the norm (decreasingly).

(D.6)

(iv) Joint concavity: (ϕ1 + ϕ2 )σ (ψ1 + ψ2 ) ≥ (ϕ1 σ ψ1 ) + (ϕ2 σ ψ2 ). Proof (i) Let ω := ϕ2 + ψ2 . It is easy to see that Tϕ1 /ω ≤ Tϕ2 /ω and Tψ1 /ω ≤ Tψ2 /ω . Hence by Lemma D.4 we have hϕ1 σ ψ1 = hω1/2 (Tϕ1 /ω σ Tψ1 /ω )hω1/2 ≤ hω1/2 (Tϕ2 /ω σ Tψ2 /ω )hω1/2 = hϕ2 σ ψ2 . (ii) This will be proved below after we give the integral expression of ϕσ ψ. (iii) The joint monotonicity in (i) says that ϕn σ ψn is monotone decreasing as n → ∞. Hence a positive linear functional φ on M is defined by φ(x) := limn→∞ (ϕn σ ψn )(x), x ∈ M. Since ϕn σ ψn ≥ φ, it is clear that φ is normal, so φ ∈ M∗+ and ϕn σ ψn − φ = (ϕn σ ψn − φ)(1) → 0 as n → ∞. To prove that φ = ϕσ ψ, let ω := ϕ1 + ψ1 . It follows that Tϕn /ω T1 and Tψn /ω T2 for some T1 , T2 ∈ M+ . For every x ∈ M+ one has ϕn (x) = tr(hω1/2 xhω1/2 )Tϕn /ω tr(hω1/2 xhω1/2)T1 1/2

1/2

1/2

1/2

as n → ∞, so that ϕ(x) = tr xhω T1 hω , showing hϕ = hω T1 hω . This implies that T1 = Tϕ/ω and similarly T2 = Tψ/ω . Hence by Lemma D.4 one has (ϕn σ ψn )(x) = tr(hω1/2 xhω1/2)(Tϕn /ω σ Tψn /ω )

tr(hω1/2 xhω1/2)(Tϕ/ω σ Tψ/ω ) = (ϕσ ψ)(x) as n → ∞. Therefore, φ = ϕσ ψ. (iv) With ω := ϕ1 + ϕ2 + ψ1 + ψ2 , it is easy to see that T(ϕ1 +ϕ2 )/ω = Tϕ1 /ω + Tϕ2 /ω and T(ψ1 +ψ2 )/ω = Tψ1 /ω + Tψ2 /ω . Hence by Lemma D.4 one has h(ϕ1 +ϕ2 )σ (ψ1 +ψ2 ) = hω1/2 ((Tϕ1 /ω + Tϕ2 /ω )σ (Tψ1 /ω + Tψ2 /ω ))hω1/2 ≥ hω1/2 ((Tϕ1 /ω σ Tψ1 /ω ) + (Tϕ2 /ω σ Tψ2 /ω ))hω1/2 = hϕ1 σ ψ1 + hϕ2 σ ψ2 = hϕ1 σ ψ1 +ϕ2 σ ψ2 , where the inequality above is due to the joint concavity in [85, Theorem 3.5] (also [51, Corollary 3.2.3]).

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D Operator Connections of Normal Positive Functionals

Corresponding to (D.1) we have the integral expression of ϕσ ψ as follows: Theorem D.7 For every ϕ, ψ ∈ M∗+ ,  (ϕσ ψ)(x) = αϕ(x) + βψ(x) + (0,∞)

1+s ((sϕ) : ψ)(x) dμ(s), s

x ∈ M.

Proof Let ω := ϕ + ψ. By applying (D.2) to Tϕ/ω and Tψ/ω we have (ϕσ ψ)(x) = tr(hω1/2 xhω1/2 )(Tϕ/ω σ Tψ/ω ) = αtr(hω1/2 xhω1/2 )Tϕ/ω + βtr(hω1/2 xhω1/2)Tψ/ω  1+s tr(hω1/2 xhω1/2 )((sTϕ/ω ) : Tψ/ω ) dμ(s) + s (0,∞)  1+s ((sϕ) : ψ)(x) dμ(s), = αϕ(x) + βψ(x) + s (0,∞) as asserted. The following variational expression is the variant of (D.4) for the parallel sum ϕ : ψ. This expression and the integral expression in Theorem D.7 together may serve as a definition of ϕσ ψ. Theorem D.8 For every ϕ, ψ ∈ M∗+ and any x ∈ M, (ϕ : ψ)(x ∗ x) = inf{ϕ(y ∗ y) + ψ(z∗ z) : y, z ∈ M, y + z = x}.

(D.7)

Proof Let ω := ϕ + ψ. Note that inf {ϕ(y ∗ y) + ψ(z∗ z)} = inf inf {(ϕ + εω)(y ∗ y) + (ψ + εω)(z∗ z)}

y+z=x

y+z=x ε>0

= inf inf {(ϕ + εω)(y ∗ y) + (ψ + εω)(z∗ z)}. ε>0 y+z=x

From this and (D.6) it suffices to prove (D.7) in the case ϕ ∼ ψ. Let e := s(ϕ) = 1/2 1/2 1/2 s(ω) and choose an A ∈ eMe such that hϕ = Ahω = hω A∗ . Then A is ∗ invertible in eMe and Tϕ/ω = A A. Since Tϕ/ω + Tψ/ω = e, we find that hϕ:ψ = hω1/2 (Tϕ/ω : Tψ/ω )hω1/2 = hω1/2 Tϕ/ω (Tϕ/ω + Tψ/ω )−1 Tψ/ω hω1/2 = hω1/2 A∗ A(e − A∗ A)hω1/2 = hω1/2 A∗ (e − AA∗ )Ahω1/2 = hϕ1/2 (e − AA∗ )hϕ1/2 .

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181

When y, z ∈ M and y + z = x, one has ϕ(y ∗ y) + ψ(z∗ z) − (ϕ : ψ)(x ∗ x) = tr(x − z)∗ (x − z)hϕ + tr z∗ zhψ − tr x ∗ x(hϕ − hϕ1/2 AA∗ hϕ1/2 ) = tr z∗ zhϕ + tr z∗ zhψ − tr x ∗ zhϕ − tr z∗ xhϕ + tr x ∗ xhϕ1/2 AA∗ hϕ1/2 = tr z∗ zhω + tr A∗ hϕ1/2 x ∗ xhϕ1/2A − 2Re tr x ∗ zhϕ = zhω1/2 22 + xhϕ1/2 A22 − 2Re tr A∗ hϕ1/2 x ∗ zhϕ1/2 A∗−1 ! = zhω1/2 22 + xhϕ1/2 A22 − 2Re xhϕ1/2A, zhω1/2 thanks to hω = hϕ A∗−1 . Therefore, the Schwarz inequality gives 1/2

1/2

ϕ(y ∗ y) + ψ(z∗ z) − (ϕ : ψ)(x ∗ x) ≥ 0. 1/2

1/2

Moreover, since Mhω = L2 (M)e and xhϕ A ∈ L2 (M)e, one can choose a 1/2 1/2 sequence {zn } ⊂ M such that zn hω − xhϕ A → 0. Letting yn := x − zn one has ϕ(yn∗ yn ) + ψ(zn∗ zn ) − (ϕ : ψ)(x ∗ x) = zn hω1/2 22 + xhϕ1/2 A22 − 2Re xhϕ1/2 A, zn hω1/2  −→ 0, so that (D.7) follows. Now we are in a position to prove Theorem D.6 (ii). Proof (Theorem D.6 (ii)) For the transformer inequality, by Theorem D.7 it suffices to prove (D.5) for the parallel sum ϕ : ψ. For every x, a ∈ M, using (D.7) twice we have (a ∗ (ϕ : ψ)a)(x ∗ x) = (ϕ : ψ)((xa ∗ )∗ (xa ∗ )) = inf{ϕ(y ∗ y) + ψ(z∗ z) : y, z ∈ M, y + z = xa ∗ } ≤ inf{ϕ((ya ∗)∗ (ya ∗ )) + ψ((za ∗ )∗ (za ∗ ) : y, z ∈ M, y + z = x} = inf{(a ∗ ϕa)(y ∗y) + (a ∗ ψa)(z∗ z) : y, z ∈ M, y + z = x} = ((a ∗ ϕa) : (a ∗ ψa))(x ∗ x), showing inequality (D.5). Next, assume that s(ϕ + ψ) ≤ s(aa ∗), and prove that the equality holds in (D.5). Let ω := ϕ + ψ, h := hω , and a ∗ h1/2 = vk 1/2 be the polar decomposition with k 1/2 = |a ∗ h1/2 |, so that k = h1/2 aa ∗h1/2 ∈ L1 (M)+ and s(k) = s(h) since s(h) = 1/2 T 1/2  ∈ M∗+ be such that h s(ω) ≤ s(aa ∗ ). Further, let  ω,  ϕ, ψ ω = k, h ϕ =k ϕ/ω k

182

D Operator Connections of Normal Positive Functionals

and hψ = k 1/2Tψ/ω k 1/2 . Since h1/2 (Tϕ/ω + Tψ/ω )h1/2 = hϕ + hψ = h, it follows 1/2 s(h)k 1/2 = k, so that that Tϕ/ω + Tψ/ω = s(h) and h  =k ϕ + hψ ,  ω= ϕ+ψ

T ϕ / ω = Tϕ/ω ,

Tψ/ ω = Tψ/ω .

One has ∗ 1/2 1/2 ∗ T v = a ∗ h1/2 Tϕ/ω h1/2 a = a ∗ hϕ a vh ϕ v = vk ϕ / ωk

and similarly vhψ v ∗ = a ∗ hψ a. Moreover, one has ∗ 1/2 1/2 ∗ vh (T v  v = vk / ϕ / ω σ Tψ ϕσ ψ ω )k

= a ∗ h1/2 (Tϕ/ω σ Tψ/ω )h1/2 a = a ∗ hϕσ ψ a. From these with Theorem A.33, ϕ v∗ , a ∗ ϕa = v

v ∗ , a ∗ ψa = v ψ

)v ∗ . a ∗ (ϕσ ψ)a = v( ϕσ ψ

Hence it suffices to show that  ∗ = (v v ∗ ). v( ϕ σ ψ)v ϕ v ∗ )σ (v ψ

(D.8)

1/2 ∗ 1/2 = vk 1/2 v ∗ , and it is v ∗ = v ω v ∗ and hv Note that v ϕ v∗ + vψ ωv ∗ = (vkv ) immediate to see that ∗ Tv ϕ v ∗ /v ωv ∗ = vT ϕ / ωv ,

∗ Tv ψv ∗ /v / ωv . ωv ∗ = vTψ

Therefore, we find that 1/2

1/2

h(v v ∗ ) = hv v ∗ /v ϕ v ∗ /v ω v ∗ σ Tv ψ ϕ v ∗ )σ (v ψ ωv ∗ )hv ωv ∗ (Tv ωv ∗ ∗ ∗ 1/2 ∗ = vk 1/2 v ∗ ((vT v / ϕ / ω v )σ (vTψ ω v ))vk ∗ 1/2 ∗ = vk 1/2 v ∗ v(T v / ϕ / ω σ Tψ ω )v vk 1/2 ∗ ∗ = vk 1/2 (T v = vh / v , ϕ / ω σ Tψ ω )k ϕσ ψ

where the second equality above follows from Lemma D.3. Thus, (D.8) has been shown. Let H and K be Hilbert spaces. For any operator connection σ and any positive linear map α : B(H) → B(K), the following inequality is well-known: α(Aσ B) ≤ α(A)σ α(B),

A, B ∈ B(H),

(D.9)

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183

which is due to Ando [4] though stated only for the geometric mean and the parallel sum. Inequality (D.9) for general σ can be seen, for instance, by using the integral expression in (D.2), which is considered as an extension of the transformer inequality. Proposition D.9 Let γ : N → M be a normal positive linear map between von Neumann algebras. Then for every ϕ, ψ ∈ M∗+ , (ϕσ ψ) ◦ γ ≤ (ϕ ◦ γ )σ (ψ ◦ γ ). Proof Although γ is not necessarily unital here, with ω := ϕ + ψ one can define a normal positive map γω∗ : M → e0 Me0 (where e0 := s(ω ◦ γ )) as in Lemma 8.3 (also Proposition 6.6). Then the result can be shown as in Lemma 8.6. Indeed, we have h(ϕσ ψ)◦γ = γ∗ (hϕσ ψ ) = γ∗ (hω1/2 (Tϕ/ω σ Tψ/ω )hω1/2 ) 1/2 ∗ 1/2 = hω◦γ γω (Tϕ/ω σ Tψ/ω )hω◦γ 1/2 1/2 ≤ hω◦γ (γω∗ (Tϕ/ω )σ γω∗ (Tψ/ω ))hω◦γ 1/2 1/2 = hω◦γ (Tϕ◦γ /ω◦γ σ Tψ◦γ /ω◦γ )hω◦γ

= h(ϕ◦γ )σ (ψ◦γ ). In the above, the third and the fourth equalities are due to Lemmas 8.3 and 8.5, respectively, which hold without the unitality assumption of γ , and the inequality above follows from (D.9). The last result is the relation between the connection ϕσ ψ and the maximal f divergence, which is useful in Sect. 8.2. Proposition D.10 For every operator connection σ and every ϕ, ψ ∈ M∗+ ,  S−kσ (ψϕ) = −(ϕσ ψ)(1). Proof When ϕ ∼ ψ, by Definition 4.1 and Lemma D.4,    S−kσ (ψϕ) = tr hϕ1/2 (−kσ )(Tψ/ϕ )hϕ1/2 = −(ϕσ ψ)(1). For general ϕ, ψ ∈ M∗+ , by Definition 4.3 and (D.6), with ω := ϕ + ψ we have  S−kσ (ψϕ) = lim  S−kσ (ψ + εωϕ + εω) ε 0

= − lim ((ϕ + εω)σ (ψ + εω))(1) = −(ϕσ ψ)(1), ε 0

as asserted.

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Index

Symbols α-Rényi divergence, 3, 19 α-sandwiched Rényi divergence, 26–28, 33 α-z-Rényi relative entropy, 27 C ∗ -algebra, 1 C ∗ -algebra, 132 f -divergence classical, 2, 9, 44, 51, 54 maximal, 2, 41, 42, 55, 121 measured, 2, 51, 65, 115 minimal, 55 projectively measured, 2, 51, 56 quantum, 2, 3, 55 standard, 2, 8, 55, 88, 97 Lp -norm, 148, 151 Lp -space Haagerup’s, 22, 150 Kosaki’s, 27, 159 left, 159 right, 159 symmetric, 159 tracial, 148 M -support, 141 M-support, 141 n-positive map, 79 σ -weak topology, 132 ∗-representation, 132 τ -measurable operator, 146 tr-functional, 119, 150 2-positive map, 3, 95 W ∗ -algebra, 132

A Affiliated with, 146

Approximate reversibility, 4, 113, 115 Araki and Masuda’s Lp -norm, 27 Araki–Lieb–Thirring (ALT) inequality, 36 Araki–Masuda divergence, 28

B Balanced functional, 154 Belavkin and Staszewski’s relative entropy, 2, 49 Borel functional calculus, 163

C Canonical trace, 149 Center, 133 Centralizer, 136 Chain rule, 157 Classical f -divergence, 2, 9, 44, 51, 54 Cocycle identity, 156 Commutant, 132 Commute, 49, 158 Completely positive map, 2, 79 Complex interpolation method, 159 Complex interpolation space, 159 Conditional expectation, 137 generalized, 82 Connection ϕσ ψ, 120, 176 Connes’ cocycle derivative, 5, 75, 76, 88, 154 Connes’ inverse theorem, 157 Converge in the strong resolvent sense, 167 Converge strongly, 131 Converge strongly*, 131 Converge weakly, 131 Converge σ -weakly, 132

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 F. Hiai, Quantum f-Divergences in von Neumann Algebras, Mathematical Physics Studies, https://doi.org/10.1007/978-981-33-4199-9

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192 CP map, 79 Crossed product, 149 Cyclic vector, 134

D Data-processing inequality (DPI), 2 Double commutation theorem, 132 Downward continuity connection ϕσ ψ, 178 operator connection, 175 Dual action, 149 Dual weight, 149

E Exact interpolation space, 170

F Factor, 133 Fidelity, 34, 76 Flow of weights, 134, 149 Form sum, 142, 168 Furuta’s inequality, 159

G Generalized conditional expectation, 82 Generalized Powers–Størmer inequality, 26, 35, 39 Generalized s-number, 147 Geometric interpolation space, 170 GNS construction, 1 GNS cyclic representation, 134 Golden–Thompson inequality, 63

H Haagerup’s Lp -space, 22, 150 Hilbert–Schmidt inner product, 138 Hilbert–Schmidt operator, 138 Hölder’s inequality, 26, 148, 151

J Joint concavity connection ϕσ ψ, 179 Joint convexity maximal f -divergence, 42 standard f -divergence, 14 Joint lower semicontinuity maximal f divergence, 46 measured f -divergence, 54

Index Rényi divergence, 20 sandwiched Rényi divergence, 34 standard f -divergence, 14 Joint monotonicity connection ϕσ ψ, 178 operator connection, 175 K KMS (Kubo–Martin–Schwinger) condition, 153 Kosaki’s Lp -space, 27, 159 Krieger factor, 133 Kubo–Martin–Schwinger (KMS) condition, 136 Kullback–Leibler divergence, 2 L Localizable measure space, 147 M Martingale convergence maximal f -divergence, 46 standard f -divergence, 15 Maximal f -divergence, 2, 41, 42, 55, 121 Max-relative entropy, 34 Measured f -divergence, 2, 51, 65, 115 Measured relative entropy, 62 Measured α-Rényi divergence, 62 Measurement, 2, 51, 101 projective, 2, 51 Measure topology, 147 Minimal f -divergence, 55 Minimal reverse test, 48 Modular automorphism group, 135 Modular conjugation, 135 Modular operator, 135 Monotone quantum f -divergence, 55 Monotonicity, 2 maximal f -divergence, 42 measured f -divergence, 54 Rényi divergence, 21 sandwiched Rényi divergence, 34 standard f -divergence, 14 Multiplicative domain, 84 Murray–von Neumann equivalence, 133 N Natural positive cone, 7, 137 Normal functional, 132 Normal map, 14, 79 Norm one projection, 133, 137

Index O Operator algebra, 131 Operator concave function, 10 Operator connection, 6, 119, 175, 176 Operator convex function, 10 Operator monotone function, 10 Operator perspective, 2, 42, 119 Optimal measurement, 65 Optimal reverse test, 48 Order in the form sense, 166

P Pair of compatible Banach spaces, 159 Parallel sum, 175 Peierls–Bogolieubov inequality, 14 Perspective Pφ (ψ, ϕ), 120 Perturbed functional, 63 Petz’ recovery map, 4, 80 Positive quadratic form, 165 Positive self-adjoint operator, 163 Predual, 132 Projection, 133 abelian, 133 central, 132 finite, 133 Projectively measured f -divergence, 2, 51, 56 Projective measurement, 2, 51

Q Quantum channel, 2, 79 Quantum-classical channel, 2, 54, 114 Quantum f -divergence, 2, 3 monotone, 55 Quantum information, 1 Quantum operation, 3, 79 Quasi-entropy, 2, 8

R Radon–Nikodym derivative, 148, 151, 156 Radon–Nikodym theorem linear, 74 quadratic, 76, 158 Relative entropy, 2, 4, 10, 13, 20, 34 Belavkin and Staszewski’s, 49 max-, 34 measured, 62 Rényi, 3, 19 Umegaki’s, 10 Relative modular operator, 2, 7, 141 Rényi divergence, 3, 19 measured, 62

193 sandwiched, 3, 26–28, 33, 100 Rényi relative entropy, 3, 19 Representing function operator connection, 175 Reverse test, 2, 48 Reversibility, 3, 95

S Sandwiched Rényi divergence, 3, 26–28, 33, 100 Schwarz map, 2, 14, 21, 79, 88 Separating vector, 134 Standard f -divergence, 2, 8, 55, 88, 97 Standard form, 7, 138, 152 Stone’s representation theorem, 164 Strong operator topology, 131 Strong* operator topology, 131 Sufficiency, 3, 95

T Tomita’s fundamental theorem, 135 Tomita–Takesaki theory, 1, 134 TPCP map, 79 Trace-class operator, 131, 145 Trace scaling property, 149 Transformer inequality connection ϕσ ψ, 178 operator connection, 175 Transition probability, 4, 34, 95 Transpose, 10 operator connection, 175

U Umegaki’s relative entropy, 10 Unital map, 79

V Vector representative, 139 Von Neumann algebra, 1, 132 σ -finite, 134 abelian, 9, 43, 147 approximately finite dimensional (AFD), 46, 133 atomic, 133 discrete, 133 finite, 133, 135 hyperfinite, 133 injective, 133 of type III, 133 properly infinite, 133

194 purely infinite, 133 reduced, 15, 17, 47, 138 semifinite, 133, 149 Von Neumann’s density theorem, 132

Index W Weak operator topology, 131