Quantum Information Processing [draft ed.]

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Introduction to Quantum Information Processing — Draft W. L¨ ucke SS 2005

Institute for Physics and Physical Technologies Clausthal University Of Technology Leibnizstraße 4 D-38678 Clausthal–Zellerfeld

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Preface Quantum information processing is one of the most fascinating and active fields of contemporary physics. Its central topic is the coherent control of quantum states in order to perform tasks — like quantum teleportation, absolutely secure data transmission and efficient factorization of large integers — that do not seem possible by means of classical systems alone. The vast possibilities of physical implementations are currently being extensively studied and evaluated. Various proof-of-principle experiments have already been performed. However, in the present note only some possiblities can be indicated. Main emphasis will be on quantum optical methods, indispensable for transmission of quantum information. For more complete information on achivements and latest proposals concerning quantum information processing the Los Alamos preprint server http://xxx.lanl.gov/archive/quant-ph is highly recommended.

Recommended Literature: (Alber et al., 2001; Bowmeester et al., 2000; Ekert et al., 2000; Nielsen and Chuang, 2001; Preskill, 01; Shannon, 1949; Bertlmann and Zeilinger, 2002; Audretsch, 2002; Bruß, 2003)

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Contents I

Idealized Quantum Gates and Algorithms

1 Basics of Quantum Computation 1.1 Classical Logic Circuits . . . . . . . 1.2 Quantum Computational Networks 1.2.1 Quantum Gates . . . . . . . 1.2.2 Quantum Teleportation . . 1.2.3 Universality . . . . . . . . .

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2 Quantum Algorithms 2.1 Quantum Data Base Search . . . . . . . 2.1.1 Grover’s Algorithm . . . . . . . . 2.1.2 Network for Grover’s Algorithm . 2.1.3 Details and Generalization . . . . 2.2 Factoring Large Integers . . . . . . . . . 2.2.1 Basics . . . . . . . . . . . . . . . 2.2.2 The Quantum Fourier Transform 2.2.3 Quantum Order Finding . . . . .

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3 Physical Realizations of Quantum Gates 3.1 Quantum Optical Implementation . . . . . . . . . . . 3.1.1 Photons . . . . . . . . . . . . . . . . . . . . . 3.1.2 Photonic n-Qubit Systems . . . . . . . . . . . 3.1.3 Nonlinear Optics Quantum Gates . . . . . . . 3.1.4 Linear Optics Quantum Gates . . . . . . . . . 3.2 Measurement-Based Quantum Computation . . . . . 3.3 Cold Trapped Ions . . . . . . . . . . . . . . . . . . . 3.3.1 General Considerations . . . . . . . . . . . . . 3.3.2 Linear Paul Trap . . . . . . . . . . . . . . . . 3.3.3 Implementing Quantum Gates by Laser Pulses 3.3.4 Laser Cooling . . . . . . . . . . . . . . . . . .

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II

CONTENTS

Fault Tolerant Quantum Information Processing

4 General Aspects of Quantum Information 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Open Quantum Systems and Quantum Operations 4.2.2 Quantum Noise and Error Correction . . . . . . . . 4.3 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . 4.3.1 General Apects . . . . . . . . . . . . . . . . . . . . 4.3.2 Classical Codes . . . . . . . . . . . . . . . . . . . . 4.3.3 Quantum Codes . . . . . . . . . . . . . . . . . . . . 4.3.4 Reliable Quantum Computation . . . . . . . . . . . 4.4 Entanglement Assisted Channels . . . . . . . . . . . . . . 4.4.1 Quantum Dense Coding . . . . . . . . . . . . . . . 4.4.2 Quantum Teleportation . . . . . . . . . . . . . . . 4.4.3 Entanglement Swapping . . . . . . . . . . . . . . . 4.4.4 Quantum Cryptography . . . . . . . . . . . . . . . 5 Quantifying Quantum Information 5.1 Shannon Theory for Pedestrians . . . . 5.2 Adaption to Quantum Communication 5.2.1 Von Neumann Entropy . . . . . 5.2.2 Accessible Information . . . . . 5.2.3 Distance Measures for Quantum 5.2.4 Schumacher Encoding . . . . . 5.2.5 A la Nielsen/Chuang . . . . . . 5.2.6 Entropy . . . . . . . . . . . . .

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127 . 127 . 131 . 131 . 135 . 137 . 144 . 145 . 145

6 Handling Entanglement 6.1 Detecting Entanglement . . . . . . . . . . . . . 6.1.1 Entanglement Witnesses . . . . . . . . . 6.1.2 Examples . . . . . . . . . . . . . . . . . 6.1.3 Other Criteria . . . . . . . . . . . . . . . 6.2 Local Operations and Classical Communication 6.2.1 General Aspects . . . . . . . . . . . . . . 6.2.2 Entanglement Dilution . . . . . . . . . . 6.2.3 Entanglement Distillation . . . . . . . . 6.3 Quantification of Entanglement . . . . . . . . .

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165 165 166 167 171 171

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A A.1 A.2 A.3 A.4

Turing’s Halting Problem . . . . . . . . . . . . Some Remarks on Quantum Teleportation . . . Quantum Phase Estimation and Order Finding Finite-Dimensional Quantum Kinematics . . . . A.4.1 General Description . . . . . . . . . . . .

CONTENTS

7

A.4.2 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 A.4.3 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . 176 Bibliography

183

Index

199

8

CONTENTS

Part I Idealized Quantum Gates and Algorithms

9

Chapter 1 Basics of Quantum Computation 1.1

Classical Logic Circuits

The smallest entity of classical information theory (Shannon, 1949) is the bit (binary digit), i.e. the decision on a classical binary alternative. Usually bits are identified with the numbers 0 (for wrong) or 1 (for true) and typically correspond to the position of some simple switch. Every definite statement may be encoded into a sufficiently long but finite sequence (b1 , . . . , bn ) of bits.1 In this sense the essence of a calculations may be described as the transformations of a finite sequence of input bits (encoding the task) into a finite sequence of output bits (encoding the result). This suggests the following model for actual calculators: 1. An input register (array of switches) will be put into a state corresponding to the n1 -tuple (b1 , . . . , bn1 ) ∈ {0, 1}n1 encoding the task. 2. A computational circuit, the elementary components of which are called gates,2 transforms (b1 , . . . , bn1 ) into an n2 -tuple (b′1 , . . . , b′n2 ) of bits encoding the result to be stored into an output register. From the mathematical point of view it is only important which element of Fn1 ,n2 , denoting the set of all mappings from {0, 1}n1 into {0, 1}n2 , is implemented by the circuit. Therefore, computational circuits implementing the same mapping are called equivalent. Every element of Fn1 ,n2 can be implemented by some assembly of gates listed in Table 1.1: Lemma 1.1.1 For arbitrary positive integer n1 , n2 all elements of Fn1 ,n2 can be represented as compositions of tensor products of functions from Tabular 1.1. Proof: See below. DRAFT, October 17, 2007

1

An important consequence of this fact is the halting problem (see Appendix A.1). 2 For simple hardware implementations see (P¨ utz, 1971, pp. 244–252).

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12

CHAPTER 1. BASICS OF QUANTUM COMPUTATION Name

Symbol

Class

ID FANOUT

r

NOT

❍ ❍❜ ✟ ✟

Action

F1,1

b

7→

b

F1,2

b

7→

(b, b)

F1,1

b

7→

1−b

AND

&

F2,1

(b1 , b2 ) 7→

b1 b2

OR

≥1

F2,1

(b1 , b2 ) 7→

b1 + b2 − b1 b2

Table 1.1: Elementary gates

Thus every classical logic circuit corresponds to a graph consisting of symbols from Tabular 1.1. For instance, the graph

✉

&

....... .......... .. . .......... .......

≥1

& corresponds to the mapping def

SWITCH = OR ◦ (AND ⊗ AND) ◦ (IDNOT ⊗ ⊗ID ⊗ ID) ◦ (ID ⊗ FANOUT ⊗ ID) , acting as (b0 , s, b1 ) 7−→ Another example is the graph



b0 b1

✉

if s = 0 , if s = 1 .

& &

....... .......... .. . .......... .......

✉

✉

&

≥1

corresponding to def

XOR = OR  ◦ (AND ⊗ AND) ◦ (ID ⊗  FANOUT ⊗ ID) ◦ ID ⊗ (NOT ◦ AND) ⊗ ID ◦ (FANOUT ⊗ FANOUT)

and acting as

def

(b1 , b2 ) 7−→ b1 ⊕ b2 =



NOT(b2 ) if b1 = 1 b2 if b1 = 0

= b1 + b2 − 2b1 b2 = b1 + b2 mod 2 .

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1.1. CLASSICAL LOGIC CIRCUITS

Name

CNOT

TCNOT

SWAP

Symbol Class s

F2,2

(b1 , b2 )

7→

(b1 , b1 ⊕ b2 )

s

F2,2

(b1 , b2 )

7→

(b1 ⊕ b2 , b2 )

❭✜ ✜❭

F2,2

(b1 , b2 )

7→

(b2 , b1 )

❤

❤

s

CSWAP3

Action

❭✜ ✜❭

F3,3

(0, b1 , b2 )

7→

(0, b1 , b2 )

(1, b1 , b2 )

7→

(1, b2 , b1 )

(b1 , b2 , b3 )

7→

s

CCNOT4

s ❤

F3,3

Table 1.2: Some reversible gates

(b1 , b2 , b1 b2 ⊕ b3 )

14

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Of course, also for the gates listed in Table 1.2 there are equivalent networks, e.g.:5 CNOT = (ID ⊗ XOR) ◦ (FANOUT ⊗ ID) , ❤ s

≡

❭✜ ✜❭

≡

❭✜ ✜❭

s

❤

❭✜ ✜❭

s

❤

s

❤

s

❤

s

(1.1)

,

(1.2)

.

(1.3)

s

≡

❭✜ ✜❭

❤

s

❤

s

❤

s

(1.4)

Now we are prepared for the Proof of Lemma 1.1.1: Thanks to FANOUT and SWAP it is sufficient to proof the lemma for decision functions, i.e. for n2 = 1 . Obviously, then, the statement of the lemma holds for n1 = 1 , since the four elements of F1,1 are ID, def

TRUE = OR ◦ (ID ⊗ NOT) ◦ FANOUT , and their compositions with NOT (applied last). Now, assume that the statement of the lemma has already been proved for n1 = n and consider an arbitrary f ∈ Fn+1,1 . Then both f0 and f1 , where def

fs (b1 , . . . , bn ) = f (b1 , . . . , bn , s) , can be represented as compositions of tensor products of functions from Tabular 1.1. There is a composition of FANOUTs and SWAPs acting as (b1 , . . . , bn , s) 7−→ (b1 , . . . , bn , s, b1 , . . . , bn ). Composing this with SWITCH ◦ (f0 ⊗ ID ⊗ f1 ) (to be applied last) gives f . This proves the statement of the lemma for n1 = n + 1 .

According to Lemma 1.1.1 we may perform arbitrarily complex computations by composing simple hardware components of very small variety. Of course, given f ∈ Fn1 ,n2 , there are infinitely many representations of f as composition of tensor products of elementary components. Therefore, the interesting problem arises how to simplify a given gate (logic circuit) without changing its action.6 DRAFT, October 17, 2007

3

The CSWAP gate is also called Fredkin gate. 4 The CCNOT gate is also called Toffoli gate. 5 See (Tucci, 2004) for more equivalences of classical and/or quantum networks. 6 See (Lindner et al., 1999, Sect. 8.2.3) for n1 ≤ 6 , n2 = 1 and (Lee et al., 1999; Shende et al., 2003) for quantum gates.

1.1. CLASSICAL LOGIC CIRCUITS

15

From the technological point of view it is also of interest that def

NAND = NOT ◦ AND is universal in the sense that it can replace NOT, AND, and OR as elementary gates:7 NOT = NAND ◦ FANOUT , AND = NOT ◦ NAND , OR = NAND ◦ (NOT ⊗ NOT) . In the same sense

def

NOR = NOT ◦ OR is universal:

NOT = NOR ◦ FANOUT , AND = NOR ◦ (NOT ⊗ NOT) , OR = NOT ◦ NOR .

Alternatively, in order to minimize dissipation of energy (Landauer, 1961; Landauer, 1998; Plenio and Vitelli, 2001; Bub, 2001; Parker and Walker, 2003), one may execute all calculations using only reversible networks8 (Toffoli, 1980a): Since9 CCNOT3 (b1 , b2 , 1) = NAND(b1 , b2 ) ∀ b1 , b2 ∈ {0, 1} and



CCNOT1 (b, 1, 0) CCNOT3 (b, 1, 0)



= FANOUT(b) ∀ b ∈ {0, 1} ,

the CCNOT gate is universal for reversible classical computation in the following sense: For every mapping φ ∈ Fn1 ,n2 there is a reversible n-bit network composed of only CCNOT gates,10 SWAP gates, and ID gates (wires) implementing a mapping f ∈ Fn,n (n ≥ n1 , n2 ) fulfilling 



f1 (b1 , . . . , bn1 , cn1 +1 , . . . , cn )   ..  = φ(b1 , . . . , bn1 ) ∀ b1 , . . . , bn1 ∈ {0, 1}  . fn2 (b1 , . . . , bn1 , cn1 +1 , . . . , cn ) for suitably chosen constant bits cn1 +1 , . . . , cn . DRAFT, October 17, 2007

7

I.e., every classical logic circuit corresponds to a composition of tensor products of IDs, FANOUTs, and NANDs. 8 Reversible classical networks (logic circuits) are those corresponding to bijections f ∈ Fn,n for some n ∈ IN . 9 Toffoli called his gate the AND/NAND gate to indicate that also CCNOT3 (j, k, 0) = AND(j, k) holds. Correspondingly, he called CNOT the XOR/FANOUT gate. 10 The CNOT gate cannot fulfill this purpose for classical computation.

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CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Of course, it is a nontrivial task to optimize such networks.11 Theorem1.1.2 (Toffoli) For all n1 , n2 ∈ IN and for every φ ∈ Fn1 ,n2 there is some n ∈ max {n1 , n2 } , . . . , n1 + n2 and some bijection f ∈ Fn,n with 



f1 (b1 , . . . , bn1 , 0, . . . , 0)   ..  = φ(b1 , . . . , bn1 )  . fn2 (b1 , . . . , bn1 , 0, . . . , 0)

∀ b1 , . . . , bn1 ∈ {0, 1} .

Proof: See (Toffoli, 1980b, Theorem 4.1). Exercise 1 Show that CSWAP acts as indicated and, therefore, is universal for classical reversible computation: b1

s

b

s

b2

❭✜ ✜❭

1

❭✜ ✜❭

0

b1 b2

0

1⊕b

b

s

b

0

❭✜ ✜❭

b

1

Exercise 2 Show that the following networks act as indicated:12 a) Multiplication by 2: 0

a1

❭✜ ✜❭

a2

a1 ❭✜ ✜❭

.. . an−1

a2 a3

··· ···

an

❭✜ ✜❭

.. . an 0

b) Adder13 for a, b ∈ {0, 1} :

+

≡

0

❤

a

s

❤ s

b cin

s

cout = a cin ⊕ b (a ⊕ cin )

❤

a s

s

b

s

❤

a ⊕ b ⊕ cin

DRAFT, October 17, 2007

11

See (Shende et al., 2003) and (Tsai and Kuo, 2001; Younes and Miller, 2003; Shende et al., 2004a), in this connection. 12 See also (Vedral et al., 1996; Draper, 2000; Tsai and Kuo, 2001; Cheng and Tseng, 2002). 13 Note that a cin ⊕ b (a ⊕ cin ) = 0 iff a + b + cin ≤ 1 .

17

1.2. QUANTUM COMPUTATIONAL NETWORKS c) Adder for a, b ∈ {0, 1}n : 0

···

a1

···

b1

···

0

···

.. . .. . cn−2

an−1

+

bn−1 cn−1

0

bn

+

ν=1

|

aν 2n−ν + {z x

}

n X

ν=1

|

c1 .. .

y

}

|

a1 ⊕ b1 ⊕ c1 .. . .. .

···

an−2 ⊕ bn−2 ⊕ cn−2

···

an−1

···

bn−1

···

an−1 ⊕ bn−1 ⊕ cn−1

···

an

···

bn

bν 2n−ν = c0 2n + {z

a1 b1

an ⊕ bn ⊕ cn |{z}

···

0

n X

+

.. .

0

an

c0

n X

ν=1

=0

(aν ⊕ bν ⊕ cν ) 2n−ν . {z

x+y

}

Exercise 3 Show that for every reversible classical 2-bit network there is an equivalent one composed only of CNOTs, TCNOTs and NOTs.14

1.2 1.2.1

Quantum Computational Networks Quantum Gates

If the computational registers are made smaller and smaller you will finally have to take into account the quantum behavior of the devices. Then a n-bit register has to be considered as an array of quantum mechanical systems to be ‘switched’ — for classical computation — into one of two selected (pure) states corresponding to two orthonormal state vectors, usually denoted |0i and |1i . This way the n-bit information (b1 , . . . , bn ) will be encoded into the state vector |b1 , . . . , bn i ≡ |b1 i ⊗ · · · |bn i corresponding to the situation: DRAFT, October 17, 2007

14

Hint: Check the action of (ID ⊗ NOT) ◦ TCNOT on the ordered set of 2-bits.

(1.5)

18

CHAPTER 1. BASICS OF QUANTUM COMPUTATION ‘Switch’ number ν being in the state corresponding to |bν i for ν = 1, . . . , n .

Then reversible classical n-bit gates correspond to permutations of the 2n states corresponding to the orthonormal computational basis {|bi : b ∈ {0, 1}n } of the registers state space. Since such permutations are special unitary transformations there is a chance to implement them by mathematically simple quantum ¨ dinger equation). Of course the mechanical evolution (governed by some Schro interpolating states do no longer correspond to some element of the computational basis even if this is the case for both the input state and the output state. Moreover, quantum mechanics allows coherent superpositions as input states,15 corresponding to complex linear combinations of the elements of the computational basis. Then the gate causes the transition X

b∈{0,1}

n

λb |bi 7−→

|{z} ∈C

X

b∈{0,1}

n

λb |f (b)i ,

where f ∈ Fn,n is the mapping corresponding to the classical action of the gate. This means that, in a way, the gate is able to perform all the 2n transitions |bi 7−→ |f (b)i ,

b ∈ {0, 1}n ,

simultaneously — thanks to quantum mechanical evolution. Of course, one would like to exploit this massive quantum parallelism for more efficient computation. Unfortunately quantum mechanics imposes severe restrictions: 1. Unknown coherent superpositions cannot be copied with arbitrary precision (Wootters and Zurek, 1982; Peres, 2002). Otherwise a device for superluminal communication could be constructed (Werner, 2001, Chapter 3). 2. Every measurement of an unknown state destroys most of the information carried by that state (quantum state collapse). 3. It is extremely difficult to correct errors caused by unwanted interaction with the environment. Nevertheless quantum computational networks can be devised, at least in principle, which are much more efficient, for certain tasks, than classical computational networks. Their general structure is as follows: • The information is usually processed on one and the same quantum register16 realized as an array of qubits,17 i.e. quantum mechanical systems with a preselected simple quantum alternative corresponding to orthonormal state vectors, usually denoted |0i and |1i . DRAFT, October 17, 2007

15

See (Long and Sun, 2001) for an efficient preparation of these superpositions. 16 Thanks to the SWAP gate this is not a necessity but this point of view simplifies the treatment. 17 Usually qubits are treated as distinguishable, due to their localization in (essentially) disjoint regions; see (Eckert et al., 2002) for a refined description.

1.2. QUANTUM COMPUTATIONAL NETWORKS

19

• The possible (pure) states of such an n-qubit register correspond to the (normalized) complex linear combinations of the elements (1.5) of the computational basis. • ‘Simple’ quantum computational steps are depicted in the network model by quantum gates with an equal number (≤ n) of quantum wires (horizontal lines) attached on both sides. These quantum wires represent the qubits on which the gate acts. • Several quantum gates may be assembled as in the classical reversible case (without loopbacks, of course). • The whole network itself is a (more complicated) n-qubit quantum gate acting corresponding to some unitary operator Uˆnet . • The action of this operator on the initial state vector |b1 , . . . , bn i representing the task (encoded in the bit sequence (b1 , . . . , bn )) has to be checked — i.e. the output state Uˆnet |b1 , . . . , bn i has to be measured — to yield a result. Even though only the probability of a certain outcome of a quantum computation is predictable (according to the rules of quantum mechanics) quantum computation may be very useful for problems of the type “solution easy to check but difficult to find”. This will be demonstrated by several quantum algorithms in Chapter 2. Remarks: 1. We use the naive tensor product formalism of quantum mechanics to describe coupled systems. The latter is very problematic if the interaction of matter ucke, nlqo). with radiation is to be described; see Section 6.2.1 of (L¨ 2. Of course, the network model described above is just the simplest model for quantum computing. Some possible generalizations, not to be discussed in this chapter, are: • Computation with non-unitarily evolving mixed states (Tarasov, 2002). • Quantum computation via application of sequences of one-qubit projective measurements to suitably prepared initial states (Raussendorf et al., 2002). • Use of non-deterministic gates, i.e. those succeeding only with probability (considerably) less than 1 (Ralph et al., 2002; Bartlett et al., 2002). 3. We are not going to discuss oddities like18 “quantum computation even before its quantum input is defined” (Brukner et al., 2003) or “counterfactual computation” (Mitchison and Jozsa, 2001). For quantum programming we refer to (Bettelli et al., 2001). DRAFT, October 17, 2007

18

See 3.1.4, however.

20

CHAPTER 1. BASICS OF QUANTUM COMPUTATION Name

Symbol Operator

Uˆ gate



Uˆ

ˆ U

Matrix u11 u21

u12 u22

Action 

|0i 7→ u11 |0i + u21 |1i |1i 7→ u12 |0i + u22 |1i



Table 1.3: General one-qubit gate u1j u1k + u2j u2k = δjk Name

Symbol

Operator

Matrix 

1 0  0 1

ˆ1

ID gate NOT gate

❍ ❍❜ ✟ ✟

phase shift gate

δ

Sˆδ

Hadamard gate

H

UˆH

¬ ˆ

√1 2

Action



0 1 1 0



1 0 0 eiδ





1 1 1 −1





|bi 7→ |bi |0i ⇀ ↽ |1i |0i 7→ |0i

|1i 7→ eiδ |1i

|0i 7→ |1i 7→

√1 2 √1 2

(|0i + |1i) (|0i − |1i)

Table 1.4: Special one-qubit gates

The elements |b1 , . . . , bn i of the computational basis of an n-qubit system are naturally ordered by the corresponding integers def

I(b) =

n X

bν 2n−ν

ν=1

∀ b ∈ {0, 1}n .

(1.6)

It is relative to this ordering that the actions of quantum gates are usually represented by unitary matrices as in Tables 1.3–1.5. Remark: Note that every (complex) unitary 2 × 2 matrix corresponds to a spin ucke, tdst) and Sect. 4.2.1 of (L¨ ucke, qft). rotation; see, e.g., Exercise 19 of (L¨

Note that a quantum gate may be used for classical computation iff the entries of its matrix take only values from {0, 1} . Whenever this is the case we use the same symbol and name for the quantum gate as for its classical analog. In this sense we have, e.g., V

¬) 1 (ˆ

= CNOT :

s ¬ ˆ

=

s ❤

,

21

1.2. QUANTUM COMPUTATIONAL NETWORKS Name

Symbol

Matrix

Action

s V  ˆ n U gate



.. . s

1l2n 0 0 Uˆ

|b1 , . . . , bn , ci



7→

ˆ U

(

|b1 , . . . bn i ⊗ Uˆ |ci if b1 = . . . = bn = 1 |b1 , . . . , bn , ci else

Table 1.5: Special (n + 1)-qubit gates

V

s

¬) 2 (ˆ

s

s

= CCNOT :

=

s

,

(1.7)

❤

¬ ˆ

i.e. ❤

=

¬ ˆ

.

(1.8)

Obviously, the phase shift gate (for δ 6= 0 mod π) and the Hadamard gate have no classical analog. Sometimes it is more convenient to sketch the action of a gate as done in Figure 1.1 for the f -CNOT gate. |b1 i

.. . |bn i |ci

|b1 i f ❤

.. . |bn i |c ⊕ f (b)i

Figure 1.1: f -CNOT gate for f ∈ Fn,1

Quantum computational networks are called equivalent if they implement the same mapping up to a phase factor.

Exercise 4 a) For arbitrary a ∈ {0, 1}n , n ∈ IN , show that

22

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

.. fa .. . .

≡

ˆa N 1

s

ˆa N 1

.. .

.. . s

.. .

ˆa N n

❤

ˆa N n

,

❤

where

def

fa (b) = δa,b and ˆb def N =



∀ b ∈ {0, 1}n

¬ ˆ if b = 0 , ˆ1 else .

b) Show for arbitrary f, g ∈ Fn,1 that

h=f ⊕g

.. .

=⇒

h

.. .

≡

❤

.. .

f ❤

.. .

g

❤

.. . .

A first example showing the superiority of quantum networks is the DeutschJozsa problem: Assume you are given an (n + 1) qubit gate which is only known to be the f -CNOT gate of some f ∈ Fn,1 that is either constant or balanced, i.e. fulfills X (−1)f (b) = 0 . b∈{0,1}n

Find out by ‘asking’ this Deutsch-Jozsa oracle whether f is balanced or constant. Note that in classical computation |b1 i

.. . |bn i |0i

|b1 i f ❤

.. . |bn i |f (b)i

you may have to ask the Deutsch-Jozsa oracle 2n−1 + 1 times (in the worst case) to find the answer. Already for n = 60 that would take more than 259 years > 18 years 60 · 60 · 24 · 365 · 109

23

1.2. QUANTUM COMPUTATIONAL NETWORKS

to get the answer if the oracle is asked at a frequency of 1 GHz. In quantum computation, however, we may take advantage of coherent superpositions: |b1 i

√1 2

.. . |bn i

(|0i − |1i)

Since

|0i

|b1 i f ❤

.. . |bn i

f (b)

(−1) √

2

UˆH |bi =

.. . |0i

√1 2



H

|0i + (−1)b |1i



′ 1 X = √ (−1)b b |b′ i 2 b′ ∈{0,1}

and hence

UˆH⊗n |bi = 2−n/2

′

X

b ∈{0,1}

′

n

we have ⊗(n+1) ⊗(n+1) UˆH ◦ f -CNOT ◦ UˆH |0, . . . , 0, 1i = 2−n

⊗(n+1) ⊗(n+1) UˆH ◦ f -CNOT ◦ UˆH |0, . . . , 0, 1i =

.. .

❤

H



X ∼ (−1)f (b) |bi  n b∈{0,1} |1i .

∀ b ∈ {0, 1}

(−1)b ·b |b′ i

For the Deutsch-Jozsa oracle this means

f H

|1i

(|0i − |1i) ,

 

H

∀ b ∈ {0, 1}n X

b,b′ ∈{0,1}n

(1.9)

′

(−1)f (b)+b ·b |b′ , 1i . (1.10)

∼ |0, . . . , 0, 1i if f is constant, ⊥ |0, . . . , 0, 1i else.

Therefore, the following quantum gate has to be used only19 once in order to solve the Deutsch-Jozsa problem (Cleve et al., 1998, Sect. 3): |0i

.. . |0i

|1i

H .. . H

f

H

❤

H .. . H

  

H

|1i

Ψ=  



∼ |0, . . . , 0i if f is constant, ⊥ |0, . . . , 0i else.

Remark: (1.10) holds for every f ∈ F2,1 and can therefore be applied also to the Bernstein-Vazirani oracle, i.e. the f -CNOT gate with f (b) = a · b for some a ∈ {0, 1}n . Then Ψ becomes 2−n

X

b,b′ ∈{0,1}n

′ (−1)(a+b )·b |b′ i = |ai .

DRAFT, October 17, 2007

19

Actually, since there is always some tiny probability for getting the wrong answer, the quantum test should be repeated a few times. For a physical realization of the algorithm see (Gulde et al., 2003).

24

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

1.2.2

Quantum Teleportation

Obviously, the Bell network H

s ❤

acts according to20 |0, 0i + |1, 1i √ 2 def def |0, 1i + |1, 0i √ = Ψ+ = 2 def def |0, 0i − |1, 1i √ = Φ− = 2 def def |0, 1i − |1, 0i √ = Ψ− = 2 def

=

|0, 1i 7−→ Ψ0,1

=

|1, 0i 7−→ Ψ1,0 |1, 1i 7−→ Ψ1,1 where

def

|0, 0i 7−→ Ψ0,0 = Φ+ =

Uˆ0,0 Uˆ0,1 Uˆ1,0 Uˆ1,1

def

= = def = def =

def

+ |0ih0| + |1ih1| + |1ih0| + |0ih1| + |0ih0| − |1ih1| − |1ih0| + |0ih1|

= = = =

= =











Uˆ0,0 ⊗ ˆ1 Ψ0,0 , 

Uˆ0,1 ⊗ ˆ1 Ψ0,0 , 

Uˆ1,0 ⊗ ˆ1 Ψ0,0 , 

Uˆ1,1 ⊗ ˆ1 Ψ0,0 ,

ˆ1 , ¬ ˆ, Sˆπ , Sˆπ ¬ ˆ.

This indicates the possibility of quantum dense coding:21 Bob prepares the entangled 22 state Ψ0,0 by applying the Bell network to the easily available state |0, 0i and sends his first qubit to Alice (arbitrarily far away). Now Alice may transfer a 2-bit message to Bob by applying one of the operators Uˆ0,0 , Uˆ0,1 , Uˆ1,0 , Uˆ1,1 to this single qubit and sending it back to Bob. Bob can ‘read’ this message by performing DRAFT, October 17, 2007

20

The states on the r.h.s are usually called Bell They exhibit maximal correlation  states.  α −β def . Thus between the two qubits. We use the notation = +α β ⊥ 1 Ψ− = √ 2 21 22



Ψ Ψ⊥ Ψ⊥ Ψ ⊗ − ⊗ kΨk kΨ⊥ k kΨ⊥ k kΨk



∀ Ψ ∈ C2 \ {0} .

See (Mermin, 2002) for an interesting discussion of dense coding. Entanglement of vector states Ψ means i.g.

ˆ |Ψi 6= hΨ| Aˆ ⊗ ˆ1 |Ψi hΨ| ˆ1 ⊗ B ˆ |Ψi , hΨ| Aˆ ⊗ B i.e. (non-classical) correlations between the subsystems. See (Brukner et al., 2001) in this context.

25

1.2. QUANTUM COMPUTATIONAL NETWORKS a Bell measurement, i.e. checking whether the state of the 2-qubit system is Ψ0,0 , Ψ0,1 , Ψ1,0 , or Ψ1,1 . The whole procedure is described by the following network action: s

|b1 i

|b1 i

s

|b2 i

s

H

|0i

|b2 i

❤

s

π

❤

|0i

H

❤

|b1 i |b2 i .

Remarks: 1. As pointed out in (Mermin, 2002), the network s s H

s

s

❤

s

❤

❤

π

s

≡

H

❤

s ❤ ❤

— although unsuitable for dense coding — has the same effect on the considered special input. 2. We use abbreviations like s for

❭✜ ✜❭

Aˆ

s

❭✜ ✜❭

Aˆ

without explicit definition.

Moreover, checking the special cases Ψ ∈ {|0i , |1i}, we see that the teleportation network s H

s

H

❤

❤

acts on ψ ⊗ |0, 0i in the following way: ψ ⊗ |0, 0i 7−→

1 X |bi ⊗ Uˆb−1 ψ . 2 b∈{0,1}2

26

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

(for every one-qubit state vector ψ). This indicates the possibility of quantum teleportation:23 Qubits 2 and 3 are prepared in the state Φ0,0 (indicated by the Bell subnetwork on the left acting on |0, 0i). Then qubit 2 is sent to Alice and qubit 3 to Bob (far apart). Since, now, Alice and Bob share an entangled pair of qubits, Alice may teleport the unknown state ψ of qubit 1 to Bob in the following way: Alice performs a Bell measurement on the system formed by qubits 1 and 2 and sends Bob the classical 2-bit information b if the result is Ψb (corresponding to the output |bi of the teleportation network for qubits 1 and 2). After receiving this information Bob transforms the (collapsed) state of qubit 3 into ψ by applying Uˆb . Note that the actions taken by Alice and Bob, sharing the entangled pair, have the same effect on qubits 1–3 as the following post selection scheme:24 s ❤

s

H s ¬ ˆ

⌢ / ⌢ /

π

In this sense the following scheme describes teleportation of entanglement, also called entanglement swapping:25 |0i

H

|0i |0i |0i

H

s ❤

s

s

❤

❤

s

H s ¬ ˆ

⌢ / ⌢ /

π

This possibility is very important for creating entanglement for teleportation over very large distances. Exercise 5 Show that the entanglement swapping scheme prepares the subsystem formed by qubits 1 and 4 (arbitrarily far apart) in the state Ψ0,0 . DRAFT, October 17, 2007

23

The possibility of teleportation, further discussed in Appendix A.2, was first pointed out in (Bennet et al., 1993). Generalization to n-qubit states is straightforward (D´ıaz-Caro, 2005). Concerning the experimental realization of quantum teleportation see (Giacomini et al., 2002). 24 The symbol ⌢ / represents an ideal test (projective measurement) whether the corresponding qubit is in state |0i or |1i . 25 See (Bowmeester et al., 2000, Sect. 3.10) and (Gisin and Gisin, 2002).

27

1.2. QUANTUM COMPUTATIONAL NETWORKS

1.2.3

Universality

A 2-qubit gate corresponding to the unitary operator Uˆ (2) is called universal if for every quantum network there is an equivalent one composed only of one-qubit gates and 2-qubit gates corresponding to Uˆ (2) . Ordering the computational basis vectors def

|I(b)in = |bi

∀ b ∈ {0, 1}n

(1.11)

as |0in , |1in , . . . |2n − 1in then λn−1 (ˆ ¬) just interchanges the last two of these vectors, the latter being |1, . . . , 1, 0i and |1, . . . , 1, 1i . Also cyclic permutations of the computational basis vectors can be achieved by suitable composition of Λν (ˆ ¬) gates, as the following exercise shows. Exercise 6 Show that the n-qubit network ❤ s

❤

.. . s

.. . s

s

s

s

s

s

s

··· ··· ··· ··· ··· ···

.. . ❤ s

❤

s

s

¬ ˆ

acts according to |xin 7−→ |(x + 1) mod 2n in where

+ n X ν−1 bν 2 ν=1

n

def

= |bi

∀ x ∈ {0, . . . , 2n − 1} , ∀ b ∈ {0, 1}n .

(1.12)

Therefore: For every quantum gate that has a classical analogue there is an equivalent quantum network composed only of Λν (ˆ ¬) (and ID) gates. This together with the following theorem shows that the CNOT gate is universal if the following holds for every ν ∈ IN : For every Uˆ ∈ U (2) there is a network composed only of single qubit gates and CNOT gates (and ID gates) that is equivalent to the Λν (Uˆ ) gate.

(1.13)

28

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Theorem 1.2.1 Let 2 < N ∈ IN . Then every unitary N × N -matrix may be represented as a product of permutation matrices and N × N -matrices of the form  ˆ 1 0 , Uˆ ∈ U (2) . 0 Uˆ Outline of proof:26 Given 2 < N ∈ IN , choose some orthonormal basis {e1 , . . . , eN } of CN . Then, for arbitrary   u11 u12 ˆ U= ∈ U (2) , u21 u22

N ′ ∈ {2, . . . , N } , and ν ∈ {1, . . . , N } define ( u11 eN ′ −1 + u21 eN ′ def ˆN ′ eν = u12 eN ′ −1 + u22 eN ′ U

for ν = N ′ − 1 , for ν = N ′ , else .

eν

Then, for arbitrary normed z=

N X

ν=1

N ′ ∈ {2, . . . , N } , and ζ ∈ C with

N X

2

|ζ| + we have



z ν eν ∈ CN ,

ν=N ′ +1

0 .. .



2

|z ν | = 1 

0 .. .



             0   0      ′ ′ 0   ζ  ˆ (N′ )  U   =  N′  N  ζ′   z ′   N +1   N +1  z  z   .   .   ..   ..  zN

zN

ˆ (N ′ ) ∈ U (2) and ζ ′ ∈ C . Thus, by iteration, we see that there are for suitable U (N ) (2) ˆ ˆ ∈ U (2) with U ,...,U ˆ (2) · · · U ˆ (N ) eN z=U 2

and hence



ˆ (N ) U N

−1

N

 −1 ˆ (2) ··· U z = eN . 2

ˆˆ Identifying z with the last column of an arbitrarily given unitary N × N -matrix U we get  −1  −1  ˆˆ ˆ (N ) ˆ (2) U · · · U U = δν,N . 2 N ν,N

Thanks to unitarity, the latter also implies  −1  −1  ˆˆ ˆ (N ) ˆ (2) U · · · U U 2 N

= δN,ν . N,ν

Iteration of this argument, if necessary, proves the theorem. DRAFT, October 17, 2007

26

Compare (Reck et al., 1994; Dit¸a ˘, 2001).

29

1.2. QUANTUM COMPUTATIONAL NETWORKS

That (1.13) holds for ν = 1 is a simple consequence of the following lemma (Barenco et al., 1995): ˆ B, ˆ Cˆ ∈ SU(2) with Lemma 1.2.2 For every 27 Uˆ ∈ SU(2) there are A, ˆ Cˆ = ˆ1 , AˆB

ˆ¬B ˆ¬ Aˆ ˆ Cˆ = Uˆ .

ˆ ∈SU(2) . Then one may easily show (see Exercise 28 of (L¨ Proof: Let U ucke, eine)) that there are angles ψ, θ, φ with ˆ =R ˆ 3 (ψ) R ˆ 2 (θ) R ˆ 3 (φ) , U where ˆ 3 (ψ) def R =



ψ

e+i 2 0

With the definitions   θ def ˆ ˆ ˆ , A = R3 (ψ) R2 2

0

ψ

e−i 2



,

ˆ 2 (θ) def R =



+ cos θ2 − sin θ2

    θ ˆ ψ+φ def ˆ ˆ B = R2 − , R3 − 2 2

+ sin θ2 + cos θ2



.

  φ−ψ def ˆ C = R3 2

this gives ˆ Cˆ Aˆ B



 ψ+φ   φ−ψ  ˆ 3 (ψ) R ˆ2 θ R ˆ2 − θ R ˆ3 − = R R3 2 2 2 2 ˆ 3 (ψ) R ˆ 3 (−ψ) = R ˆ = 1

and ˆ¬B ˆ¬ Aˆ ˆ Cˆ

ˆ 3 (ψ) R ˆ2 = R ˆ 3 (ψ) R ˆ2 = R ˆ 3 (ψ) R ˆ2 = R

 
 ψ+φ  ˆ3 − ˆ2 − θ R ¬ ˆ R3 φ−ψ ¬ ˆR 2 2 2     

 θ ˆ2 − θ ¬ ˆ 3 − ψ+φ ¬ ¬ ˆR ¬ ˆR ˆ R3 φ−ψ 2 2 ˆ 2 2  

 ψ+φ  φ−ψ θ ˆ3 ˆ2 θ R R ˆ R 3 2 2 2 2 θ 2




ˆ 3 (ψ) R ˆ 2 (θ) R ˆ 3 (φ) = R ˆ. = U

Remark: Note that ψ, θ, φ in the above proof correspond to the well-known Euler angles; see Sect. 2.1.1.3 of (L¨ ucke, mech) and — for generalization — also (D’Alessandro, 2001).

Because of s

|0i |bi

Cˆ

❤

s ˆ B

❤

|0i

Aˆ

ˆ Cˆ |bi AˆB

and DRAFT, October 17, 2007

27

As usual, we denote by U(2) the set of all (complex) unitary 2 × 2-matrices and by SU(2) the ˆ ∈U(2) with det U ˆ = 1. set of all U

30

CHAPTER 1. BASICS OF QUANTUM COMPUTATION s

|1i

❤

Cˆ

|bi

s

|1i

❤

ˆ B

ˆ¬B ˆ¬ Aˆ ˆ Cˆ |bi

Aˆ

this has the following consequence: ˆ B, ˆ C, ˆ Uˆ according to Lemma 1.2.2 we have: Corollary 1.2.3 For A, s

s

≡

ˆ U

s

❤

Cˆ

❤

ˆ B

Aˆ

Corollary 1.2.3 together with s

δ

s

≡

Aˆ

Aˆ′ = eiδ Aˆ ,

,

Aˆ′

shows that that (1.13) holds for ν = 1 , indeed. Exercise 7 Show that ❤

≡

s

H

s

H

H

❤

H

and28 ❭✜ ✜❭

≡

s

H

s

H

s

❤

H

❤

H

❤

.

Moreover, because of |0i

s

|0i

s

|bi

Cˆ

|0i

❤

s s

|1i

s

|bi

Cˆ

❤

ˆ B s ❤

s s ˆ B

Recall (1.3).

|0i |0i

Aˆ

|bi

s

|0i

❤

DRAFT, October 17, 2007

28

s

|1i

Aˆ

ˆ Cˆ |bi B

.

31

1.2. QUANTUM COMPUTATIONAL NETWORKS |1i

s

|0i

s

|bi

Cˆ

|1i

s

❤

s

s

ˆ B

|bi

Cˆ

|0i

ˆ |bi AˆB

Aˆ s

❤

|1i

❤

s

|1i

s

s

s

|1i

❤

ˆ B

|1i

AˆCˆ |bi

Aˆ

we have (Sleator and Weinfurter, 1994): s s Vˆ 2

s

≡

s

❤

Vˆ

s s

s

❤

Vˆ ∗

Vˆ

This Sleator-Weinfurter construction may be generalized for arbitrary n ∈ IN :29 s

s

s s .. . s Vˆ 2

Since

s

≡

❤

s s

ˆ 2 ) with

n (V

s

❤

s

s

s

.. . s

.. . s

.. . s

Vˆ

Vˆ ∗

Vˆ

n

V

o

U (2) = Vˆ 2 : Vˆ ∈ U(2)

this that (1.13) holds for every ν ∈ IN , hence:

The CNOT gate is universal.30 DRAFT, October 17, 2007

29

Of course, one should look for more efficient implementations; see, e.g. Exercise 8 and (Aho and Svore, 2003; Vatan and Williams, 2004; Shende et al., 2004b). For a nice introduction into the general theory of computational complexity see (Mertens, 2002). 30 Therefore, if CNOT is implementable together with all one-qubit gates and projective measurements w.r.t. the computational basis, all observables can actually be measured.

32

CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Exercise 8 Show that the following network acts as indicated: s

|b1 i

s

|b2 i

s

|b3 i |b4 i

.. .

s .. .

.. .

.. .

···

···

···

···

···

···

···

···

|bn−1 i

···

|bn i

···

❤

|0i

s ❤

|0i

s ❤

|0i

.. .

.. .

.. . s s

s

···

···

|0i

···

s

|0i

···

|0i

···

❤

s s ❤

···

ψ=

.. .

s s

❤

❤

ˆ U (

s .. .

.. .

.. .

···

s s

❤

❤

❤ .. .

❤

π

π

≡

|0i

···

|0i

···

ψ

H

π s

acts according to |j, ki 7−→ (−1)δj,1 δk,1 |j, ki . DRAFT, October 17, 2007

31

.. .

···

and that s

|0i

|0i

s H

|0i

···

Exercise 9 Show that31

≡

.. .

|0i

Uˆ |bn+1 i if b1 = . . . = bn = 1 , |bn+1 i else.

s

|b3 i

|bn i

··· .. .

|b3 i

···

··· .. .

s

|b2 i

|bn−1 i

···

.. .

s

|b1 i

···

···

···

|bn+1 i

s

s

This equivalence is exploited in most suggestions for physical realization of CNOT.

Chapter 2 Quantum Algorithms1 So far, we have only discovered a few techniques which can produce speed up versus classical algorithms. It is not clear yet whether the reason for this is that we do not have enough intuition to discover more techniques, or that there are only a few problems for which quantum computers can significantly speed up the solution. (Shor, 2000)

2.1

Quantum Data Base Search

2.1.1

Grover’s Algorithm

Let us assume that the b ∈ {0, 1}n are the indices of the entries of some unstructured data base. Moreover let us assume we are given a search machine that provides an implementation of the fa -CNOT gate, where fa (b) =



1 if b = a 0 else

∀ b ∈ {0, 1}n ,

when fed with a unique characterization of some entry indexed by a . Now consider the following problem: Find a with probability ≥ 50% by testing the behavior of the fa -CNOT gate. In classical computing (recall 1.2.2) one has to test fa -CNOT at least 2n−1 -times2 in order to find a with probability of 50% . A substantial speedup,3 exploiting quantum DRAFT, October 17, 2007

1

Algorithms are general, step-by-step procedures for solving general problems. 2 Moreover, expectation value for the necessery number of tests for finding a is n

2 X 2n + 1 ν = . N= N 2 ν=1 3

See (Aaronson and Gottesman, 2004), however.

33

34

CHAPTER 2. QUANTUM ALGORITHMS |ai

. ..... .......... .. .... .. ... ... .. . ... ... . . ... . .... . . . .. ... . . .... ... .. ... . . . . ... . .... . .. .. ... . . . ... .. . . . .. .... . ................ . ..................... .... . ....... ............. ............. ....... .. .. .. ............. ....... . . ............. ....... ............... .... .. ....... ................. ....... ........... ....... . .. ....... .. ....... .. ....... . ....... .......... .... .....................

(n) ♣♣♣♣♣♣♣♣♣♣ ˆ ♣♣♣♣♣♣ ♣♣♣♣ Ga Φ0 ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ 2θ ♣♣♣♣♣♣ ♣♣ ♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣ (n) ♣♣♣♣ ♣♣♣♣ ♣♣ Φ

θ

0

⊥ |ai

ˆ on Φ(n) Figure 2.1: Action of G 0 . parallelism, was suggested in (Grover, 1996). The basic ingredients of Grover’s algorithm are the initial state X

(n) def

Φ0

= UˆH⊗n |0, . . . , 0i = 2−n/2

b∈{0,1}n

|bi = 2−n/2 Φ1,1

(2.1)

and the unitary operator ˆa def ˆ (n) R ˆ |ai , G = −R Φ

(2.2)

0

where

ˆ Ψ def R = ˆ1 − 2 PˆΨ ∀ Ψ ∈ Hn . ˆ ˆ |ai and R ˆ (n) leave the |ai-Φ(n) Since both reflections R 0 -plane invariant, Ga acts as Φ0 a rotation in this plane. To determine this rotation it suffices to check its effect on (n) Φ0 . As explained in Figure 2.1 this action is a rotation by the angle 2 θ towards |ai , (n) where π/2 − θ is the angle between Φ0 and |ai . Therefore: (n)

ˆa an appropriate number of times to Φ0 and testing the Applying G result with respect to the computational basis solves the posed problem.

2.1.2

Network for Grover’s Algorithm

Exercise 10 Show that the following (n + 1)-qubit networks act as indicated:4  

.. .

|bi

H

Ψ

fa ❤

.. .

 

H

|bi ,

ˆ |ai Ψ δ0b Ψ + δ1b R



DRAFT, October 17, 2007

4

Recall Exercise 4a).

35

2.1. QUANTUM DATA BASE SEARCH

 

H .. . H

|bi

H

Ψ

 

H .. . H

f0 ❤

ˆ (n) Ψ δ0b Ψ − δ1b R Φ



π

H

|bi .

Using the obvious notation and generalization of Uˆ , we get from Exercise 10

.. G ˆa .. . .

.. .

≡

s

0

V

fa

ˆ for n-qubit unitary operators

1 (U )

H .. . H

f0

❤

H

H .. . H

❤

π

H

and therefore: |0i

.. . |0i

H .. . H

|

.. G ˆ . a

···

s

|1i

2.1.3

.. .

ˆa G

···

··· {z

µ-times

s

 

(n)

ˆ Φ G  a 0 µ

|1i .

}

Details and Generalization

For actual application of Grover’s one has to know the number of times Ga should be applied. This can be read off from5 







(n) Gaµ Φ0 = sin (2µ + 1) θ |ai + cos (2µ + 1) θ √

where

1 θ = arcsin √ n 2 def

!

.

Since DRAFT, October 17, 2007

5

θ ≈ 2−n/2

for 2−n/2 ≪ 1

Formula (2.3) was presented first in (Boyer et al., 1998).

1

X

2n − 1 b6=a

|bi ,

(2.3)

(2.4)

36

CHAPTER 2. QUANTUM ALGORITHMS

we have µ

Ga

(n) Φ0

≈ |ai

for µ =

 √ π

2n



and 2n ≫ 1 .

4 In this sense Grover’s algorithm provides a quadratic speedup compared to classical computation. Let us now consider the case that there are exactly t data base entries,6 indexed by a1 , . . . , at ∈ {0, 1}n , meeting the search criteria and that the search engine, therefore, provides an implementation of the (fa1 + . . . fat )-CNOT gate. In order to ˆ a by find at least one of these aν we just have to replace G ˆ (n) R ˆ a1 ,...,at def ˆ |a i · · · R ˆ | at i , = −R G 1 Φ 0

which may be implemented as described in 2.1.2 with the fa -CNOT gate replaced by the (fa1 + . . . fat )-CNOT gate. Correspondingly, (2.3)/(2.4) have to be replaced by   |a i + . . . |a i   X 1 1 t ˆ µa ,...,a Φ(n) √ √ = sin (2µ + 1) θ G +cos (2µ + 1) θ |bi t t 0 t 1 2n − t b6∈{a1 ,...,at } t (2.5) and ! t def θt = arcsin √ n . (2.6) 2 



Choosing µ such that sin2 (2µ + 1) θt is close to 1 we get a state that is essential a superposition of only those states of the computational basis which correspond to data base entries meeting the search criteria. Performing a test we select one solution at random. Unfortunately, we only know how to choose qµ nif we  know t . If t is unknown we 2 −1 find a solution after an expected number of O applications of the described t ′ procedure with suitably chosen µ s (Boyer et al., 1998, Sect. 4).

For interesting modifications of Grover’s algorithm see (Ambainis, 2005; Korepin and Grover, 2 Tulsi et al., 2005) and references given there. For a few-qubit experimental implementation of the algorithm see (Walther et al., 2005) and references given there. For application to robots see (Dong et al., 2005). Final Remark: Presumably Grover’s algorithm will not be useful for searching a standard database, because transferring the database to quantum memory would require too much effort (Deutsch and Ekert, 1998).

DRAFT, October 17, 2007

6

For instance, we may be (Grover and Radhakrishnan, 2004)

interested

in

only

a

subset

of

bit-values

37

2.2. FACTORING LARGE INTEGERS

2.2

Factoring Large Integers

2.2.1

Basics

The greatest common divisor gcd(n0 , n1 ) for given n0 , n1 ∈ ZZ may be efficiently determined via Euclid’s algorithm:7 Defining



ν nν − nν+1 ⌊ nnν+1 ⌋ if nν+1 6= 0 0 else successively for ν = 0, 1, 2, . . . we get

def

nν+2 =

gcd(n0 , n1 ) = ns

for s = sup {ν ∈ IN : nν 6= 0} < ∞ .

(2.7)

(2.8)

Thus, factoring a given product N = p1 p2 of unknown large integers p1 , p2 is a task of the type “A solution is easy to check but extremely difficult to find.” Classical encryption schemes rely on this fact. The most popular public-key encryption algorithm8 is RSA, named after its three inventors Ron Rivest, Adi Shamir, and Leonhard Adleman. It works as follows (Rivest et al., 1978): • Messages and keys are represented by natural numbers corresponding to binary strings. • Messages M are encoded as C = M e mod N , where N > M and e are two public keys created in the following way: 1. Two large prime numbers p and q of comparable size are randomly chosen9 and kept secret. Only their product N =p·q is publicly announced. 2. e is chosen as a large random number having 1 as largest common divisor with (p − 1) · (q − 1) — to be checked by Euclid’s algorithm. DRAFT, October 17, 2007

7

As usual, we use the notation def

⌊x⌋ = sup {n ∈ ZZ : n ≤ x}

∀ x ∈ IR .

Thus, for nν+1 6= 0 , nν+2 is the remainder of the integer division of nν by nν+1 . For details concerning Euclid’s algorithm see Section 2.2.3. 8 See (Singh, 2002; Kahn, 1967) for the history of classical cryptography and (Schneier, 1996) for applications. 9 See (Rivest et al., 1978, Section VII.B) how to find large prime numbers without testing primality by factorization.

38

CHAPTER 2. QUANTUM ALGORITHMS • The message may be decrypted in the form M = C d mod N . where d ∈ {1, . . . , (p − 1) · (q − 1) − 1} is the private key to be determined from10 e · d = 1 mod (p − 1) · (q − 1) . Of course, d has to be kept secret as well as the prime numbers p, q which then may be forgotten.

The task of factorizing N in the form N =p·q

with p, q ∈ {2, . . . , N − 1}

is essentially solved if a factorization n+ · n− = 0 (modN )

(2.9)

of an integer multiple of N is found that fulfills the conditions n± 6= 0 (modN )

(2.10)

gcd (n± , N ) ∈ {2, . . . , N − 1}

(2.11)

Then11 and these factors may be efficiently determined using Euclid’s algorithm. Outline of proof for (2.11): Obviously, every prime factor of N must be a factor of either n+ or n− (or both) and neither n+ nor n− can be the product of all these (not necessarily pairwise different) factors.

Finding a factorization of type (2.9),(2.10) is facilitated by the following theorem. Theorem 2.2.1 (Euler’s Theorem) Let x, N ∈ IN . If x and N are coprime, i.e. if gcd(x, N ) = 1 , then xϕ(N ) = 1 mod N holds, where Euler’s ϕ function is defined as 12 def ϕ(N ) = {y ∈ IN : y < N , gcd(y, N ) = 1} .

DRAFT, October 17, 2007

10

The essential point is that

(p − 1) · (q − 1) = ϕ(p · q) , where ϕ denotes the Euler function introduced in Theorem 2.2.1, below. d may be determined modulo (p − 1) · (q − 1) running Euclid’s algorithm (2.7) for n0 = (p − 1) · (q − 1) and n1 = e and resubstituting iteratively the expressions for ns = 1 , s given by (2.8), using (2.7) to yield the representation 1 = e · x + y · N with certain integers x, y . 11 As usual, we denote by gcd(n1 , n2 ) the greatest common divisor of two integers n1 , n2 .

39

2.2. FACTORING LARGE INTEGERS Proof: See, e.g., (Schroeder, 1997, Sect. 8.3) or (Nielsen and Chuang, 2001, Theorem A4.9).

By Euler’s theorem, for N ∈ IN and x ∈ {2, . . . , N − 1} , the function

def

f (ν) = xν

gcd (x, N ) = 1

(2.12)

∀ν ∈ ZZ

(2.13)

has a minimal period def

r = inf {a ∈ IN : xa = 1 (modN )} ,

(2.14)

called the order 13 of x modulo N . If r is even and xr/2 6= −1 (modN )

(2.15)

then the conditions (2.9) and (2.10) are fulfilled for n± = xr/2 ± 1 . Outline of proof: (2.9) is a consequence of (2.14) and    xr/2 + 1 xr/2 − 1 = xr − 1 .

(2.10) follows from (2.15) and the corresponding property xr/2 6= +1 (modN ) implied by (2.14).

Summarizing, we have the following factoring algorithm: 1. Randomly choose some x ∈ {2, . . . , N − 1}. 2. If gcd(x, N ) 6= 1 then x is already a nontrivial factor of N . 3. If (2.12) holds, determine the order r of x modulo N . 4. If r is odd or xr/2 = −1 (modN ), restart the algorithm.



5. If r is even and xr/2 6= −1 (modN ), determine the factors gcd xr/2 ± 1, N using Euler’s algorithm.



DRAFT, October 17, 2007

12

By |M | we denote the number of elements of a finite set M . 13 The numbers x ∈ {1, . . . , N − 1} form a group w.r.t. multiplication modulo N . Every element x of this group generates a cyclic subgroup of order r .

40

CHAPTER 2. QUANTUM ALGORITHMS

Thanks to the following theorem (and Euler’s algorithm) the efficiency of this factoring algorithm depends solely on the available techniques for determining (2.14). Theorem 2.2.2 Let m be the number of different prime factors of the positive integer N and let x ∈ {1, . . . , N − 1} be randomly chosen. If gcd(x, N ) = 1 , then the (conditional) probability for (2.14) being even and xr/2 6= −1 (modN ) is not smaller than 1 − 2−m . Proof: See, e.g., (Nielsen and Chuang, 2001, Theorem A.4.13).

In order to gain exponential speed up for the determination of (2.14), Peter W. Shor suggested the following (Shor, 1994): Instead of calculating xa (modN ) for a = 1, 2, . . . until the result is 1 (modN ), transform the state 2L 1 2 X−1 |ai2L ⊗ |0iL , 2L a=0

where

n

def

o

L = min l ∈ IN : N ≤ 2l , into the state14

2L 1 2 X−1 |ai2L ⊗ |xa (modN )iL 2L a=0

(exploiting quantum parallelism) and evaluate the latter by means of the quantum Fourier transform applied to the first 2L qubits..

2.2.2

The Quantum Fourier Transform

In order to plot, over the interval [−Ω, +Ω] , the Fourier transform 1 Z T0 fe(ω) = √ f (t) ei ω t dt 0 2π

of a (sufficiently well-behaved) signal restricted to the time interval [0, T0 ] it is sufficient to have the discrete values 

fe k

DRAFT, October 17, 2007

14

2π T0



,

k ∈ ZZ ,

2π k ≤ Ω,

T0

An efficient implementation is described in (Vedral et al., 1996).

41

2.2. FACTORING LARGE INTEGERS

if T0 is large enough (depending on the required precision). In order to determine these values approximately it is sufficient to know the sampling values 

T0 N

f j



j ∈ {0, 1, . . . , N − 1} ,

,

for sufficiently large N ∈ IN (depending on T0 and Ω): 

fe k

2π T0



 −1  2π 1 NX T0 T0 ei k N j ≈√ f j N N 2π j=0





2π for k ≤ Ω . T0

Remark: In order to estimate the quality of this approximation note that  Z T0 N −1  X 2π T0 f j ei k N j = ∆T0 /N (t) f (t) ei ω t dt , N 0 j=0 where

 X  T0 δ t−ν N

def

∆T0 /N (t) =

ν∈ZZ 15

and hence

e T /N (ω) = ∆ 0

√ N X  2π δ ω − µπ T0 µ∈ZZ



2N T0 |{z}

.

Nyquist-frequency

Hence, using the so-called discrete Fourier transform {xj }j∈{0,...,N −1} we get



 

−1  2π 1 NX def xj ei k N j 7 → xek = √ −   N j=0

k∈{0,...,N −1}

  T0 2π e ≈√ f k

T0

2π N

for the sampling values



xj = f j Since

N −1 X

ei k

2π N

m

=

k=0

(2.16)

T0 N



,

1 − ei 2π m 2π

1 − ei N

m

xek( mod N )

2π if k ≤ Ω

T0

j ∈ {0, . . . , N − 1} . = 0 ∀ m ∈ {1, . . . , N − 1} ,

(2.17)

the inverse of the transformation (2.16) is {xek }k∈{0,...,N −1} 7−→

DRAFT, October 17, 2007

15

(

xj

−1 2π 1 NX = √ xek e−i k N j N k=0

def

See, e.g., (L¨ ucke, musi, Anhang A.1).

)

j∈{0,...,N −1}

.

(2.18)

42

CHAPTER 2. QUANTUM ALGORITHMS

Especially for N = 2n we define the quantum Fourier transform Fˆn by X

Fˆn

b∈{0,1}

X

def

n

x(b) |bi =

where16

b∈{0,1}

n

n

∀ (x0 , . . . , x2n −1 ) ∈ C2 ,

xe(b) |bi 

def x(b) = xI(b)  def xe(b) = xeI(b) 

(2.19)

∀ b ∈ {0, 1}n .

n Obviously, Fˆn is a linear operator on C2 and, therefore,



x(a) = δa,b

∀ a ∈ {0, 1}

n



2π 1 xe(a) = √ n ei I(a) 2n I(b) 2

=⇒

(2.16)

implies17 √

2n Fˆn |bi = =

X

∀ a ∈ {0, 1}

2π

n

ei I(a) 2n I(b) |ai

a∈{0,1} n  X O

n−ν a

ei 2

2π ν 2n

n

!

(2.20) I(b)

a∈{0,1}n ν=1

|aν i



∀ b ∈ {0, 1}n .

The latter implies18 n   1 O i 2π 2−ν I(b) ˆ √ |0i + e |1i Fn |bi = 2n ν=1

Thanks to

n−µ−ν

ei 2π bµ 2 we may rewrite this as

∀ b ∈ {0, 1}n .

(2.21)

= 1 for µ ≤ n − ν

 n  Pn 1 O bµ 2n−µ−ν i 2π ˆ µ=n−ν+1 |0i + e |1i Fn |bi = √ n 2 ν=1

∀ b ∈ {0, 1}n

(2.22)

or as  Y ν n 1 O −(α−1) b (n−ν)+α |0i + |1i ei 2π 2 Fˆn |bi = √ n 2 ν=1 α=1

!

showing, by the way, that Fˆn is isometric. Exercise 11 Using (2.23), show that the n-qubit network DRAFT, October 17, 2007

16

Recall the definition of I(b) in (1.6). 17 Note the ordering of def ⊗nν=1 χν = χ1 ⊗ . . . ⊗ χn .

18

ˆ ⊗n |0i . An import special case is Fˆn |0i = U H

∀ b ∈ {0, 1}n

(2.23)

43

2.2. FACTORING LARGE INTEGERS H

sˆ2 s

.. .

···

sˆn−1

sˆn

···

···

H

···

sˆn−2 sˆn−1

s

s s

.. .

···

s

H

sˆ2 s

···

m

H

with def

sˆν =



1 0 i 2π/2ν 0 e



|b1 i .. . |bn i

,

m

|bn i .. , . |b1 i

implements the quantum Fourier transform.19 Moreover, using Corollary 1.2.3, show that sˆν s

≡

Cˆν

❤ Cˆ −1 ν s

❤ s

δν

ˆ 3 (δν ) . holds for δν = −π/2ν and Cˆν = R Remarks: 1. If the crossings are ignored then the above implementation of the quantum Fourier transform uses n/2 SWAP gates, n Hadamard gates and n2 /2 Λ1 (ˆ sν ) gates. 2. Since x eI(a)

= (2.19) =

X

b∈{0,1}n X

b∈{0,1}n

D E xI(b) a Fˆn b

E D xI(b) Fˆn−1 a b

n

∀ a ∈ {0, 1} ,

the above network implementation for Fˆn yields a nice factorization of the matrix corresponding to the discrete Fourier transformation. This factorization is the core of the radix-2 version of the fast Fourier transform (FFT) algorithm.20 DRAFT, October 17, 2007

19

20

Note that


1 ˆH |bi √ |0i + ei π b |1i = U 2

See, e.g., (Brigham, 1974; Nussbaumer, 1982).

∀ b ∈ {0, 1} .

44

CHAPTER 2. QUANTUM ALGORITHMS

Exercise 12 Prove the identities21 √ 2 Fn′ (0, b2 , . . . , bn ) = +Fn′ −1 (b2 , . . . , bn′ , 0, bn′ +1 , . . . , bn ) +ei 2π √

Pn′

b ν=2 ν

2−ν

Fn′ −1 (b2 , . . . , bn′ , 1, bn′ +1 , . . . , bn ) ,

2 Fn′ (1, b2 , . . . , bn ) = +Fn′ −1 (b2 , . . . , bn′ , 0, bn′ +1 , . . . , bn ) −ei 2π

Pn′

b ν=2 ν

2−ν

Fn′ −1 (b2 , . . . , bn′ , 1, bn′ +1 , . . . , bn )

for the partial discrete Fourier transform Fn′ defined by def

(Fn′ x) (b) =

′

X

b ∈{0,1}

n′

for n′ < n ∈ IN and



x(b′1 , . . . , b′n′ , bn′ +1 , . . . , bn ) exp+i 2π def

n′ −ν−ν ′

bν bν ′ 2

ν,ν ′ =1

 

def

(Fn x) (b) = xe(b) ,

2.2.3

′

n X

(F0 x) (b) = x(b) .

Quantum Order Finding

As already mentioned at the end of 2.2.1, to find the order (2.14) of a given integer x ∈ {2, . . . , N − 1} , Shor suggested to exploit the effectively implementable22 state 2L  1 2 X−1  ˆ F |ai ⊗ |xa (modN )iL 2L 2L L 2 a=0

def

ΨShor

=

2L 1 2 X−1 i a 2π e 22L c |ci2L ⊗ |xa (modN )iL . 2L 2 a,c=0

=

(2.19),(2.16)

where

def

o

n

L = min l ∈ IN : N ≤ 2l .

The essential point is the following: ′

xa (modN ) = xa (modN ) Therefore,

⇐⇒

a′ = a mod r .



D E 2 def p(a, c) = |ci2L ⊗ |xa (modN )iL ΨShor

1 24L

= DRAFT, October 17, 2007

21

Let us point out that ei 2π

22

Pn′

ν=2

X ′ ei a a′ ∈Ma

bν 2−ν

iπ

=e

2π 22L

2 c

I(b2 ,...,b ′ ) n ′ 2n −1

See also (Coppersmith, 1994), in this connection.

n

o

∀ a, c ∈ 0, . . . , 22L − 1 ,

∀ b2 , . . . , bn′ ∈ {0, 1} .

45

2.2. FACTORING LARGE INTEGERS where def

Ma =



n

′

2L

a ∈ 0, . . . , 2

o

− 1 : a = a mod r ,

is negligible unless all the phases 2π with



′

brc (mod2π) ∈ [0, 2π) 22L

n

b ∈ 0, . . . ,

j



22L − 1 − min Ma /r

ko

,

are predominantly almost the same, i.e. (assuming L sufficiently large) unless   rc r [0, 2π) ∋ 2π 2L (mod2π) = O 2L . 2 2 23 The latter means that

holds for some integer d .

  c 1 d 2L − = O 2 r 22L

More precisely, one can show:24 A projective measurement of ΨShor w.r.t. the computational basis is likely to find the first 2L-qubit register in a state |ci2L with c 1 d , 2L − ≤ 2 r 2 r2

gcd(d, r) = 1

(2.24)

being fulfilled for integer d .

If we have found a fraction c/22L fulfilling (2.24) for some (unknown) integer d with gcd(c, d) = 1 , then r may be efficiently determined using the continued fraction algorithm:25 The motivation for the definition (2.7) — given n0 , n1 ∈ IN — is the observation that % %! $ $ 1 nν nν nν + nν − nν+1 = ∀ nν , nν+1 ∈ IN nν+1 nν+1 nν+1 nν+1 DRAFT, October 17, 2007

|

{z

remainder 2L

}

If, by chance, r divides 22L then b runs from 0 to 2 r − 1 and, therefore, an adaption of (2.17) shows that p(a, c) vanishes exactly unless 2r2Lc = d holds for some integer d . 24 See (Shor, 1997, Section 5) for details and Appendix A.3 for an improved search algorithm. Note that r ≤ ϕ(N ) < N . ′ 25 Otherwise the prescription will yield an integer r′ that fails the test xr = 1(modN ) almost certainly. Then the whole procedure has to be repeated. 23

46

CHAPTER 2. QUANTUM ALGORITHMS

since, then, we have n0 = n1



=



=





n2



n2



n2

1 n0 + n  1 n1

1 n0 + jn k 1 n1 +1 n2 n3

1 n0 + jn k 1 n1 + j k1 n2 n3

+

1

( nn43 )

etc.,

hence the continued fraction expansion26 



ν−2 X 1 n0 n0 j + = n1 n1 µ=1

nµ nµ+1

with s given by (2.8).

|

|

1

k +

nν−1 nν

∀ ν ∈ {1, . . . , s}

(2.25)

Remark: Formally, i.e. without specification of the aµ and bµ , finite continued fractions27 a1 | a2 | aν | b0 + + + ... + , | b1 | b2 | bν are recursively defined by a1 a1 | def = b0 + b0 + | b1 b1 and a1 | a2 | aν+1 | def a2 | aν | a1 | = b0 + . + + ... + + . . . b0 + a | b1 | b2 | bν+1 | b1 | b2 b + ν+1 ν

bν+1

Due to

nν+1 6= 0

=⇒

(2.7)

s must be finite and fulfill the equation gcd (nν−1 , nν ) 6= 0

nν+2 < nν+2 < nν+1 j

ns−1 ns

k

⇐⇒

=

ns−1 ns

. Since

gcd (nν+1 , nν )

we see that gcd(n0 , n1 ) divides ns and that ns , dividing ns−1 , also divides n0 and n1 . Hence (2.8) holds, indeed. Given c, L ∈ IN fulfilling (2.24), the set of possible fractions stricted by the following lemma.

d c

is strongly re-

DRAFT, October 17, 2007

26

Note that the continued fraction expansion does not change if n0 and n1 are replaced by n′0 = p n0 and n′1 = p n1 , where p ∈ IN . 27 See, e.g., (Perron, 1954; Perron, 1957) or (Brezinski, 1991) for the general theory of continued fractions. See also (Baladi and Vallee, 2003).

47

2.2. FACTORING LARGE INTEGERS Lemma 2.2.3 Given n0 , n1 , d, r ∈ IN fulfilling

n 1 d 0 , − < n1 r 2 r2

using definition (2.7) and (2.8), we have28 

for some t ∈ {1, . . . , s − 1} .



t X 1 n0 d j + = r n1 µ=1

nµ nµ+1

|

k

Proof: See (Nielsen and Chuang, 2001, Theorem A4.16).

Therefore, using (2.7) for n0 = c ,

n1 = 22L

and determining — for t = 1, 2, . . . , s — the numbers At , Bt ∈ IN characterized by 

we have



t X 1 n0 j + n1 µ=1

nµ nµ+1

|

k =

At Bt

gcd(At , Bt ) = 1 ,

r = Bt for some t ≤ s . Also these At , Bt can be efficiently determined via Euclid’s algorithm: If b0 + is well-defined then b0 +

a2 | an | a1 | + + ... + , | b1 | b2 | bn

ν+1 X

ν−1 X aµ | aµ | aν bν+1 | = b0 + + | bν bν+1 + aν+1 µ=1 | bµ µ=1 | bµ

∀ ν = 1, . . . , n − 1

and, therefore, b0 +

a2 | aν | Aν (a1 , . . . , aν ; b0 , . . . , bν ) a1 | + + ... + = | b1 | b2 | bν Bν (a1 , . . . , aν ; b0 , . . . , bν )

∀ ν = 1, . . . , n ,

DRAFT, October 17, 2007

28

Note, however, that at = 1

=⇒

t−1 t X 1 | 1 | X 1 | = + | aµ | aµ | at−1 + 1 µ=1 µ=1

∀ a0 , . . . , at ∈ IN .

48

CHAPTER 2. QUANTUM ALGORITHMS

if the Aν and Bν are recursively defined by29  

def

= bν Aν−1 + aν Aν−2 for ν = 1, 2, . . . , n , = bν Bν−1 + aν Bν−2 

Aν Bν

def

where

def

A−1 = 1 ,

def

A0 = b 0 ,

def

B−1 = 0 ,

def

B0 = 1 .

Fortunately, these definitions also imply aµ = 1 ∀ µ ∈ {1, . . . , n} bµ ∈ IN ∀ µ ∈ {0, . . . , n}

)

=⇒

gcd (Aν , Bν ) = 1 ∀ ν ∈ {1, . . . , n} .

Outline of proof: Aν+1 Bν

= bν+1 Aν Bν + Aν−1 Bν = bν+1 Aν Bν + Aν−1 Bν−2 + bν Aν−1 Bν−1

and the corresponding equation with A, B interchanged imply Aν+1 Bν − Bν+1 Aν = − (Aν Bν−1 − Bν Aν−1 ) . By induction, starting from A0 B−1 − B0 A−1 = −1 , this gives Aν Bν−1 − Bν Aν−1 = (−1)

ν+1

∀ ν ∈ {0, . . . , n} .

(2.26)

In case Aν = dν cν ,

Bν = eν cν

the positive integer cν would divide the l.h.s. of (2.26), hence also the r.h.s. The latter, however, is only possible for cν = 1 .

Example30 N = 899 , L = 10 : If, for instance, the projective measurement gives |267137i2L for the first 2L qubits, then we get31 c 22L

= [0, 3, 1, 12, 2, 1, 1, 1, 26, 1, 1, 22, 2] def

= 0+

1| 1 | 1| 1| 1| 1| 1 | 1| 1| 1 | 1| 1| + + + + + + + + + + + | 3 | 1 | 12 | 2 | 1 | 1 | 1 | 26 | 1 | 1 | 22 | 2

DRAFT, October 17, 2007

29

Thus (bν bν+1 + aν+1 ) Aν−1 + (aν bν+1 ) Aν−2 = bν+1 Aν + aν+1 Aν−1

and similarly for the Bµ . 30 Compare (Ros´e et al., 2004, Section III). 31 In Maple, after the command ”with(numtheory);” this result will be produced by the command ”convert(267137/220 , cfrag);” and the subconvergent [0, 3, 1, 12, 2, 1, 1, 1] can be evaluated by the command ”nthconver([0,3,1,12,2,1,1,1],7);”.

49

2.2. FACTORING LARGE INTEGERS and succeed with t = 7 : [0, 3, 1, 12, 2, 1, 1, 1] =

107 d = , 420 r

i.e. testing r = 420 , we get32 the factors gcd(11210 + 1, 899) = 29 ,

gcd(11210 − 1, 899) = 31 =

899 . 29

If, for instance, we unfortunately measure c = 801411 then we get 801411 = [0, 1, 3, 4, 7, 1, 80, 1, 1, 7, 6] 220 with the subconvergent 107 [0, 3, 4, 7, 1] = 140





321 = . 420

In this case, although the condition c d 1 2L − ≤ 2 r 2 r2

(but not gcd(d, r) = 1) is fulfilled (with d = 321) r = 420 will presumably not be detected and the whole procedure will be repeated, maybe with a different random value for x .

DRAFT, October 17, 2007

32

E.g., by the Maple commands ”gcd(112 10,899);” and ”gcd(112 10,899);”.

50

CHAPTER 2. QUANTUM ALGORITHMS

Chapter 3 Physical Realizations of Quantum Gates1 A set of necessary conditions to be fulfilled for the physical implementation of quantum computation is given by DiVincenzo’s checklist (DiVincenzo, 2000): 1. Qubits have to be well characterized and scalable.2 2. The standard states |0, . . . , 0i must be preparable.3 3. The duration of a gate operation must be much smaller than the decoherence time. 4. A universal set of gates must be implementable. 5. The qubits must be measurable in order to be able to ‘read out the result’ of a quantum computation. Remark: For quantum communication the following two requirements have to be added: • It must be possible to convert static qubits into flying qubits (typically photons). • It must be possible to protect flying qubits against decoherence.

3.1

Quantum Optical Implementations Optical systems currently constitute the only realistic proposal for long-distance quantum communication and underly implementations of quantum cryptography. (Knill et al., 2001) DRAFT, October 17, 2007

1

See the special volume Fortschr. Phys. 48 (2000) No. 9–11. 2 Thus, 1-photon realizations of n-qubit systems should be considered as qudits with d = 2n rather than n-qubit systems proper. 3 This is still difficult for n-photon systems.

51

52

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

3.1.1

Photons

In the Coulomb gauge the free electromagnetic field E(x, t) , B(x, t) is given in the form ∂ B(x, t) = curl A(x, t) , E(x, t) = − A(x, t) , ∂t 

A(x, t) = A(+) (x, t) + A(+) (x, t) by some complex vector potential A(+) (x, t) =

Z

 

∗



2 X

dk ǫj (k) fˇj (k) e−i(c|k|t−k·x) q , 2 |k| j=1

where, for every k 6= 0 , the vectors ǫj (k) form a right handed orthonormal basis

(

k ǫ1 (k), ǫ2 (k), |k|

)

and thus guarantee div A(+) = 0 . ucke, edyn). Remark: We use SI conventions; see Apendix A.3.3 of (L¨

In the Heisenberg picture of the quantized theory the classical fields E(x, t) and B(x, t) have to be replaced by corresponding observables on the state space Hfield , ˆ ˆ i.e. by operator-valued (generalized) functions E(x, t) and B(x, t) to be interpreted in the following way: If Φ ∈ Hfield is a sufficiently well-behaved (and kΦk = 1) then D



E

ˆ Φ E(x, t) Φ

resp.

D



E

ˆ Φ B(x, t) Φ

is the expectation value for E(x, t) resp. B(x, t) in the Heisenberg state (corresponding to) Φ . Up to unitary equivalence these observables are given by ˆ ˆ B(x, t) = curl A(x, t) ,

∂ ˆ ˆ E(x, t) = − A(x, t) , ∂t 

ˆ ˆ (+) (x, t) + A ˆ (+) (x, t) A(x, t) = A

†

,

53

3.1. QUANTUM OPTICAL IMPLEMENTATION where4 (+)

ˆ A

Z

q

def

(x, t) = (2π)−3/2 µ0 h ¯c

 



2 X

dk ǫj (k) a ˆj (k) e−i(c|k|t−k·x) q 2 |k| j=1

(3.1)

and the a ˆj (k) are annihilation operators5 fulfilling the commutation relations6 

†

[ˆ aj (k), a ˆj ′ (k′ ) ]− = δjj ′ δ(k − k′ )

[ˆ aj (k), a ˆj ′ (k′ )]− = 0 ,

(3.2)

on a suitable dense subspace D0 of the Hilbert space H containing a cyclic normalized vacuum state vector Ω characterized (up to a constant phase factor) by a ˆj (k) Ω = 0

(3.3)

(in the distributional sense). This also fixes the inner product on H . The operators aˆ of the form a ˆ=

X Z

∗



a ˆj (k) fˇj (k)

j=1,2

with7

h

a ˆ, a ˆ†

i

dk ,

−

fˇ1 , fˇ2 ∈ L2 (IR3 )

= ˆ1

(3.4)

(3.5)

characterize modes of the quantized electromagnetic field corresponding to the classical complex vector potentials (+)

A

(x, t)

= = =

(3.4),(3.1),(3.2)

*

(+)

ˆ A

(x, t)

†

+ † Ω a ˆ Ω + 

 ˆ (+) (x, t) , a Ω A ˆ† Ω −   Z 2 q X dk (2π)−3/2 µ0 h ¯ c  ǫj (k) fˇj (k) e−i(c|k|t−k·x) q *

j=1

. 2 |k| (3.6)

DRAFT, October 17, 2007

4

Thanks to the special choice of the factor in front of the integral we get the desired expression  Z  1 1 ˆ ˆ ˆ ˆ ˆ H= : B(x, t) · B(x, t) : dx ǫ0 : E(x, t) · E(x, t) : + 2 µ0

for the Hamilton operator, characterized (up to an additive constant) by i ∂ ˆ (+) i h ˆ ˆ (+) A (x, t) = H , A (x, t) . ∂t ¯h − 5 6

7

See (Mizrahi and Dodonov, 2002), however. Of course, the notation † includes the requirement  †  a ˆj (k) Φ Φ′ . hΦ | a ˆj (k) Φ′ i =

† ˆ Ω = 1 . Condition (3.5) is equivalent to a

54

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

With these modes the subspaces H(n) ⊂ H of n-photon state vectors may be defined recursively by def H(0) = {λ Ω : λ ∈ C} and

def

H(n+1) =

n

o

a ˆ† Φ(n) : Φ(n) ∈ H(n) , a ˆ mode

for n = 0, 1, 2, . . . .

Physical characterization of n-photon states: n (ideal) detectors but no more can be made fire by an incoming n-photon state.

Modes a ˆν , a ˆµ are called orthogonal, iff the states a ˆ†ν Ω , a ˆ†µ Ω are orthogonal, i.e. iff [ˆ aν , a ˆµ ]− = 0 . Fortunately, already classical electrodynamics tells us how photons are affected by passive linear optical components: The change of the mode a ˆ of a photon, caused by a passive linear optical component, is such that the corresponding complex vector potential changes as predicted by classical electrodynamics.

3.1.2

Photonic n-Qubit Systems

Single-Photon Realization For every n ∈ IN , using sufficiently many beam splitters a single-photon state aˆ† Ω can be changed into a coherent superposition8 X

b∈{0,1}n

λb a ˆ†b Ω

of (essentially) orthogonal 1-photon states def

|bi = a†b Ω which may be chosen as elements of the computational basis of a (simulated) n-qubit system. Remarks: 1. Since the single qubits of the 1-photon realization cannot exist independently from each other we should better call the system a qu2n it system. DRAFT, October 17, 2007

8

For the special case n = 3 an example is illustrated in Figure 3.1.

55

3.1. QUANTUM OPTICAL IMPLEMENTATION qqqqqqqqq

.......... .

√1 8

|0, 0, 0i

qqqqqqqqqq

.......... .

√1 8

|0, 0, 1i

qqqqqqqqq

.......... .

√1 8

|0, 1, 0i

qqqqqqqqqq

.......... .

√1 8

|0, 1, 1i

qqqqqqqqqq

.......... .

√1 8

|1, 0, 0i

qqqqqqqqqq

.......... .

√1 8

|1, 0, 1i

qqqqqqqqqq

...........

√1 8

|1, 1, 0i

qqqqqqqqqq

.......... .

√1 8

|1, 1, 1i

.. ..... ..... ..... ..... . . . . ..... ........... . .... ........ ..... ..... ..... ..... ..... ..... . . . . ... . . . . ... . . . . .... . . . .................... . ..... ......... ..... ..... ..... ..... . . . . ..... .... . ..... . ... . ... ..... ..... . . . . ..... ..... ... . . . ..... ........ . ... . . . . . . . . . . . ..... .......... . ..... ..... ..... ..... ..... ..... ..... ..... ..... .... . . ..... .. . . ... ....... . . . . ......... ... . . . . ..... . ..... ........ ..... ............ . ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . . ..... . . .................. . ..... ..... .. ... ..... ..... ......... ..... ..... ..... ..... ..... ..... . . . . ..... . ... . . ..... . . ... ..... . . . . ..... .. ..... ........ ..... ............. . ..... ..... ..... ..... ..... .... ..... ..... ..... ..... ..... ..... ......... . ..... ............. . ..... ..... ..... ..... ...

(3)

Figure 3.1: Preparation of Φ0 in a single-photon realization.

2. Cascades of beam splitters may also be used to implement approximate measurement of the number of photons:9 E.g., replace the single-photon input in Figure 3.1 by a 2-photon state and direct the output rays into separate ideal detectors. Then the probability that exactly two of these detectors ‘fire’ is 15/16 (≈ 94 %).

For such a choice of computational bases all unitary transformations can be (essentially) effected by linear optical components. Thanks to Theorem 1.2.1 it is sufficient to show this for n = 1 : Assume, for instance, that that |0i and |1i describe horizontally polarized (almost monochromatic) photons. Then the 1-qubit gates may be implemented by linear optical elements corresponding to Jones matrices Uˆ in the following way: λ1 |0i

..... ........................... ..... .....

♣♣♣♣♣ ♣♣

♣ ♣♣ ♣♣ ♣♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣ ♣♣♣♣♣

ˆ π ..... ........................... ..... .....♣♣.♣..♣....♣♣♣♣♣♣♣ D .. ♣♣ 2

.. ... . ... . .. ....... ......... ... ... . ... . ... . ... 2...... . ..... ..................... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

λ |1i

♣♣♣♣

♣♣♣♣♣ ♣♣♣♣♣ χ ...... ♣♣♣♣♣ ♣♣♣♣♣ .............. ♣♣♣

............ .........

λ′ |0i .............. .....1 ˆ . . . . . . . . . . . . . . . . . . .... D− π2

.. ...... ... ...

Uˆ

′ χ .......

.. ...........

♣ ♣♣♣ λ′2 |1i ♣♣ ♣♣ ..♣..♣. ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....................... ..... ♣ ♣ ♣♣♣ ♣♣ ♣♣♣

All unitary transformations of the polarization state of a photon (with almost sharp momentum) can be (almost accurately) performed by proper use of only λ/2-blades DRAFT, October 17, 2007

9

See (Bartlett et al., 2002) for details. See also (Haderka et al., 2003; Waks et al., 2003).

56

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

and polarization-dependent phase shifters.

General remark: Instead of performing the n-qubit transformations Uˆ(n) one may choose a fixed n-qubit transformation Vˆ(n) perform the n-qubit transformations Uˆ(n) Vˆ(n) and interprete the result w.r.t. the n o def new computational bases |bi′ = Vˆ(n) |bi : b ∈ {0, 1}n . Usually, in practice, this freedom is tacitly made use of. Exercise 13 Consider the single-photon realization of a 2-qubit system with: b a ˆI(0,0) =

b a ˆI(0,1) =

b a ˆI(1,0) =

b a ˆI(1,1) =

(

(

(

(

upper path and vertical polarization , upper path and horizontal polarization , lower path and vertical polarization , lower path and horizontal polarization .

a) Show that the CNOT gate may be implemented by placing a 90◦ polarization rotator into the lower path. b) Show that the TCNOT gate may be implemented by applying a polarization beam splitter — reflecting the vertically polarized components and transmitting the horizontally polarized components — in the following way: α00 |0, 0i + α01 |0, 1i

α10 |1, 0i + α11 |1, 1i

♣♣♣ qqqqqqqqqqq qqqq ♣♣♣ ♣♣♣ ♣ ♣♣♣♣♣♣♣♣♣ qqqqqqqqqqqqq ♣♣♣♣♣ ♣♣♣♣ ♣♣ ♣♣♣♣♣ ♣♣ ♣♣♣♣♣ ♣♣♣♣ ♣♣ ♣ ♣♣♣♣♣ ♣♣♣♣♣♣♣ ♣♣♣♣♣ ♣♣ ♣♣♣ ♣ ♣ ♣♣♣♣♣ ♣ ♣♣ ♣♣♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ qqq ♣♣ ♣ qqqqqqqqqqqqqq q qqqqqqqqqqq ♣♣♣♣♣♣♣ ♣♣♣ ♣♣

.......... .. ..... ... ....... ..... ..... ..... ..... ..... ..... . . . ..... . ..... ..... ..... ..... ..... ..... ..... ........ ........ ........ ..... ......... ..... ..... ..... ..... ..... ..... . . . . ..... .... . . ..... . ... ..... . . . . ..... .......... ....... .. .......

.......... .......

α00 |0, 0i + α11 |0, 1i

.......... .......

α10 |1, 0i + α01 |1, 1i .

The single-photon ‘realization’ has a serious disadvantage spoiling the eventual speed-up of quantum computation: The number of optical devices required for the single-photon simulation of n-qubit systems grows exponentially with n (Cerf et al., 1998; Kwiat et al., 2000). Therefore, we will consider only many-photon realizations in the following.

57

3.1. QUANTUM OPTICAL IMPLEMENTATION Multi-Photon Realization For given n ∈ IN we may choose a fixed10 set n

a ˆν,j : ν ∈ {1, . . . , n} , j ∈ {0, 1}

o

of 2n pairwise orthogonal modes and consider the n-photon states def

|bi = a ˆ†1,b1 · · · a ˆ†n,bn Ω ∀ b ∈ {0, 1}n as elements of the computational basis of a true n-qubit system.11 Here, while the 1-qubit gates may still be easily implemented by linear optical components, the physical realization of universal 2-qubit gates is quite a technological challenge.12 Fock Realization Another possibility is to choose some fixed set {ˆ a1 , . . . , a ˆn } of pairwise orthogonal modes and consider the states |bi = |biF

∀ b ∈ {0, 1}n ,

(3.7)

 ν 1  ν n 1 a ˆ†1 · · · a ˆ†n Ω ∀ ν1 , . . . , νn ∈ ZZ+ , ν1 ! · · · ν n !

(3.8)

where def |ν1 , . . . , νn iF = √

as elements of the computational basis of the n-qubit system. The states a ˆ†b Ω of the single-photon realization for n-qubit systems form the subset n o |biF : b1 + . . . + bn = 1

of the computational basis of the Fock realization for 2n -qubit systems using the 2n modes a ˆI(b)+1 = a ˆb , b ∈ {0, 1}n .

Obviously, for n=1 the Fock realization does not coincide with the single-photon realization. Actually, 1-qubit gates like the Hadamard gate cannot be implemented by linear optical components, since |0iF represents the vacuum state. Nevertheless, the Fock realization has certain advantages as, e.g., those to be discussed in 3.1.4. DRAFT, October 17, 2007

10

Recall the above general remark, however. 11 Of course, for n = 1 the n-photon realization coincides with the single-photon realization. 12 Concerning recent progress in detector technologies see (Rosenberg et al., 2005).

58

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

3.1.3

Nonlinear Optics Quantum Gates

For both the n-photon and the Fock realization of n-qubit systems the (universal) CPHASE gate Λ1 (Sˆπ ) s

s

≡

π

❤

H

H

is nonlinear in the sense that — contrary to the action of linear optics components — the modes are are not transformed independently of each other. If a non-linear sign gate NS is available,13 i.e. an optical one-way gate acting according to14 



α (ˆ a∗ )0 + β (ˆ a∗ )1 + γ (ˆ a∗ )2 Ω .......................... NS

.........................





α (ˆ a∗ )0 + β (ˆ a∗ )1 − γ (ˆ a∗ )2 Ω

then the CPHASE gate can be easily implemented by proper use of linear optical components like the Hadamard beam splitters (with deflecting mirrors) characterized in Figure 3.2. |0i......

..... ............. ......... .

√1

..... ............

.... ....... ........

. .... ..... . ..... . . . . ..... .... . .......................... .......................... ..... .......................... .......................... ..... ..... ..... ........ ...... .... .....

|0i

2 ..... ..................... ..... .....

.... ....... ........

.....

√1

|1i |1i

......... ..................2..... ...

.. ..... .................. ........

. .... ..... . . . . . .. .... .......................... .......................... .......................... ........ .......................... ..... . .... ..... ..... ..... ..... ...... ............. ......... ... .....

+ √12 |0i

..... ..................... ..... .....

− √1 |1i

..... 2 ..... ..................... .....

Figure 3.2: Hadamard beam splitter. One such implementation,15 suggested in (Ralph et al., 2002), is sketched in Figure 3.3. The essential idea is to exploiting two-photon interference (see Section 5.2.3 of (L¨ ucke, nlqo)) at two Hadamard beam splitters forming a balanced MachZehnder interferometer as sketched in Figure 3.4.

One would like to realize the necessary NS gates by means of optical nonlinearities. Unfortunately, sufficiently strong nonlinearities of crystals are accompanied DRAFT, October 17, 2007

13

See (Sanaka et al., 2003) for an experimental realization. The gate is non-linear in the sense that its action cannot be reduced to a linear transformation of the modes a ˆ. 15 The modes a ˆν,µ are assumed to differ only by vertical translation and to describe photons with (almost) sharp momenta. 14

59

3.1. QUANTUM OPTICAL IMPLEMENTATION a ˆ.....1,0 .................... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ................... .........

.........

a ˆ1,1 .. ....

NS

......... ..... .......... ...... ..... ..... .......... ......... ..... ..... ............ ...... ..... ..... .... ..... ..... . ....................................... .. ....................................... ....................................... ....................................... ....................................... ..... ..... . ....................................... .... ..... 2,1 .......... ..... ..... ................... ..... . . . . . . . . . . . ..... .......... ..... ..... ..... ..... ........

a ˆ

NS

..... ..... ........................ ..... ..........

......... .......... ..... ..... ................ ..... ..... .... ..... ..... . .. ....................................... ....................................... ....................................... ....................................... ....................................... ..... ..... . ....................................... .... ..... . .......... ..... ..... ................. ..... . . . . . . . . . . . . . . ..... ..... ............. ..... ..... ........

a ˆ2,0 ..........

..... .................... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .........................

Figure 3.3: Optical implementation of a CPHASE gate. α |0i ......

..... ............... ...... .

β |1i ...... ... ..... ............... ..

..... ............ .

.....

.....

.....

.....

.....

.....

..........

..... ..... ..... ..... .......................... .......................... .. .......................... .......................... .... .... ..... . .... ..... ..... ..... . . . . . ..... ........... ..... .....

.......

.....

.....

.....

.... ....... ........

α |0i

..... ........ ...... .. ..... .....

..... ..... ..... ..... ..... ..... .......................... ......................... .. .......................... ......................... .... .... ..... . .... ..... ..... ..... . . . . . ......... ..... ...... .... ..... .....

β |1i

......... ................ .. ...

Figure 3.4: Action of a balanced Mach-Zehnder interferometer.

by too strong absorption and therefore are not suitable. A way out may eventually be provided by electromagnetically induced transparency16 (EIT), discussed in Section 8.3.2 of (L¨ ucke, nlqo). Using beam splitters characterized by |0i ..... ..... ................. ...... ......

....

.....

...... ............ ..

.....

.....

. ....

.....

.....

ϑ

..... ..... . .... ..... .... . ..... ..... ..... ..... .....

... ........... .....

.....

cos ϑ |0i

................ ..... ..... ..... ..... .......

. ....

ϕ

.....

e−i ϕ sin ϑ |1i |1i ..... ..... ................. .......

.. ....................... ..... ..... .....

......

.....

..... ..... . .... . .... ......... ..... ..... . ..... . ... ..... ..... ..... ..... ..... .. .... . . . . . . . . ......... ................. .... .. ..... ..... .....

ϑ

..... ................. .... .

... ........... .....

−e+i ϕ sin ϑ |0i

................ ..... ..... ..... ..... .......

ϕ

cos ϑ |1i

. ....................... ..... ..... .....

indeterministic17 NSx -gates, i.e. gates acting according to 



α ˆ1 + β a ˆ† + γ a ˆ†

2 

Ω

7−→





α ˆ1 + β a ˆ† + x γ a ˆ†

2 

Ω

DRAFT, October 17, 2007

16

See (Ottaviani et al., 2005) and concluding discussion of (Munro et al., 2005a). See also (Munro et al., 2005b) concerning the use of ‘weak’ cross Kerr nonlinearities. 17 A gate is called indeterministic if it acts correctly with nonzero probability < 1 not depending on the input state and if, after its action, it is known whether the action was correct or not. An indeterministic gate is called near-deterministic if its probability of success is near to 1.

60

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES NSx ..... ......................... ..... ..... ..... ...

ϕ4 ..... ..... ..... ......................... ..... ..... ..... . ..... .

....

′ †

.....

.....

....... ......... ......

(ˆ a ...)....... Ω . ..........

..... ..... ..... ..... ...... ................. ..... ..... ..... ..... ..... ..... . ..... .... ..... . .... ..... . ..... ..... . .... ........ ... 1 ..... ..... .....

ϑ

.....

.....

.....

.....

.....

. ....

. ....

..... .... ............ ..... .

.... ......... . . 2 . ..... ..... .. 2 ... .... ..... ..... . . . . . .. ... . .... ................ .......... ..... .. .......

ϑ

.....

.....

.....

.. ....... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ...................... ..

ϕ

. ..... ..... ..... .................. . ..... .. 1 . ... ....

D

.....

.....

.....

.....

. ...... ............... .

..... ..... ..... ..... . 3 ..... .... ..... ... 1 3 .. . ..... .... ..... ..... ..... ... ..... ...... . ............... ............... ..... . ...... .. ........ ..... ..... ........ . .... ...... . ..... ..... .... ..... ..... ..... .................. ..... 2 . ..... ..... ..

ϑ

ϕ

.....

.....

ϕ

D

Figure 3.5: An indeterministic NSx -gate

if successful, can be implemented18 as sketched in Figure 3.5. According to (Knill et al., 2001, Fig. 1) these gates act successfully iff a single-photon state is detected by D1 and the vacuum state is detected by D2 . For 

ϑ1





◦

+22.5



     ϑ2  =  65.5302◦ 

ϑ3

−22.5◦









0 ϕ1  0  ϕ     2  =   0   ϕ3  180◦ ϕ4

and

we have x=-1 and the probability of success is 0.25. For 

ϑ1





◦

+36, 53



     ϑ2  =  62.25◦ 

ϑ3

◦

−36.53

and









88, 24◦ ϕ1  −66, 52◦  ϕ     2  =   −11, 25◦   ϕ3  102, 24◦ ϕ4

we have x=i and the probability of success is 0.18082.

3.1.4

Linear Optics Quantum Gates19

Indeterministic Gates Fortunately, using photonic memory and quantum teleportation, (deterministic) CPHASE gates may by implemented using indeterministic ones: DRAFT, October 17, 2007

18

See also (Rudolph and Pan, 2001; Ralph et al., 2002; Hofmann and Takeuchi, 2002) and especially (Gilchrist and Milburn, 2002) for other possibilities. 19 See (Dowling et al., 2004).

61

3.1. QUANTUM OPTICAL IMPLEMENTATION

The essential observation, due to (Gottesman and Chuang, 1999), is the equivalence

Φ+

Φ+

s

B





⌢ /

s

⌢ /

❤

π

s

❤

π

π

s

s

≡

(3.9)

,

π

⌢ /

B

s

⌢ /

where s

≡

B

H

❤

is the inverse of the Bell network and Φ+ the state defined in 1.2.2. This equivalence is obvious from the discussion of quantum teleportation in 1.2.2 and, thanks to s

s

❤

s

s

≡

π

❤

π

π

implies the equivalence20

Φ+

Φ+

 

s

B

s

s

π ❤

π

❤ π

⌢ / π

≡

π

s B

⌢ / s π

.

⌢ / s

⌢ /

Therefore: If the auxiliary 4-qubit state (4) def

ΦB

=





ˆ1 ⊗ CPHASE ⊗ ˆ1 (Φ+ ⊗ Φ+ )

is available then the (deterministic) CNOT gate can be replaced by appropriately modified double quantum teleportation. DRAFT, October 17, 2007

20

See also (Brukner et al., 2003) in this connection.

62

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES (4)

If ΦB can be stored for later use21 then it is sufficient to have a nondeter(4) ministic gate producing ΦB with nonzero probability from standard input. One such possibility, obviously, is to replace the deterministic CNOT gates in the above (4) characterization of ΦB by nondeterministic ones, if the latter are available. In principle (Kim et al., 2001) , (deterministic) teleportation is possible by exploiting sum frequency generation of both type I and type II for a complete Bell measurement (see, e.g., Section 3.2.2 of (L¨ ucke, nlqo) for the explanation of sum frequency generation). However, also indeterministic teleportation by only partial Bell measurement — as, e.g. sketched in Figure 3.6 and explained by Exercise 14 — may be useful for improving the probability of success, at least , of an indeterministic CPHASE gate. ..... ........................ ..... ..... .

.....

.....

. ....

...... ..... ..... ................. ..... . ..... ..... ......

..... ..... ... ..... .. .. ....... ................................. .................................. . . ................................. .................................. ... ... ................................. .. ..... ..... ..... ..... . . . .. ...... ..... . ..... ........................ ..... .... ..... ..... ................. ..... . .... ......

Figure 3.6: Partial Bell measurement.

Exercise 14 Consider a Hadamard beam splitter for the orthogonal modes aˆ1 a ˆ2 , i.a a beam splitter acting as 1 a1 + a ˆ2 ) , a ˆ1 7→ √ (ˆ 2

1 a ˆ2 7→ √ (ˆ a1 − a ˆ2 ) . 2

Show that the beam splitter, applied to the Bell states  1  Φ± = √ Ω ± a ˆ†1 a ˆ†2 Ω , 2

 1  † Ψ± = √ a ˆ1 ± a ˆ†2 Ω 2

in the corresponding Fock representation as sketched in Figure 3.6, acts as follows:

Ψ01

 1  1 |2, 0iF − |0, 2iF , Φ± 7−→ √ |0, 0iF ± 2 2 F = Ψ+ 7−→ |1, 0i ,

Ψ11 = Ψ− 7−→ |0, 1iF .

According to (3.9) and Exercise 14 a so-called CZ 1 gate, i.e. an indeterministic 4 CPHASE gate working with probability of success 1/4 , can be implemented in the Fock realization as sketched in Figure 3.7 This gate succeeds if the (photon number

63

3.1. QUANTUM OPTICAL IMPLEMENTATION ..... ..... ........................ ..... ..... ..... ..... ..... ..... .... .

Ψ01

Ψ01

(

D1

.....

.....

....... .. ..... ...... .... .... .... .... .... .... .... ..................... . .... .... ...... ... . ....

..... .... ... ..... .. ........... . ....................................... . . . ....................................... . . ....................................... .. ... ....................................... ....................................... ....................................... .. ..... ... .... ..... .... . . . . . .... . . ...... .... ... 2 ...... .... .... .... .... .... .... .... .................... . .... ..... ......................... ..... ..... ..... ..... ..... .. . . ....... . . . . . ..... ........................ ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....................... .....

D

✉

✉

( ..... .......................... π

π

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

..... ......................... ..... ..... ..... .....

.....

π

..... ..... ..... ..... ..... ..... ........................ .....

. . . . . . ..... ................ ......... .... ..................... . ..... . . . . . . . . . . . . .. ....... ..... .. . . .. .......

✉

D2′

.... ... ..... .... . . . ..... . ....................................... ....................................... ....... ... ....................................... ....................................... ....................................... . ....... ... .. ..... ... .... ..... .... ..... .... ...... .... ..... . . . . ...... .... .... .... .... .... .... .... ...................... . .... . . . . . . . . ..... ..... ................ ..... ..... ..... ..... ..... ..... ...... .. ....... .....

D1′

Figure 3.7: Implementation of a CZ 1 gate 4

resolving) detectors D1 , . . . , D2′ indicate that the common state of the first two qubits as well as that of the last two qubits is a single-photon state.22 The ancillary states |0, 1i + |1, 0i a ˆ†1 + a ˆ†2 √ Ψ01 = = √ Ω 2 2 may be easily prepared using a single-photon source and a Hadamard beam splitter:23 a ˆ†1 Ω.......... . .... .... .................... .... ...  ..... ... ........ ..... .... ..... ..... ... . ..... . . . ..... .... ..... ..... ... ..... .... ........ ....................................... ....................................... ....................................... . ... ....................................... ....................................... ....................................... . ..... . ... . .... . .... . . .... . .... . . ...... .... ..................... .... ... . . . . . . . . . .. ...... ........

Ω

         

Ψ01

Near-Deterministic Quantum Gates The construction sketched in Figure 3.7 can be generalized in the following way using 4n ancillary qubits24 instead of only 4 (Knill et al., 2000; Knill et al., 2001): DRAFT, October 17, 2007

21

See (Pittman and Franson, 2002) in this connection. 22 The dotted vertical lines indicate that the effect of the corresponding CPHASE gates on the output qubits can be achieved via LOCC (Local Operations and Classical Communication). 23 Note that the single-photon state Ψ01 is entangled when interpreted as Fock realization of a 2-qubit state. 24 The case n = 2 is of special interest (Nielsen, 2004).

64

CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

Let a ˆ0 , . . . , a ˆ2n be pairwise orthonormal modes describing photons with (almost) sharp momenta and prepare the first 2n ancillary qubits in the state (n)

Φtele







j 2n n Y X Y 1 def  = √ a ˆ†µ  Ω , a ˆ†ν  n + 1 j=0 ν=1 µ=n+j+1

(instead of Ψ01 ). Let Fˆ(n) be the linear operator on Haˆ0 ,...,ˆa2n characterized by Fˆ(n) Ω = Ω and −1 Fˆ(n) a ˆ Fˆ(n) =

(

a ˆ √1 n+1

Pn

j=0

−i k

e

2π n+1

j

a ˆj

for a ˆ ⊥ {ˆ a0 , . . . , a ˆn } , for a ˆ=a ˆk , k ∈ {0, . . . , n} ,

(3.10)

where — as in Section 1.2.3 of (L¨ ucke, nlqo) — we denote by Haˆ0 ,...,ˆa2n the smallest closed subspace of Hfield that contains Ω and is invariant under aˆ†0 , . . . , a ˆ†2n . Then, as explained in connection with the single-photon simulation of n-qubit systems, the transformation −1 a ˆν 7→ Fˆ(n) a ˆν Fˆ(n) ∀ ν ∈ {0, . . . , 2n} (3.11)

can be implemented using by linear optics. Remark: For n + 1 = 2m and25

|kim = a ˆ†k Ω ∀ k ∈ {0, . . . , 2m − 1} the corresponding state transformation |kim

7−→

1 Fˆ(n) |kim = √ n+1

n X j=0

2π

e+i k n+1 j |jim

∀ k ∈ {0, . . . , 2m − 1}

simulates the m-qubit quantum Fourier transform.

Exercise 15 Using the Campbell-Hausdorff formula26 ˆ ˆ −A ˆ ˆ, eA Be = exp(adAˆ ) B

where

(3.12)

def ˆ ˆ adAˆ Cˆ = [A, C]− ,

show the following: a) †

†

e−i aˆj aˆj ϕ a ˆk e+i aˆj aˆj ϕ = e+i ϕ δj,k a ˆk

∀ j, j ∈ {0, . . . , 2n} , ϕ ∈ IR .

b) −i(a ˆ†1 a ˆ2 +ˆ a†2 a ˆ1 )θ

+i(a ˆ†1 a ˆ2 +ˆ a†2 a ˆ1 )θ

a ˆ1 e e † † −i(a ˆ†1 a ˆ2 +ˆ a†2 a ˆ1 )θ a ˆ2 e+i(aˆ1 aˆ2 +ˆa2 aˆ1 )θ e DRAFT, October 17, 2007

25

Recall (1.12). 26 Compare Footnote 50.

 = cos θ a ˆ1 + i sin θ a ˆ2  = i sin θ a ˆ + cos θ a ˆ  1

2

∀ θ ∈ IR .

65

3.1. QUANTUM OPTICAL IMPLEMENTATION c) Fˆ(n) is a unitary operator on Haˆ0 ,...,ˆa2n . Since j  Y

−1 Fˆ(n) a ˆν Fˆ(n)

ν=1





j Y

= (3.10)



n X

e+i ν

ν=1 lν =0 j n X Y

=



l1 ,...,lj =0 ν=1

2π l n+1 ν

2π



a ˆ†lν 

e+i ν n+1 lν a ˆ†lν



∀ j ∈ {1, . . . , n}

and j  Y

−1 Fˆ(n) a ˆν Fˆ(n)

ν=0



j Y

n X

=

2π

ˆ†lν e+i ν n+1 lν a

l0 ,...,lj =0 ν=0 j+1 n X Y

=

2π

l1 ,...,lj+1 =0 ν=1 j+1 n X Y −i

e

=

l0′ ,...,lj′ =0

2π

e−i n+1 lν e+i ν n+1 lν a ˆ†lν 2π ′ l n+1 ν

ν=1

2π

′

e+i ν n+1 lν a ˆ†lν′

∀ j ∈ {0, . . . , n − 1} ∀ j ∈ {0, . . . , n − 1} ,

we get 



(n) Fˆ(n) α ˆ1 + β a ˆ†0 Φtele

√

=

+α



1 α n+1 n X

 Y 2n

ν=n+1

N0 ,...,Nn =0 0 1/3 (4.15) cannot be positive, since, e.g. 

    ¯ )(W ˆ λ ) |0, 0i + |1, 1i = 1 − 3λ |0, 0i + |1, 1i . (1 ⊗ T 4

DRAFT, October 17, 2007

25

Note that, for ρˆ′ ∈ S(H1 ) , transposition is equivalent to complex conjugation ρˆ′ =

n1 X

j,k=1 26 27

n1  ED ∗ ED X (1) (1) (1) (1) φk . ρj k φj φk 7−→ ρj k φj j,k=1

Recall (4.9). These states are distinguished by their invariance property 

  † ˆ ⊗U ˆ W ˆ ⊗U ˆ =W ˆλ U ˆλ U

ˆ ∈ L(H) for all unitary U

(Werner, 1989, Section II). Note that the flip operator |0, 0ih0, 0| + |1, 0ih0, 1| + |0, 1ih1, 0| + |1, 1ih1, 1| coincides with ˆ 1⊗ˆ 1 − 2 |Ψ− ihΨ− | .

(4.15)

101

4.2. QUANTUM CHANNELS

4.2.2

Quantum Noise and Error Correction

The action of noisy quantum channels corresponds to non-invertible quantum operations C . Nevertheless such an operation may be become invertible by restriction to states with support on a suitable subspace C of H — and thus allow for error correction on the code space C . Theorem 4.2.4 Let H be Hilbert space, C ∈ Q(H, H) and let C be a linear subspace of H . Then the following three statements are equivalent: ˆ 1, . . . , K ˆ N for C and a1 , . . . , aN ≥ 0 with 1. There are Kraus operators K ˆ j† K ˆ k PˆC = δjk aj PˆC PˆC K

∀ j, k ∈ {1, . . . , N } .

(4.16)

2. There is a trace preserving quantum operation R with 



(R ◦ C) |ψihψ| ∝ |ψihψ|

∀ ψ ∈ C kψk = 1 .

(4.17)

ˆ 1′ , . . . , K ˆ N′ for C with 3. There are Kraus operators K ˆ j′ † K ˆ k′ PˆC = ajk PˆC PˆC K

∀ j, k ∈ {1, . . . , N }

(4.18)

for some self-adjoint matrix (ajk ) . Outline of proof: Assume the first statement to be true. Then, using the polar decomposition q ˆ j PˆC = U ˆj + PˆC K ˆj† K ˆ j PˆC K =

√ +

ˆj PˆC aj U

(4.19)

ucke, eine)) we have (see, e.g., Lemma 7.3.20 of (L¨

and hence For

R(ˆρ) def =

N  X j=1

where

ˆ† U ˆ ˆ δjk PˆC = PˆC U j k PC

(4.20)

PˆUˆj C PˆUˆk C = δjk PˆUˆj C .

(4.21)

  † ˆ † Pˆ ˆ ˆ † Pˆ ˆ ˆ U U + Pˆ0 ρˆ Pˆ0 j Uj C ρ j Uj C def Pˆ0 = ˆ1 −

and thus

  Pˆ0 C PˆC ρˆ PˆC Pˆ0

= (4.19) = (4.21)

N X j=1

N X j=1

∀ ρˆ ∈ S(H) ,

PˆUˆj C

ˆj PˆC ρˆ PˆC U ˆ † Pˆ0 aj Pˆ0 U j

0 ∀ ρˆ ∈ S(H) ,

(4.22)

(4.23)

102

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION this gives28 N   X ˆ † Pˆ ˆ (R ◦ C) |ψihψ| = PˆC U j Uj C j,k=1

|

=

(

4.21)

N X aj |ψihψ| = (4.20) j=1

Since

N  X j=1

ˆ † Pˆ ˆ U j Uj C

† 

this implies the second statement.29

=

δjk

√

+

4.19) {z

(

ˆ k † Pˆ ˆ U ˆ ˆ hψ| K Uj C j PC (4.24)

ˆ k |ψi K | {z } √

+

ˆk |ψi ak U

ˆk |ψi ak U

}

∀ψ ∈ C .

 ˆ † Pˆ ˆ ˆ† ˆ U j Uj C + P 0 P 0

= ˆ1 ,

(4.21)

Now assume statement to be    the second must be constant for true. Then (4.17) holds and, of course,30 trace C |ψihψ| normalized ψ ∈ C . Therefore, 
 R ◦ C PˆC ρˆ PˆC = γ PˆC ρˆ PˆC

∀ ρˆ ∈ S(H)

(4.25)

ˆ′,...,K ˆ ′ are Kraus operators for C and R ˆ1, . . . , R ˆ N are holds for some γ ≥ 0 . If K 1 N Kraus operators for R then (4.25) is equivalent to N X

j,k=1

ˆj K ˆ ′ PˆC ρˆ PˆC K ˆ′† R ˆ † = γ PˆC ρˆ PˆC R k j k

∀ ρˆ ∈ S(H) .

Then,31 by Theorem 4.2.3, there are complex numbers λjk with ˆj K ˆ ′ PˆC = λjk PˆC R k

∀ j, k ∈ {1, . . . , N }

and hence ∗ ˆ ′† R ˆ† R ˆ ˆ′ ˆ ˆ PˆC K j j Kk PC = λjl λjk PC l

∀ j, k, l ∈ {1, . . . , N } .

Since R is trace preserving, this implies the third statement with alk =

N X j=1

λ∗jl λjk

∀ k, l ∈ {1, . . . , N } .

Finally, assume the third statement to be true. Then, by the spectral theorem, there are a unitary matrix (ujk ) and real numbers a1 , . . . , aN with N X

u∗j ′ j aj ′ k′ uk′ k = aj δj,k

j ′ ,k′ =1

∀ j, k ∈ {1, . . . , N } .

DRAFT, October 17, 2007

ˆ † Pˆ ˆ . ˆ † Pˆ ˆ = U Note that PˆC U j Uj C j Uj C Recall Remark 1 to Definition 4.2.1. 30 Note that for ρˆ1 , ρˆ2 ∈ S(H) and λ1 , λ2 ∈ C we have  λ1 ρˆ1 + λ2 ρˆ2 ∝ ρˆ1 + ρˆ2 =⇒ λ1 = λ2 . ρˆ1 6= ρˆ2

28

29

31

We may add N − 1 zeros as Kraus operators to

√ ˆ γ PC .

103

4.3. ERROR CORRECTING CODES This, together with (4.18), implies the first statement with ˆj = K

N X

ˆ k′ ukj K

k=1

∀ j ∈ {1, . . . , N } .

Remark: (4.19), (4.21), and (4.22) show how errors produced by C can be corrected for the code C : Perform a projective measurement w.r.t. the orthogonal subspaces Uˆj C and apply Uˆj† according to the result of this ‘measurement’. ˆ 1, . . . , Corollary 4.2.5 Let C be a linear subspace of the Hilbert space H and let K ˆ N resp. K ˆ 1′ , . . . , K ˆ N′ be Kraus operators for C ∈ Q(H, H) resp. C′ ∈ Q(H, H) . If K ˆ j′ are complex linear combinations of the K ˆ j then, with R (4.16) holds and if the K as constructed in the proof of Theorem 4.2.4, (4.17) holds also for C replaced by C′ . Outline of proof: Assume that ˆ′ = K j

N X

ˆk λjk K

k=1

and let R be defined as in the proof of Theorem 4.2.4. Then   (R ◦ C′ ) |ψihψ| N  †   X √ + + ˆ † Pˆ ˆ ˆl PˆC |ψihψ| λ∗kr √ ˆC U ˆr† PˆC U ˆ † Pˆ ˆ = λ a U a P PˆC U kl k r j Uj C j Uj C (4.24) j,k,l,r=1 = (4.21)

4.3 4.3.1

N X

j,k=1

2

aj |λkl | |ψihψ|

∀ψ ∈ C .

Error Correcting Codes General Apects

ucke, 1996) the time evolution of closed32 According to standard formulations (L¨ quantum mechanical systems is unitary. Let us consider the closed system of one qubit together with its environment. A unitary transformation of the corresponding state space maps separated pure states to entangled pure states: 



|χi = α |0i + β |1i ⊗ |Ei 





7−→ |χ′ i = α |0i ⊗ |E0,0 i + |1i ⊗ |E0,1 i + β |0i ⊗ |E1,0 i + |1i ⊗ |E1,1 i DRAFT, October 17, 2007

32

For open quantum systems see (Alicki, 2003) and references given there.



(4.26)

104

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

(∀ α, β ∈ C). This way the originally pure partial state33 

hχ| Aˆ ⊗ ˆ1 |χi = |α|2 h0| Aˆ |0i + |β|2 h1| Aˆ |1i + 2 ℜ α β h0| Aˆ |1i may become a mixture:34 hχ′ | Aˆ ⊗ ˆ1 |χ′ i = |α|2 h0| Aˆ |0i + |β|2 h1| Aˆ |1i



∀ Aˆ ∈ Lsa (C2 )

if 35 hEj,k | Elm i = δjl δk0 δm0 .

In other words: The environment may cause decoherence. Therefore, we have to be able to undo unwanted changes caused by the environment. Now, for arbitrary vectors |Er i from the state space of the environment we have 3  X

r=0















σ ˆr |0i ⊗ |Er i = |0i ⊗ |E0 i + |E3 i + |1i ⊗ |E1 i + i |E2 i

(4.27)

and 3  X

r=0







σ ˆr |1i ⊗ |Er i = |0i ⊗ |E1 i − i |E2 i + |1i ⊗ |E0 i − |E3 i ,

(4.28)

where the σ ˆr correspond to the Pauli matrices:36 def

σ0 =



1 0 0 1



def

,

σ1 =



0 1 1 0



,

def

σ2 =



0 +i

−i 0



,

def

σ3 =



1 0 0 −1



.

(4.29)

Equations (4.26)–(4.28) imply |χ′ i = if

3  X

r=0



σr α |0i + β |1i

|E0,0 i = |E0 i + |E3 i ,

i.e. if



⊗ |Er i

(4.30)

|E0,1 i = |E1 i + i |E2 i ,

|E1,0 i = |E1 i − i |E2 i , |E1,1 i = |E0 i − |E3 i ,

|E0,0 i + |E1,1 i |E0,1 i + |E1,0 i , |E1 i = , 2 2 |E0,0 i − |E1,1 i |E0,1 i − |E1,0 i , |E3 i = . |E2 i = 2i 2 |E0 i =

DRAFT, October 17, 2007

33

2

2

We assume that |α| + |β| = 1 and hE | Ei = 1 . 34 This is why open quantum quantum mechanical systems (Davies, 1976) do not evolve unitarily. 35 Consider, e.g., the simple example |Ei = |0i , |χ′ i = CNOT |χi. 36 Hence σ ˆ0 = ˆ 1, σ ˆ1 = ¬ ˆ, σ ˆ3 = Sˆπ , σ ˆ2 = i σ ˆ1 σ ˆ3 .

105

4.3. ERROR CORRECTING CODES

(4.30) tells us that there are only three types of errors to be corrected, corresponding to37 σ ˆ1 , σ ˆ2 , σ ˆ3 . In this sense the set of possible errors for single-qubit systems is discrete. More generally, a unitary operation of the state space of an n-qubit system and its environment acts according to38 X

X

λb |bi ⊗ |Ei 7−→



X

λb

E

|b′ i ⊗ Eb,b′ ,

(4.31)

b∈{0,1}n b∈{0,1}n b′ ∈{0,1}n E where the Eb,b′ are suitable state vectors of the environment depending on |Ei

(and b, b′ , of course), but not on the λb . Now, from (4.27)/(4.28) we see that for arbitrary states |Eb,b′ i of the environment there are vectors |Er i with 3  X

r=0



σ ˆr |bi ⊗ |Er i =

X

b′ ∈{0,1}

|b′ i ⊗ |Eb,b′ i

∀ b ∈ {0, 1} .

E Straightforward induction shows that for arbitrary state vectors Eb,b′ there are corresponding state vectors |Er (b)i ; r ∈ {0, . . . , 3}n ; with X

r∈{0,1,2,3}

n





σ ˆr |bi ⊗ |Er (b)i =

where



X

|b′ i ⊗ Eb,b′

b′ ∈{0,1}n

def

E

∀ b ∈ {0, 1}n ,

∀ r ∈ {0, . . . , 3}n .

σ ˆr = σ ˆ r1 ⊗ . . . ⊗ σ ˆ rn

Together with (4.31) this shows that every unitary action on an n-qubit system39 and its environment is of the form X

b∈{0,1}n

λb |bi ⊗ |Ei 7−→

X

λb

b∈{0,1}n

X

r∈{0,1,2,3}n





σ ˆr |bi ⊗ |Er (b)i ,

∀ b ∈ {0, 1}n .

Usually, in the theory of quantum error correction, only the case |Er (b)i = |Er i

∀ b ∈ {0, 1}n , r ∈ {0, 1, 2, 3}n

DRAFT, October 17, 2007

37

Of course, the σ ˆν are not the only set of operators serving this purpose: def σ ˆr′ =

3 X s=0

=⇒

3 X r=0

38

urs σ ˆs ,

def |Er′ i =

3 X s=0

u∗rs

|Es i ,

3 X

σ ˆr′ |bi ⊗ |Er′ i σ ˆr |bi ⊗ |Er i = r=0

3 X

urq u∗qs = δrs

q=0

∀ b ∈ {0, 1} .

This is a simple consequence of linearity and the fact that the |bi form a basis of the n-qubit state space. 39 We assume that the qubits are not destroyed. For instance, if a qubit is identified with an atom in a superposition of its ground state and its first excited state, then exciting a higher level destroys this qubit.

106

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

is considered.40 Then the error action is of the form Ψ ⊗ |Ei 7−→

X

r∈{0,1,2,3}n





σ ˆr Ψ ⊗ |Er i ,

X

Ψ=

b∈{0,1}n

λb |bi ,

(4.32)

and error correction for such quantum noise should be possible along the lines indicated below. Remark: Alternatively, (4.32) may be written in the form

E

(4.33)

ˆb def X = (δ0b1 + δ1b1 σ ˆ1 ) ⊗ . . . ⊗ (δ0bn + δ1bn σ ˆ1 ) , def Zˆb = (δ0b1 + δ1b1 σ ˆ3 ) ⊗ . . . ⊗ (δ0bn + δ1bn σ ˆ3 ) .

(4.34)

Ψ ⊗ |Ei 7−→

X

a,b∈{0,1}

where:

n





ˆa Zˆb Ψ ⊗ Ea,b , X

To explain the essential idea of quantum error correction, let us assume that also for multi-qubits systems only one-qubit errors corresponding to σ ˆ3 (phase errors) occur. In order to conserve an unknown one-qubit state (disentangled from the environment) we first of all encode Ψ = α |0i + β |1i for arbitrary α, β ∈ C into the (pure) three-qubit state ˆ = α |wˆ0 i + β |wˆ1 i , Ψ where def

|w ˆ0 i =

def

|w ˆ1 i =

 1  √ |0, 0, 0i + |0, 1, 1i + |1, 0, 1i + |1, 1, 0i 4  1  √ |1, 1, 1i + |1, 0, 0i + |0, 1, 0i + |0, 0, 1i 4

(even parity) (4.35) (odd parity) .

This may be done in the following way: ❤

α |0i + β |1i |0i

H

|0i

H

s

❤ s

    

α |wˆ0 i + β |wˆ1 i .

DRAFT, October 17, 2007

40

For the general case see (Knill et al., 1999). The b-independence of the |Er (Ψ)i is easily derived if every qubit interacts only with its own environment. Note, however, that the |Er i need neither be orthogonal nor normalized nor unique!

107

4.3. ERROR CORRECTING CODES Decoding is not more difficult:   

❤ ❤

α |wˆ0 i + β |wˆ1 i  

s

s

α |0i + β |1i H

|0i

H

|0i .

If the Estate vector of the total system (three-qubit system plus environment) is ˆ Eˆ then, according to our assumption, the interaction between both subsystems Ψ⊗ can cause only transitions of the form E

ˆ ⊗ Eˆ Ψ

7−→

3  X

ν=0





(ν) ˆ ⊗ Eˆ3(ν) σ ˆˆ 3 Ψ

E

E (ν) of the environment, where with suitable state vectors Eˆ3

ˆˆ (ν) |b1 , b2 , b3 i def σ = 3



|b1 , b2 , b3 i if ν = 0 bν (−1) |b1 , b2 , b3 i else

∀ b ∈ {0, 1}3 .

The essential point is that the subspaces o

n

def ˆ (ν) Hν = σ ˆ3 α ˆ |wˆ0 i + βˆ |wˆ1 i : α ˆ , βˆ ∈ C

are pairwise orthogonal. Therefore, to restore the original encoded state vector α |0i + β |1i, it suffices to perform an optimal test (measurement of first kind ) to (ν) which of the four subspaces Hν this state vector belongs and apply σˆ3 according to the outcome. Exercise 17 Show that the (n + 1)-qubit network s

s

❤ ❤ .. .





···

s

··· ···

.. . ❤

transforms λ0 |0i + λ1 |1i ⊗ |0, . . . , 0i into λ0 |0, . . . , 0i + λ1 |1, . . . , 1i and discuss its possible use for error correction.

108

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

4.3.2

Classical Codes

The general idea of classical error correction (Hamming, 1950; Pless, 1989) is the following: • Consider a channel transmitting n-bit words without changing more than m bits of any word. • Then the original words can be uniquely reconstructed from the received ones if only special code words w = (w1 , . . . , wn ) ∈ {0, 1}n are sent which are chosen such that the Hamming distance ′

def

d(w, w ) =

n X

ν=1

|wν −

wν′ |



′ 2

= kw − w k



between any two code words w, w′ is > 2 m . Obviously, 2n must be larger than the number of code words (the more the larger m is) for error correction to work this way.41 Actually: “Error-correcting coding is the art of adding redundancy efficiently so that most messages, if distorted, can be correctly decoded.” (Pless, 1989, p. 2)

Especially convenient are the [n, k] linear classical codes, for which a set C ⊂ {0, 1}n ˆ of 2k code words — the code — is selected by means of an (n − k) × n-matrix H 42 as n o ˆ def ˆb=0 . C = ker(H) = b ∈ {0, 1}n : H

ˆ have to be Of course, the n − k rows of the so-called parity check 43 matrix H independent in order to have 



ˆ = k. dim ker(H) Warning: The code C may contain transposed row vectors of the parity ˆ. check matrix H Without restriction of generality the parity check matrix can be assumed to be of the form44   ˆ = Aˆ 1ln−k , H DRAFT, October 17, 2007

41

Of course this reduces the capacity of the communication channel. 42 Here, we identify matrices with the corresponding linear maps. Note that the components of ˆ are in {0, 1} and that all arithmetic is to be understood modulo 2. Hence, e.g., H ˆ ≡ −H ˆ. H n X ? 43 hν bν = 0 on the substring of those bits Every row (h1 , . . . , hn ) gives rise to a parity check ν=1

of b in places where the row has 1’s . 44 This is easily seen using Gaußian elimination. Eventually the bits have to be relabeled.

109

4.3. ERROR CORRECTING CODES where Aˆ is some (n − k) × k-matrix. In this form we easily see that ˆG ˆ=0 H holds with the n × k-matrix hence45

ˆ= G



1lk Aˆ



,

n

o

ˆ a : a ∈ {0, 1}k . C= G

Remark: A possible coding would be {0, 1}k ∋

a |{z}

word of message

7−→

ˆ

Ga |{z}

corresp. code word

∈ {0, 1}n .

ˆ may directly be used as a generator of the code, H ˆ is more convenient While G for error detection: ˆ \ E be an Let E ⊂ {0, 1}n be the set of possible ‘errors’ and let H/ injection. Then the distortion w 7→ w′ = w + e

of a code word w ∈ C by an error e ∈ E can be identified by checking the error syndrome ˆ w′ = H ˆe H and corrected by adding (=subtracting) e . Exercise 18 The rows of the r × (2r − 1) parity check matrix characterizing the so-called binary Hamming code Ham[r, 2] are the nonzero elements of {0, 1}r , ordered46 according to the value of the corresponding binary numbers.47 a) Show for every r ≥ 2 that Ham[r, 2] is suitable for correcting errors on single bits. b) Discuss Ham[2, 2] in detail. Let C be a a [n, k] linear classical code. Then def

C ⊥ = {b ∈ {0, 1}n : b · b′ = 0 mod 2 ∀ b′ ∈ C}

is called its dual code.

ˆ and Exercise 19 Let C be a [n, k] linear classical code with parity check matrix H 48 ˆ . Show the following: generator G DRAFT, October 17, 2007

ˆ are all independent. Note that, thanks to 1lk , the rows of G Actually, different orderings give rise to equivalent codes. 47 See (4.53) for r = 3 . 48 As usual, we denote the number of elements of a finite set C by |C| . 45

46

110

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

ˆ⊥ = G ˆ T and a) C ⊥ is a [n − k, n] linear classical code with parity check matrix H ˆ⊥ = H ˆT . generator G b)



c) X

(−1)

b′ ∈C

4.3.3

b·b′

=

C⊥

(

⊥

=C.

|C| if b ∈ C ⊥ , 0 if b ∈ {0, 1}n \ C ⊥ .

Quantum Codes

In classical communication the received message may be inspected and corrected according to the error syndrome. In quantum communication, however, we should carefully avoid too detailed ‘measurement’ (associated with uncontrollable ‘collapse’) of the state before reconstruction. Therefore, in order to be able to correct all errors corresponding to error operations σˆ ∈ E we have to look for quantum codes49 of the following form: • The n-qubit state space H containing the quantum code words is a direct sum of specified subspaces Hd . • Every σ ˆ ∈ E is of a definite type d , i.e. σ ˆ |wi ˆ ∈ Hd holds for all quantum 50 code words |wi ˆ . • If σ ˆ, σ ˆ ′ ∈ E are of the same type d then σ ˆ |wi ∼ σ ˆ ′ |wi holds for all quantum code words |wi (but not necessarily for other state vectors). Under these conditions — if only errors corresponding to operations σ ˆ ∈ E are superimposed51 — quantum error correction is possible as indicated in 4.3.1: • The ‘received’ state is forced — via corresponding ‘measurement’ — to ‘collapse’ into a state described by an element of one of the subspaces Hd . • Since the ‘collapsed’ state is just the sent code word distorted by an error of type d we only have to apply the inverse of some unitary error operation of type d to reconstruct the correct code word. DRAFT, October 17, 2007

49

By n-qubit quantum code we always mean the set of pairwise orthogonal n-qubit state vectors used as quantum code words. Note, however, that many authors mean by quantum code the complex linear span of quantum code words. 50 ˆ 0 i , |w ˆ1 i of the computational base states as In 4.3.1 we already used linear superpositions |w quantum code words, in order to indicate additional possibilities in quantum coding. 51 Of course, E should include the trivial ‘error operation’ σ ˆ = ˆ1 .

111

4.3. ERROR CORRECTING CODES

Exercise 20 a) Show that for Shor’s 9-qubit code words

      E def ˆ ˆ 0 = 2−3/2 |0, 0, 0i + |1, 1, 1i ⊗ |0, 0, 0i + |1, 1, 1i ⊗ |0, 0, 0i + |1, 1, 1i , w       E def −3/2 ˆ ˆ1 = 2 |0, 0, 0i − |1, 1, 1i ⊗ |0, 0, 0i − |1, 1, 1i ⊗ |0, 0, 0i − |1, 1, 1i w

every superposition of single-qubit errors, i.e. every distortion of the form E E E  E X ˆ ˆ ˆ ˆ α w ˆ 1 ⊗ |Er i , ˆ r w ˆ0 + β σ ασ ˆ r w ˆ 1 ⊗ |Ei 7−→ ˆ 0 + β w def

R9 =

r∈R9

3 n [

o

permutations of (r, 0, 0, 0, 0, 0, 0, 0, 0) ,

r=0

may be corrected as described above.



E

E

ˆˆ 0 + β w ˆˆ 1 may be achieved as b) Show that encoding Ψ = α |0i + β |1i into α w follows: s

Ψ

s

❤

|0i

s

|0i

❤

|0i

❤

|0i

H

❤

H

s

|0i

❤

|0i

s ❤

H

s

|0i

❤

|0i

s

s ❤

                                          



E



ˆˆ 1 ˆˆ 0 + β w α w

E

Recall that, according to (4.33), for every n ∈ IN the possible error n-qubit error operations are elements of the Pauli group 52 def

n

ˆ b Zˆb : ν ∈ {0, . . . , 3} , b1 , b3 ∈ {0, 1}n Sn = i ν X 1 3

o

(4.36)

That the latter is a group w.r.t. operator multiplication follows immediately from the algebra of Pauli matrices: σ ˆν σ ˆν = ˆ1 ∀ ν ∈ {0, 1, 2, 3} , σ ˆj σ ˆk = − σ ˆk σ ˆj ∀ j, k ∈ {1, 2, 3} , j 6= k , (4.37) σ ˆ1 σ ˆ2 = i σ ˆ3 , σ ˆ2 σ ˆ3 = i σ ˆ1 , σ ˆ3 σ ˆ1 = i σ ˆ2 . DRAFT, October 17, 2007

52

ˆ and Zˆ were defined by (4.34) and (4.29). The X b b

112

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

The following statements also follow directly from these relations: σ ˆν = σ ˆν∗ = σ ˆν−1 Sn ⊃

def S0n =

n

∀ ν ∈ {0, 1, 2, 3} ,

σ ˆ r1 ⊗ · · · ⊗ σ ˆrn : r1 , . . . , rn ∈ {0, . . . , 3}

ˆ b Zˆb ∈ S0n ⇐⇒ i ν = i b1 ·b3 iν X 1 3 σ ˆσ ˆ ′ ∈ {+ˆ σ′σ ˆ , −ˆ σ′σ ˆ} ∀ σ ˆ , σˆ′ ∈ Sn .

(4.38)

o

is not a group ,

(4.39)

∀ ν ∈ {0, 1, 2, 3} , b1 , b3 ∈ {0, 1}n , (4.40) (4.41)

Theorem 4.3.1 Let W ⊂ H be an n-qubit quantum code and let E ⊂ Sn be a set of error operations including the trivial operation ˆ1 . Assume that the linear span HW of W is stabilized by the subset SW of Sn , i.e. that o

n

ˆ ∈ H : gˆ Ψ ˆ =Ψ ˆ ∀ gˆ ∈ SW . HW = Ψ Moreover, assume

σ ˆ∗ σ ˆ′ ∈ / N(SW ) \ SW

∀σ ˆ, σ ˆ′ ∈ E ,

(4.42) (4.43)

where N(SW ) denotes the normalizer of SW : N(SW ) = {ˆ σ ∈ Sn : σ ˆ gˆ σ ˆ ∗ = gˆ ∀ gˆ ∈ SW } . Then there is a unique mapping dσˆ from SW into {+1, −1} such that: 



ˆ gˆ σ ˆΨ

σ ˆ HW ⊂ Hdσˆ





ˆ dσˆ (ˆ g) σ ˆΨ

=

n

def

ˆ ∈ HW , ∀ gˆ ∈ SW , σ ˆ ∈ E, Ψ o

ˆ ∈ H : gˆ Φ ˆ = dσˆ (ˆ ˆ Φ g) Φ

=

′

∀ˆ σ ∈ E,

σ, σ ˆ ∈ E, dσˆ 6= dσˆ ′ =⇒ Hdσˆ ⊥ Hdσˆ ′ ∀ˆ ˆ =σ ˆ ∀ˆ ˆ ∈ HW . dσˆ = dσˆ ′ =⇒ σ ˆΨ ˆ′ Ψ σ, σ ˆ′ ∈ E , Ψ

(4.44) (4.45) (4.46) (4.47)

Outline of proof: (4.44) is a direct consequence of (4.41) and (4.42). (4.44) directly implies (4.45). Since53 gˆ = gˆ∗ ∀ gˆ ∈ SW (4.48) — and since eigenvectors corresponding to different eigenvalues of a self-adjoint operator are always orthogonal — (4.44) also implies (4.46). Finally, (4.44) and (4.41) imply gˆ σ ˆ (∗) = dσˆ (ˆ g) σ ˆ (∗) gˆ ∀ gˆ ∈ SW , σ ˆ∈E and hence ∗

(ˆ σ∗ σ ˆ ′ ) gˆ (ˆ σ∗ σ ˆ′)

∗

=

g ) gˆ (ˆ σ∗ σ ˆ ′ ) (ˆ σ∗ σ ˆ′) dσˆ (ˆ g ) dσˆ ′ (ˆ

=

dσˆ (ˆ g ) dσˆ ′ (ˆ g ) gˆ ∀ gˆ ∈ SW , σ ˆ, σ ˆ′ ∈ E .

(4.38) DRAFT, October 17, 2007

53

This is because for all σ ˆ ∈ Sn we have σ ˆ 2 = ˆ1 ⇐⇒ σ ˆ=σ ˆ∗ .

113

4.3. ERROR CORRECTING CODES Therefore (4.47) follows according to dσˆ = dσˆ ′

=⇒ =⇒

∗

(ˆ σ∗ σ ˆ ′ ) gˆ (ˆ σ∗ σ ˆ ′ ) = gˆ ∀ gˆ ∈ SW σ ˆ∗ σ ˆ ′ ∈ N(SW )

=⇒

σ ˆ∗ σ ˆ ′ ∈ SW

=⇒

ˆ =Ψ ˆ ∀Ψ ˆ ∈ HW σ ˆ∗ σ ˆ′ Ψ

=⇒

ˆ =σ ˆ ∀Ψ ˆ ∈ HW . ˆΨ σ ˆ′ Ψ

(4.43) (4.42) (4.38)

Remarks: 1. In view of (4.43), SW should be chosen as large as possible. 2. The maximal SW fulfilling the requirements of Theorem 4.3.1 for given HW is an abelian group called the stabilizer of HW .

3. Quantum codes W fulfilling the requirements of Theorem 4.3.1 are called stabilizer codes. 4. If there are σ ˆ, σ ˆ ′ ∈ E and |wi ˆ ∈ W with σ ˆ |wi ˆ =σ ˆ ′ |wi ˆ

but σ ˆ 6= σ ˆ′

then the code is called degenerate w.r.t. E . 5. Shor’s 9-qubit code, described in Exercise 20, is degenerate w.r.t. the set of single qubit error operations. This can be easily seen by considering phase flip errors on different qubits. 6. A stabilizer code is nondegenerate w.r.t E iff σ ˆ 6= σ ˆ ′ =⇒ σ ˆ∗σ ˆ′ ∈ / SW

∀σ ˆ, σ ˆ′ ∈ E .

7. In classical coding there is no analog for degeneracy. Lemma 4.3.2 For j ∈ {1, 2} , let the Cj be a [n, kj ] linear classical codes with C2 ⊂ C1 6= C2 , and define  

1

 

X

def def W = CSS (C1 , C2 ) = |wˆb i = q |b + b′ i : b ∈ C1  . |C2 | b′ ∈C2

(4.49)

where H denotes the n-qubit state space. Then (4.42) holds for n

o

ˆ a Zˆb : a ∈ C2 , b ∈ C1⊥ . SW = X

(4.50)

114

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION Outline of proof: Let o n X ˆ = ˆ ∈ H : gˆ Ψ ˆ =Ψ ˆ ∀ gˆ ∈ SW . Ψ λb |bi ∈ Ψ b∈{0,1}n Then ˆ Ψ

= (4.50) = =

Ex. 19

=

(4.50)

= Since, obviously,

X X 1 ˆ ′ Z λb |bi b C ⊥ 1 b∈{0,1}n b′ ∈C1⊥ X X ′ 1 (−1)b·b |bi λb ⊥ C1 bX ∈{0,1}n b′ ∈C1⊥ λb |bi b∈C1 1 X ˆ X Xb′ λb |bi |C2 | ′ b ∈C b ∈C 1 X 2  λ b w ˆb . b∈C1

  ˆ a Zˆ ′ w ˆ X b b ˆb = w

∀ a ∈ C2 , b′ ∈ C1⊥ , b ∈ C1

this proves the lemma.

Remarks: 1. The quantum codes CSS(C1 , C2 ) described by Lemma 4.3.2 are called Calderbank-Shor-Steane codes . 2. The number of code words for these codes is |CSS (C1 , C2 )| =

|C1 | = 2k1 −k2 . |C2 |

Lemma 4.3.3 Let Cj , W and SW be given as in Lemma 4.3.2. If C1 as well as C2⊥ is suitable for correcting errors on up to t bits then σ ˆ 6= σ ˆ ′ =⇒ σ ˆ∗ σ ˆ′ ∈ / N(SW ) holds for

(

n

∀σ ˆ, σ ˆ′ ∈ E

ˆ e1 Zˆe3 : e1 , e3 ∈ b ∈ {0, 1}n : E= X

n X

ν=1

ν

) o

|b | ≤ t

Outline of proof: Consider σ ˆ, σ ˆ ′ ∈ E with σ ˆ 6= σ ˆ ′ . Then there are ) ( n X n |bν | ≤ t ej , e′j ∈ b ∈ {0, 1} : ν=1

(4.51)

.

(4.52)

115

4.3. ERROR CORRECTING CODES with54 {e1 + e′1 , e3 + e′3 } 6= {0}

and σ ˆ∗ σ ˆ′

=



ˆ e Zˆe X 3 1

∗

ˆ e′ Zˆe′ X 3 1

ˆ e +e′ Zˆe′ = Zˆe3 X 1 3 1 ′ e · e + e ) ( ˆ = (−1) 3 1 1 X

e1 +e′1 Zˆe3 +e′3

Hence ∗ ′ ∗ ˆ a Zˆ (ˆ (ˆ σ∗ σ ˆ′) X ˆ) b σ σ

ˆ a Zˆ ˆZˆe +e′ Xe +e′ ˆ e +e′ Zˆe +e′ X = X 3 1 3 1 b 3 1 3 1 ′ ′ n e ( 3 +e3 )·a b·(e1 +e1 ) ˆ = i i Xa Zˆb ∀ a b ∈ {0, 1} . If e3 + e′3 = 6 0 then (e3 + e′3 ) · a 6= 0 mod 2 and, consequently, ∗ ˆ a (ˆ ˆa (ˆ σ∗ σ ˆ′) X σ∗ σ ˆ ′ ) = −X

for some a ∈ C2 since the generator of C2 is the parity check matrix of C2⊥ . On the other hand, if e1 + e′1 6= 0 then b · (e1 + e′1 ) 6= 0 mod 2 and, consequently, ∗ (ˆ σ∗ σ ˆ ′ ) Zˆb (ˆ σ∗ σ ˆ ′ ) = −Zˆb for some b ∈ C1⊥ . Thus σ ˆ∗ σ ˆ′ ∈ / N (SW ).

Remark: Obviously, CSS(C1 , C2 ) is nondegenerate w.r.t. E specified by Lemma 4.3.3. 55 σ ˆr affecting at most t ∈ {0, . . . , n} qubits In general, the number of operations ! t X n 3j . Therefore, in order to correct all corresponding of an n-qubit system is56 j j=0 errors for a nondegenerate n-qubit code according to the scheme described above, that many subspaces Hd are needed. Moreover, the dimension of each of these subspaces must not be smaller than the number of code words. Therefore:

Correction of all errors on at most t qubits of a nondegenerate nqubit code spanned by 2k orthogonal code words is not possible if the quantum Hamming bound t X

j=0

!

n j k 3 2 ≤ 2n j

is violated. Note that for k = t = 1 the quantum Hamming bound becomes 2 + 6n ≤ 2n , hence n ≥ 5.

For further details on quantum codes see (Preskill, 01, Chapter 7), and (Schlingemann and Werner, 20 Schlingemann, 2001; Keyl and Werner, 2002). DRAFT, October 17, 2007

54

Recall Footnote 42. 55 Recall (4.32). 56 The index j = 0 corresponds to the trivial error operation (unit operator).

116

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

4.3.4

Reliable Quantum Computation

Let us discuss the implementation of error For simplicity,  correction in more detail.  ⊥ we consider only the quantum code CSS Ham[3, 2], Ham[3, 2] , called the Steane code. According to Exercise 18 a parity check matrix for Ham[3, 2] is 



0 0 0 1 1 1 1 def  ˆ H3 =  0 1 1 0 0 1 1  . 1 0 1 0 1 0 1

(4.53)

Exercise 21 ˆ 3 are a) Show that the code words corresponding to the parity check matrix H the same as those corresponding to the parity check matrix 



0 1 1 1 1 0 0  ˆ 3′ def = H 1 0 1 1 0 1 0 . 1 1 0 1 0 0 1 b) Show that57



1 0 ˆ 4 def H =  0 0

is a parity matrix for Ham[3, 2]⊥ .

0 1 0 0

0 0 1 0

0 0 0 1

0 1 1 1

1 0 1 1



1 1   0 1

c) Show that n

o

ˆ 3 b = 0 , (−1)b1 +...+b7 = 1 . Ham[3, 2]⊥ = b ∈ {0, 1}7 : H According to Exercise 21, the quantum code words of the Steane code are 1  |wˆ0 i = √ |0000000i + |1101001i + |1011010i + |0111100i 8  + |0110011i + |1100110i + |1010101i + |0001111i , 1  |wˆ1 i = √ + |1111111i + |0010110i + |0100101i + |1000011i 8  + |1001100i + |0011001i + |0101010i + |1110000i , DRAFT, October 17, 2007

57

Recall Exercise 19 a).

(4.54)

117

4.3. ERROR CORRECTING CODES Exercise 22 Show for (4.54) that the following network acts as indicated: ❤

α |0i + β |1i |0i

H

|0i

H

|0i

H

               

s

s

❤

s

s

s s ❤

|0i |0i

❤

|0i

❤

❤

❤

❤

❤

❤

              

❤

α |wˆ0 i + β |wˆ1 i .

Exercise 23 Show that the following 10-qubit network acts as indicated: |b1 i

s

|b1 i

s

|b2 i

|b2 i

s

|b3 i

|b3 i

s

|b4 i

|b4 i

s

|b5 i

s

|b6 i

s

|b7 i

❤

|0i |0i |0i

|b5 i

❤ ❤

❤

❤ ❤

❤

❤

❤

❤

❤ ❤

|b6 i |b7 i

  E ˆ H3 b , 

where, e.g., s .. . ❤ ❤

def

≡

s

s

.. .

.. .

❤ ❤

.

The network of Exercise 23 may be used to reduce single-qubit errors of type σ ˆ1 or/and σ ˆ2 to those of type σ ˆ0 or/and σ ˆ3 :

118

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION An ideal test for the computational basis of the last 3 (ancillary) qubits causes the 10-qubit state to collapse into some state being the direct product of • a (possibly only partially coherent) superposition of distortions of the original code word by single-qubit errors, appearing in one and the same position if of type σ ˆ1 or σ ˆ2 , and • a base state of the ancillary 3-qubit system which corresponds either to (0, 0, 0) , if the collapsed 7-qubit state is a distortion of the original code word by single-qubit errors of only type σˆ0 or/and σ ˆ3 , or else corresponds to the classical error syndrome of a bit-flip in the position, where the code word is distorted by an error of type σ ˆ1 or/and σ ˆ2 . If the collapsed state of the ancillary system does not correspond to (0, 0, 0) then σ ˆ1 should be applied to the qubit in the position where the code word is distorted. In any case, then, the resulting 7-qubit state will be a (possibly only partially coherent) superposition of distortions of the original code word by single-qubit errors of type σˆ0 or/and σ ˆ3 .

Exercise 24 Show that the following 10-qubit network flips the

X 3

ej 23−j -th

¬ ˆ

.



j=1

qubit for input of the specified type with |e1 , e2 , e3 i = 6 |0, 0, 0i and, therefore, may be used to avoid testing the error syndrome for single-qubit errors of type σ ˆ1 or/and σ ˆ2 : ❤ ❤ ❤ ❤ ❤ ❤ ❤ |e1 i |e2 i |e3 i

❝

❝

❝

s

s

s

s

❝

s

s

❝

❝

s

s

s

❝

s

❝

s

❝

s

|e1 i |e2 i |e3 i

where ❝

def

≡

¬ ˆ

s

¬ ˆ

,

❝

def

≡

¬ ˆ

s

119

4.3. ERROR CORRECTING CODES

The eventually remaining single-qubit errors of only type σˆ0 or/and σ ˆ3 . may be converted into errors of type σ ˆ0 or/and σ ˆ1 by applying UˆH⊗n . Correction these errors as just described and applying UˆH⊗n once more restores the original message. Up to now we tacitly assumed that all devices used for error correction work perfectly error free. Of course this is unrealistic and, actually, special care has to be taken to prevent these devices from making things worse. For instance, if a phase error appears for the first ancillary qubit of the error syndrome network presented in Exercise 23 then according to Exercise this error may propagate into all of the last four data qubits. To prevent this one could use s s s s ❤

|0i

❤ ❤

|0i

❤

s ❤

|0i

❤ s

❤

|0i

s

instead of s s s s |0i

❤

❤

❤

❤

and implement in a suitable way.58 While such precautions prohibit propagation of errors of the ancillary part of the network into the data part they do not guarantee a correct error syndrome. Therefore the ‘measurement’ of the error syndrome should be repeated and only used for error correction if confirmed. In order to protect calculations against quantum noise they should be performed directly on the encoded data. Of course, the encoded data should be error checked sufficiently often. Especially, the actual computation should not be started before the initial encoded state has been checked to be free of errors. DRAFT, October 17, 2007

58

See (M¨ ott¨ onen and Vartiainen, 2005, Fig. 8), in this connection.

120

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

Altogether it seems possible to implement reliable quantum computation, if sufficient care is taken. For further details see (Preskill, 1998b; Preskill, 1998a; Leung, 2000).

Entanglement Assisted Channels59

4.4

“Entanglement is monogamous — the more entangled Bob is with Alice, the less entangled he can be with anyone else.” Charles Bennett60

4.4.1

Quantum Dense Coding61

Let S = S1 ⊕ S2 ben a bipartite62 2-qubit system with state space H ⊗ H and o def . Then the so-called Bell states computational basis |ν, µi = φν ⊗ φµ ν,µ∈{0,1}

def

Φ+ =

def

Φ− = +

Ψ

def

=

def

Ψ− =

1 √ (|0, 0i + |1, 1i) , 2 1 √ (|0, 0i − |1, 1i) , 2 1 √ (|0, 1i + |1, 0i) , 2 1 √ (|0, 1i − |1, 0i) 2

(4.55)

form an orthonormal basis of H ⊗ H and may be locally transformed into each other:63   Φ∓ = σ ˆ3 ⊗ ˆ1 Φ± , Ψ∓ =

Ψ+ =







σ ˆ3 ⊗ ˆ1 Ψ± ,

(4.56)



σ ˆ1 ⊗ ˆ1 Φ+ .

Obviously, 2 classical bits of information may be encoded via the Bell states, e.g.: b Φ+ , (0, 0) =

DRAFT, October 17, 2007

59

b Φ− , (0, 1) =

b Ψ+ , (1, 0) =

b Φ− . (1, 1) =

See (L¨ ucke, 2002, Section 1.2.2) for the network models of dense coding, teleportation, and entanglement swapping. See also (Devetak and Winter, 2003; Devetak et al., 2004) for related protocols. 60 http://qpip-server.tcs.tifr.res.in/ qpip/HTML/Courses/Bennett/TIFR2.pdf 61 See also (Mermin, 2002). 62 See Appendix A.4.3. 63 As usual, we denote by σ ˆ1 , . . . , σ ˆ3 the Pauli operators, i.e. w.r.t. (|0i , |1i) :       1 0 0 −i 0 1 . , σ ˆ3 = b , σ ˆ2 = b σ ˆ1 = b 0 −1 +i 0 1 0

121

4.4. ENTANGLEMENT ASSISTED CHANNELS

Thus, if Alice and Bob (situated arbitrarily far apart) initially share a pair of qubits forming S1 ⊕ S2 in a Bell state,64 say Ψ− , then Alice may communicate 2 bits of information by sending Bob her single qubit (system S1 ) after acting on it in an appropriate way: 2-bit information operation by Alice new Bell state (0,0)

σ ˆ1 σ ˆ3

Φ+

(0,1)

σ ˆ3 σ ˆ1 σ ˆ3

Φ−

(1,0)

σ ˆ3

Ψ+

(1,1)

none

Ψ−

After receiving Alice’s qubit Bob just has to perform a projective measurement w.r.t. the Bell basis65 {Φ+ , Φ− , Ψ+ , Ψ− } in order to decode the 2-bit information. Needless to say, without entanglement Alice would not have any chance to transmit more than a single bit by sending just a single qubit. Therefore, the described procedure to communicate 2 bits by sending just 1 qubit is called quantum dense coding (of classical information). Of course, the crucial point is that Alice has to be given one partner of an entangled pair of qubits first. Note, however, that Alice and Bob may store their qubits for some time in suitable quantum memory66 before starting to communicate. Then the information carried by the sent qubit is of no use for any potential eavesdropper.

4.4.2

Quantum Teleportation

Consider, e.g., a 3-qubit n system S = S1 ⊕ S2 ⊕oS3 with state space H ⊗ H ⊗ H and def computational basis |α, β, γi = φα ⊗ φβ ⊗ φγ in the initial state67 α,β,γ∈{0,1}

1 Ψ0 = ψ ⊗ √ (φ0 ⊗ φ1 − φ1 ⊗ φ0 ) . 2

(4.57)

In order to determine the effect of a projective measurement on the subsystem S1 ⊕S2 w.r.t. its Bell basis we rewrite this state in the form Ψ0 = Φ+ ⊗ χ0 + Φ− ⊗ χ1 + Ψ+ ⊗ χ2 + Ψ− ⊗ χ3 .

(4.58)

Writing ψ = α |0i + β |1i DRAFT, October 17, 2007

64

For the creation of photon pairs in Bell states via parametric down conversion see, e.g., (Gatti et al., 2003) and references given there. 65 For the implementation of such measurements see, e.g., (Paris et al., 2000; Tomita, 2000; Kim et al., 2001). 66 For the possibility of storing optical qubits see (Gingrich et al., 2003). 67 Obviously, then, the subsystem S2 ⊕ S3 is in the Bell state corresponding to Ψ− .

122

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

and comparing √

2ψ ⊗

√1 2

(φ0 ⊗ φ1 − φ1 ⊗ φ0 )

= α |0, 0, 1i − α |0, 1, 0i − β |1, 1, 0i + β |1, 0, 1i with √

2 Φ+ ⊗ χ0 + Φ− ⊗ χ1 + Ψ+ ⊗ χ2 + Ψ− ⊗ χ3 1 = |0, 0i ⊗ χ0 + |1, 1i ⊗ χ0 + |0, 0i ⊗ χ1 − |1, 1i ⊗ χ1 2  + |0, 1i ⊗ χ2 + |1, 0i ⊗ χ2 + |0, 1i ⊗ χ3 − |1, 0i ⊗ χ3 χ2 + χ3 χ2 − χ3 χ0 − χ1 χ0 + χ1 + |0, 1i ⊗ + |1, 0i ⊗ + |1, 1i ⊗ = |0, 0i ⊗ 2 2 2 2

we get

and hence

χ0 + χ1 χ2 + χ3 χ2 − χ3 χ0 − χ1

= = = =

+α |1i , −α |0i , +β |1i , −β |0i

χ0 = α |1i − β |0i , = σ ˆ1 σ ˆ3 ψ χ1 = α |1i + β |0i , = σ ˆ1 ψ χ2 = β |1i − α |0i , = −ˆ σ3 ψ

(4.59)

χ3 = −α |0i − β |1i , = −ψ .

(4.57)–(4.59) show the possibility of quantum teleportation (of quantum information):68 If Alice and Bob (situated arbitrarily far apart) initially share a pair of qubits forming S2 ⊕ S3 in the Bell state √12 (φ0 ⊗ φ1 − φ1 ⊗ φ0 ) then Alice may communicate to Bob the quantum information contained in the unknown state ψ in the following way: Alice performs a projective measurement on S1 ⊕ S2 w.r.t. the Bell basis {Φ+ , Φ− , Ψ+ , Ψ− } and tells Bob her result via classical communication. Bob just has to perform one of the operations σˆ1 σ ˆ3 , σ ˆ1 , σ ˆ3 or none — according to the outcome of Alice’s measurement — on his DRAFT, October 17, 2007

68

See (Sanctuary et al., 2003) for critical remarks on corresponding experiments.

123

4.4. ENTANGLEMENT ASSISTED CHANNELS qubit (system S3 ) in order to have the latter in the state ±ψ : result of Alice’s measurement

Bob’s operation

Φ+

σ ˆ3 σ ˆ1 = (ˆ σ1 σ ˆ3 )−1

Φ−

σ ˆ1 = (ˆ σ1 )−1

Ψ+

σ ˆ3 = (ˆ σ3 )−1

Ψ−

none

Note that the classical information sent by Alice would be of no use to an eavesdropper and that sending classical information avoids the decoherence problems connected with sending qubits.

4.4.3

Entanglement Swapping

Consider a 4-qubit system S = S0 ⊕ S1 ⊕ S2 ⊕ S3 in the initial state ˆ 0 = Ψ− ⊗ Ψ− . Ψ

Then the calculations of Section 4.4.2 show that  1   ˆ0 = ˆ1 ⊗ ˆ1 ⊗ ˆ1 ⊗ σ ˆ1 σ ˆ3 Ψ |0i ⊗ Φ+ ⊗ |1i − |1i ⊗ Φ+ ⊗ |0i 2   1  |0i ⊗ Φ− ⊗ |1i − |1i ⊗ Φ− ⊗ |0i 1ˆ ⊗ 1ˆ ⊗ 1ˆ ⊗ σ ˆ1 2  1   ˆ3 |0i ⊗ Ψ+ ⊗ |1i − |1i ⊗ Ψ+ ⊗ |0i − 1ˆ ⊗ 1ˆ ⊗ 1ˆ ⊗ σ 2  1 − |0i ⊗ Ψ− ⊗ |1i − |1i ⊗ Ψ− ⊗ |0i . 2 Now assume that Victor has access to S0 ,

(4.60)

(4.61)

S1 ⊕ S2 ,

Alice

has access to

Bob

has access to S3 .

Then — even though Victor, Alice, and Bob may be arbitrarily far apart, the entanglement of the subsystem S0 ⊕ S1 may be swapped to the subsystem S0 ⊕ S3 in the following way: Alice performs a projective measurement on S1 ⊕S2 w.r.t. the Bell basis {Φ+ , Φ− , Ψ+ , Ψ− } and tells Bob her result via classical communication. Bob just has to perform one of the operations σˆ1 σ ˆ3 , σ ˆ1 , σ ˆ3 or none — according to the outcome of Alice’s measurement — on his qubit (system S3 ) in order to have the partial state of the subsystem S0 ⊕ S3 in the state Ψ− .

124

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

Thus Alice may act as an entanglement provider: Alice prepares pairs of entangled qubits and distributes one partner of each pair to various customers including Victor and Bob. If Victor and Bob need to share an entangled pair they instruct Alice to perform a projective measurement on S1 ⊕ S2 w.r.t. the Bell basis {Φ+ , Φ− , Ψ+ , Ψ− } and communicate the result to either Victor and Bob who then knows the type of entanglement of the pair shared with Bob.

4.4.4

Quantum Cryptography69

As explained in 2.2.1 the security of the RSA encryption scheme relies on the extreme difficulty to factorize large numbers n by classical means (and the possibility of authentication of submitted messages). Otherwise d could be determined from n and e . However, in view of Shor’s quantum factoring algorithm (Shor, 1994) and the possible implementation of quantum computers such classical cryptosystems as RSA may become insecure. Fortunately, quantum mechanics itself offers means for secure communication exploiting the Vernam cipher70 (also called one time pad ): Exploiting quantum mechanisms, Victor and Bob agree on a purely random secret key c = (c1 , . . . , cn ) ∈ {0, 1}n . Then, instead of sending Bob the pure the plaintext message c = (c1 , . . . , cn ) ∈ {0, 1}n Victor sends him the encoded message71 e = (b1 ⊕ c1 , . . . , bn ⊕ cn ) (through some public channel) which Bob may decrypt as b = (e1 ⊕ c1 , . . . , en ⊕ cn ) but appears purely random to all eventual eavesdroppers. As shown by Claude Shannon (Shannon, 1949a), this cryptosystem is absolutely secure if e is kept secret and used only once. If Victor and Bob share enough (nearly) maximally entangled pairs of qubits72 they may establish a secret key in the following way: DRAFT, October 17, 2007

69

See (Bowmeester et al., 2000, Chapter 2) for a nice introduction and (Elliott et al., 2005) for actual implementation. A commercial quantum cryptosystem is offered at: www.idquantique.com 70 Developed by Gilbert Vernam at AT&T in 1917 (first published in 1926). 71 Note that def b ⊕ b′ = b + b′ mod 2 ∀ b, b′ ∈ {0, 1} . 72

If the entanglement is not good enough even if it is fairly bad they may perform entanglement distillation resulting in a smaller number of nearly perfectly entangled pairs; see Section 6.2.3.

4.4. ENTANGLEMENT ASSISTED CHANNELS

125

Via public communication they agree on an orthonormal basis (e1 , e2 , e3 ) of IR3 and on a series of joint measurements of the following type on definite pairs: For every tested pair the momenta of the partners are directed parallel or antiparallel e2 and a check for linear polarization is performed, but Victor and Bob independently and randomly between two possibilities: Either they test whether the linear polarization of their photon is parallel or orthogonal to e1 or check whether the linear polarization is parallel or orthogonal 1 to e′1 = √ (e1 + e3 ). 2 As long as their choices are different their results for the corresponding pairs are completely uncorrelated. Whenever they choose the same type of measurement their results are (nearly) perfectly correlated. Thus they may agree via public communication on a random secret key in the following way: • Alice and Bob identify those pairs for which, by chance, they had chosen the same type of measurement and discard those pairs of which at least one barter got lost and thus did not provide a definite result. • The remaining pairs are split into two groups. • Bob and Alice (publicly) compare their results for the first group in order to check perfect correlation. • If the correlations turn out to be (nearly) perfect for the first group then Alice and Bob agree to use their results on the second group for a common key. Thank to the correlations they need not communicate the results concerning the second group. Therefore a possible eavesdropper has no access to the chosen key. • Weak deviations from perfect correlation may be error corrected after communicating results of various parity checks73 — of almost no use for any eavesdropper. This cryptosystem can only be attacked by manipulating the entangled pairs before Victor’s and Bob’s measurements. But such attack will be detected by Victor and Bob, who can eventually discard the current key and create another one.

DRAFT, October 17, 2007

73

Alternatively, if the correlations are only marginally spoiled, one could apply standard classical error correction74 to the appropriately encoded (and slightly disturbed) plaintext after use of the one time pad.

126

CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

Chapter 5 Quantifying Quantum Information

In fact, the mathematical machinery we need to develop quantum information theory is very similar to Shannon’s mathematics (typical sequences, random coding, . . . ); so similar as to sometimes obscure that the conceptual context is really quite different. (Preskill, 01, Section 5.2)

5.1

Shannon Theory for Pedestrians

For simplicity, let us consider an information source (Z, p) of the following type: 1. Letters x are randomly drawn from a finite alphabet Z = {z1 , . . . , zN } . 2. The probability for drawing the n-letter word w = (zj1 , . . . , zjn ) is1 p(zj1 , . . . , zjn ) =

n Y

ν=1

p(zjν ) ∀ zj1 , . . . , zjn ∈ Z ,

where p(z) = probability for drawing z

∀z ∈ Z .

Consistency, of course requires p(z1 ) + . . . + p(zN ) = 1 .

(5.1)

Then, for the corresponding Shannon entropy def

H(Z, p) = −

N X





p(zj ) log2 p(zj )

j=1

(5.2)

DRAFT, October 17, 2007

1

Thus we assume that the probabilities for the successively drawn letters are independent.

127

128

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

one may prove2 Shannon’s noiseless coding theorem: For arbitrarly given δ > 0 , we may associate with every n ∈ IN a set Wδ,n of typical n-letter words3 such that: 1. |Wδ,n | ≤ 2n(H(Z,p)+δ)

∀ n ∈ IN .

2. The probability for a drawing a n-letter word w ∈ / Wδ,n tends to 0 for n → ∞ .

In other words: Asymptotically, the relevant words can be indexed by ⌈n (H(Z, p) + δ)⌉ bits,4 for every fixed δ > 0 . 



In this sense, the information gained by drawing the letter z is − log2 p(z) bits. The average information gained by drawing a letter, correspondingly, is H(Z, p) bits. Remarks: 1. As to be expected, we have: H(Z, p) = 0

⇐⇒

∃ z0 ∈ Z : p(z0 ) = 1 .

(5.3)

2. Straightforward calculation shows that5 Z = Z 1 ∪ Z2 ,

Z1 6= ∅ = Z1 ∩ Z2 6= Z2 



=⇒ H(Z, p) = H {Z1 , Z2 } , p + p(Z1 ) H(Z1 , p1 ) + p(Z2 ) H(Z2 , p2 ) , (5.4) DRAFT, October 17, 2007

2

See (Shannon, 1949, Appendix 3). 3 The typical words have to include essentially all those containing the letter x approximately ⌈n p(z)⌉-times for every z ∈ Z . For large n the number of such words is of the order   n! −N N n H(Z,p)   since N ! ≈ e N for large N . ≈ 2 Q Stirling z n p(z) !   Obviously, such coding can be used for data compression, if H(Z, p) < log2 |Z| . 5 (5.4) together with (5.5) and continuity in the p(z) fixes H uniquely (Shannon, 1949, Theorem 4

2).

129

5.1. SHANNON THEORY FOR PEDESTRIANS where def

p(Zj ) =

X

p(z)

z∈Zj

pj (z)

def

= p(z)/p(Zj ) ∀ z ∈ Zj

    

∀ j ∈ {1, 2} .

3. H(Z, p) as a functional of p is maximal6 for constant p , i.e. for p(z1 ) = . . . = p(zN ) = 1/N . 4. p(z) =

1 |Z|

∀z ∈ Z

5. Since







log2 |Z| = N



H(Z, p) = log2 |Z| .

=⇒

(5.5)

if |Z| = 2N ,

compression is not possible for constant p(z) (essentially all words are typical ). Now assume Z =X ×Y

and define def

p1 (x) =

X

y∈Y def

p2 (y) =

X

x∈X



p(x, y) ∀ x ∈ X ,

(5.6)

p(x, y) ∀ y ∈ Y ,

(5.7)

p(x, y)/p2 (y) 1/ |X|  p(x, y)/p1 (x) def p2 (y|x) = 1/ |Y | def

p1 (x|y) =

if p2 (y) > 0 else if p1 (x) > 0 else

∀ (x, y) ∈ Z ,

(5.8)

∀ (x, y) ∈ Z .

(5.9)

Remark: A possible application is the following: X

= {elements drawn and sent via some channel } ,

Y

= {elements received through that channel} ,

p(x, y)

=

joint probability for sending x and receiving y ,

p1 (x)

=

probability for sending x ,

p2 (y)

=

probability for receiving y ,

p2 (y|x) =

probability for receiving y when x is sent ,

p1 (x|y) =

probability for x having been sent when y is received .

DRAFT, October 17, 2007

6

The simplest way to check this this is by means of a Lagrange multiplier λ : Determine λ ∈ IR
P and p(z1 ), . . . , p(zN ) ≥ 0 (not a priori postulating (5.1)) for which H(Z, p) + λ 1 − z∈Z p(z) is maximal.

130

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

Then, thanks to7 a 6= b =⇒ a (ln a − ln b) > a − b ∀ a, b > 0

(5.10)

H(X × Y, p) ≤ H(X, p1 ) + H(Y, p2 ) (subadditivity)

(5.11)

we have8 and

H(X × Y, p) = H(X, p1 ) + H(Y, p2 )

=⇒ p(x, y) = p1 (x) p2 (y) ∀ (x, y) ∈ X × Y .

(5.12)

Outline of proof: Replacing b by b c in (5.10) and using log2 (x) =

ln(x) ln(2)

∀x > 0,

we get   a − bc a 6= b c =⇒ a log2 (a) − log2 (b) − log2 (c) > ln(2)

∀ a, b, c > 0

and hence

H(X, p1 ) + H(Y, p2 ) − H(X × Y, p)     X X X  = − p(x, y) log2 p1 (x) − x∈X

y∈Y

y∈Y

+

X

=

(x,y)∈X×Y

≥

X

(x,y)∈X×Y

≥ 0



p(x, y) log2 



X

x∈X

X

(x,y)∈X×Y

!

  p(x, y) log2 p2 (y)   p(x, y) log2 p(x, y)

     p(x, y) − log2 p1 (x) − log2 p2 (y)

. p(x, y) − p1 (x) p2 (y) ln(2)

with equality iff p(x, y) − p1 (x) p2 (y) ∀ (x, y) ∈ X × Y .

Moreover, according to Shannon’s noiseless coding theorem, the conditional entropy9   def X H1 (X|Y ) = p2 (y) H X, p1 (.|y) (5.13) y∈Y

is the average asymptotic amount of bits of information needed in addition per Y part of the drawn z ∈ Z , if only these parts are known, in order to determine also the X-parts. Accordingly, we have H(X × Y, p) = H(Y, p2 ) + H(X|Y ) .

(5.14)

DRAFT, October 17, 2007

7

Setting x = b/a , (5.10) follows from: 0 < x 6= 1 =⇒ ln x < x − 1 . 8 This corresponds to the fact that correlations between X and Y contain additional information. 9 Its quantum analogue can be negative (Horodecki et al., 2005).

131

5.2. ADAPTION TO QUANTUM COMMUNICATION Outline of proof: H(X|Y ) = (5.13) =

− −

X

p2 (y)

y∈Y

X

p1 (x|y) log2

x∈X



p1 (x|y) | {z }

=p(x,y)/p2 (y)





 X   X p2 (y) p1 (x|y) log2 p2 (y) . p2 (y) p1 (x|y) log2 p(x, y) + {z } | y∈Y x∈X (x,y)∈Z =p(x,y) {z } | X

=1

Similarly we have for

H(X × Y, p) = H(X, p1 ) + H(Y |X) def

H2 (Y |X) =

X





p1 (x) H Y, p2 (.|x) .

x∈X

Thanks to subadditivity (5.11), the mutual information def

I(X : Y ) = H(X, p1 ) + H(Y, p2 ) − H(X × Y, p)

(5.15)

is non-negative, as required by its interpretation as amount of information contained in the correlations between X and Y . Its relation to the conditional entropy is given by I(X : Y ) = H(X, p1 ) − H1 (X|Y ) = H(Y, p2 ) − H2 (Y |X)

(5.16)

≥ 0.

5.2 5.2.1

Adaption to Quantum Communication Von Neumann Entropy10

The von Neumann entropy 11 def

S1 (ˆ ρ) = −trace (ˆ ρ log2 ρˆ)

∀ ρˆ ∈ S(H)

(5.17)

(von Neumann, 1927) can be considered as a generalization of the Shannon entropy in the following sense: DRAFT, October 17, 2007

10

See also (Wehrl, 1978; Ohya and Petz, 1993; Petz, 2001; Ruskai, 2002). 11 One can easily prove that, at least for the (normalized) statistical operator ρˆ of the microcanonical or canonical ensemble, k ln(2) S1 (ˆ ρ) is the usual thermodynamic entropy; see, e.g., (Gardiner and Zoller, 2000, Section 2.4.1). Note, however, that the von Neumann entropy — contrary to the thermodynamic entropy — is non-extensive (not additive) for homogeneous nonequilibrium systems. For generalizations of the von Neumann entropy as a measure of ‘mixedness’ see (Berry and Sanders, 2003) and references given there.

132

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION If ρˆ =

n X

ν=1

then

λν |φν ihφν | , |{z}

hφα | φβ i =

≥0



1 for α = β , 0 else ,

S1 (ˆ ρ) = H(X, p1 ) holds for

n

def

X = 



(5.18) o

|φ1 ihφ1 | , . . . , |φn ihφn | def

p1 |φν ihφν |

∀ ν ∈ {1, . . . , n} .

= λν

(5.19) (5.20)

Warning: If {φ1 , . . . , φn } is not an orthonormal system then (5.19) and (5.20) do not imply (5.18) in general!

ˆ B ˆ ∈ L(H) we have Theorem 5.2.1 (Klein’s inequality12 ) For all A, ˆ≥0, 0 ≤ Aˆ 6= B 



ˆ ⊂ ker(A) ˆ ker(B)

ˆ =⇒ trace Aˆ ln Aˆ − ln B



ˆ . > trace (Aˆ − B)

Outline of proof: Thanks to the spectral theorem there are orthonormal systems {φ1 , . . . , φn } and {φ′1 , . . . , φ′n } of H with Aˆ =

n X

aν |{z} ν=1 ≥0

ˆ= B

|φν ihφν | ,

for suitable a1 , . . . , bn . Then

ˆ = trace (Aˆ ln A)

n X

n X

bν |{z} ν=1 ≥0

|φ′ν ihφ′ν |

aν ln aν

ν=1

and ˆ = trace (Aˆ ln B) =

n X

α=1 n X

α,γ=1

hence

+ n n X X aβ |φβ ihφβ | ln bγ |φγ ihφγ | φα φα

*



 2 aα ln(bγ ) φα | φ′γ ,

ˆ − trace (Aˆ ln B) ˆ = trace (Aˆ ln A) DRAFT, October 17, 2007

12

See (Klein, 1931).

γ=1

β=1

n X

α=1

aα

ln aν −

n X

γ=1



 2 ln(bγ ) φα | φ′γ

!

.

133

5.2. ADAPTION TO QUANTUM COMMUNICATION Since, for x > 0 , ln(x) is a strictly concave function,13 the latter implies ˆ − trace (Aˆ ln B) ˆ trace (Aˆ ln A) !  n n X X

 2 aα ln aα − ln bγ φα | φ′γ ≥ α=1

γ=1

n X

≥

(5.10) α=1 n X

=

α=1

aα − aα −

n X

γ=1 n X



 2 bγ φα | φ′γ

!

(5.21) (5.22)

(5.23)

bγ

γ=1

ˆ . trace (Aˆ − B)

=

In (5.22) equality  holds only if, for every α ∈ {1, . . . , n} , at most one of products aα bγ φα | φ′γ is different from zero. Then, with suitable relabelling of the φ′γ , we n X

 2 bγ φα | φ′γ = bα and equality in (5.23) holds only if aα = bα for all have γ=1

ˆ. α ∈ {1, . . . , n} , i.e. if Aˆ = B

Remark: As an immediate consequence of Theorem 5.2.1 we have strict positivity of the quantum relative entropy 

def



S(ˆ ρkˆ ρ′ ) = trace ρˆ (ln ρˆ − ln ρˆ′ )

for all states ρˆ, ρˆ′ ∈ S(H) with ker(ˆ ρ′ ) ⊂ ker(ˆ ρ) and ρˆ 6= ρˆ′ . Corollary 5.2.2 Let H1 , H2 be Hilbert spaces and ρˆ ∈ S(H1 ⊗ H2 ) . Then 







S1 (ˆ ρ) ≤ S1 trace 2 (ˆ ρ) + S1 trace 1 (ˆ ρ) and 







S1 (ˆ ρ) = S1 trace 2 (ˆ ρ) + S1 trace 1 (ˆ ρ)

(subadditivity)

⇐⇒ ρˆ = trace 2 (ˆ ρ) ⊗ trace 1 (ˆ ρ) .

Outline of proof: Application of Klein’s inequality to Aˆ = ρˆ , gives 0 = ≤

ˆ = trace 2 (ˆ B ρ) ⊗ trace 1 (ˆ ρ) ˆ trace (Aˆ − B)

ˆ −S1 (ˆ ρ) − trace (Aˆ log2 B)

DRAFT, October 17, 2007

13

Strictly concave functions f of x > 0 are those fulfilling   λ f (x1 ) + (1 − λ) f (x2 ) < f λ x1 + (1 − λ) x2 ∀ λ ∈ (0, 1) , x1 , x2 > 0 .

134

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION with equality only for ρˆ = trace 2 (ˆ ρ) ⊗ trace 1 (ˆ ρ) . Since ˆ trace (Aˆ logB)  

 ρ) ⊗ ˆ1 + ˆ1 ⊗ log2 trace 1 (ˆ ρ) = trace Aˆ log2 trace 2 (ˆ  

 ρ) log2 trace 1 (ˆ = trace trace 2 (ˆ ρ) log2 trace 2 (ˆ ρ) ρ) + trace trace 1 (ˆ     = −S1 trace 2 (ˆ ρ) − S1 trace 1 (ˆ ρ) ,

this proves the corollary.

The von Neumann entropy of a state ρ increases if the latter is changed by a complete projective measurement operation (ignoring the results): Corollary 5.2.3 Let H be a Hilbert space, ρˆ0 ∈ S(H) , pairwise orthogonal projection operators on H with

n

Pˆ1 , . . . , Pˆl

o

a set of

Pˆ1 + . . . + Pˆl = ˆ1 and define def ρˆ′0 =

l X

Pˆk ρˆ0 Pˆk .

k=1

Then ρˆ0 6= ρˆ′0

=⇒

S1 (ˆ ρ0 ) < S1 (ˆ ρ′0 ) .

=

trace

Proof: Since trace (ˆ ρ log2 ρˆ′ )

X l

k=1

=

trace

|

X l

k=1

X l

=

trace

=

−S1 (ˆ ρ′ ) ,

[ρˆ′ ,Pˆk ]− =0

k=1

Pˆk Pˆk ρˆ log2 ρˆ′ {z

=ˆ 1

}



Pˆk ρˆ log2 (ˆ ρ′ ) Pˆk



 ′ ˆ ˆ Pk ρˆ Pk log2 (ˆ ρ)

The statement follows from Klein’s inequality (Theorem 5.2.1) applied to Aˆ = ˆ = ρˆ′ . ρˆ , B

Warning: In general, trace preserving quantum operations may decrease the von Neumann entropy.14 DRAFT, October 17, 2007

14

E.g., if

ˆ 1 = |0ih0| K

ˆ 2 = |0ih1| K

5.2. ADAPTION TO QUANTUM COMMUNICATION

5.2.2

135

Accessible Information

Assume that the ‘alphabet’ X = {ˆ ρ1 , . . . , ρˆN1 } is a set of (pairwise different) states on H . Then, by “drawing the ‘letter’ ρˆ from X” we mean the random choice of an individual from an ensemble in the state ρˆ . Again, for simplicity, we assume that for every letter ρˆ its probability p1 (ˆ ρ) for being drawn is given, positive, and does not depend on which letters have been drawn before. The best one can do, in order to acquire information about an drawn letter, o is perform a POV measurement15 corresponding to some set Y = Eˆ1 , . . . , EˆN2 of events Eˆ represented by positive bounded operators on H with16 Eˆ1 + . . . + EˆN2 = ˆ1 . ˆ , if ρˆ According to quantum mechanical rules, the probability for Eˆ is trace (ˆ ρ E) was drawn. Hence, the probability for ρˆ being drawn and Eˆ being detected is17 ˆ def ˆ . p(ˆ ρ, E) = p1 (ˆ ρ) trace (ˆ ρ E)

(5.24)

If the letter drawn is unknown, the probability for Eˆ is ˆ p2 (E)

N1 X

ˆ = p(ˆ ρj , E) (5.7) j=1 ˆ , = trace (ˆ ρ0 E)

where def

ρˆ0 =

N1 X

p1 (ˆ ρj )ˆ ρj

j=1

is the state of the source providing the letters. Of course,18 ρˆ0 does not uniquely determine (X, p1 ) . Nevertheless, the von Neumann entropy fulfills the Holevo bound I(X : Y ) ≤ S1 (ˆ ρ0 ) −

N1 X

p1 (ˆ ρj ) S1 (ˆ ρj )

(5.25)

j=1

DRAFT, October 17, 2007

are the Kraus operator of the quantum operation C acting on a qubit system we have C(ˆ1/2) = |0ih0| and hence   S1 (ˆ1/2) > S1 C(ˆ1/2) = 0 . 15

If the elements of X are linearly independent, then the best result can be achieved by projective ˆ1 , . . . , E ˆN being projection operators (Eldar, 2003). For the importance measurement, i.e. with E 2 ˆν ’s see, e.g., (Kaszlikowski et al., 2003). of considering also linearly dependent E 16 Recall Corollary A.4.3, in this connection. 17 Obviously, (5.24) is consistent with (5.6). 18 Recall Corollary A.4.3.

136

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

(Nielsen and Chuang, 2001, Theorem 12.1) and the condition ∃ j1 , j2 ∈ {1, . . . , N1 } : trace (ˆ ρj1 ρˆj2 ) 6= 0 , j1 6= j2 =⇒ S1 (ˆ ρ0 ) < H(X, p1 ) +

N1 X

p1 (ˆ ρj ) S1 (ˆ ρj )

(5.26)

j=1

(Nielsen and Chuang, 2001, Theorem 11.10). A direct consequence of (5.25) and (5.26) is the upper bound ∃ j1 , j2 ∈ {1, . . . , N1 } : trace (ˆ ρj1 ρˆj2 ) 6= 0 , j1 6= j2

=⇒ A(X, p) < H(X, p1 )

(5.27)

on the accessible information def

A(X, p) =

max I(X : Y ) .

Y =POV ˆ

(5.28)

Hence: It is impossible to get full information on the letters actually drawn unless19 ρˆ 6= ρˆ′ =⇒ ρˆ ρˆ′ = ˆ0 ∀ ρˆ, ρˆ′ ∈ X . Remark: Note that, for arbitrary ρˆ, ρˆ′ ∈ S(H) we have20 ρˆ ρˆ′ = ˆ0 ⇐⇒ trace (ˆ ρ ρˆ′ ) = 0 ⇐⇒ ρˆ H ⊥ ρˆ′ H . On the other hand, (5.27) (together with continuity of the entropies) implies the bound A(X, p) ≤ S1 (ˆ ρ0 ) . (5.29) DRAFT, October 17, 2007

19

Otherwise, distinguishability of arbitrary states could be used for superluminal communication. 20 Here, positivity of the operators ρˆ, ρˆ′ is essential! For Aˆ = Aˆ† ∈ L(H) the range Aˆ H of Aˆ is also called the support of Aˆ , since ˆ Aˆ Ψ 6= 0 ⇐⇒ 0 6= Ψ ∈ AH for such Aˆ .

137

5.2. ADAPTION TO QUANTUM COMMUNICATION

5.2.3

Distance Measures for Quantum States21

A natural distance measure for quantum states ρˆ , ρˆ′ of a finite-dimensional22 quantum system is the trace distance23 def 1 D(ˆ ρ, ρˆ′ ) = kˆ ρ − ρˆ′ k1 , (5.30) 2 where k.k1 denotes the trace norm def

kAk1 = trace

q +



Aˆ† Aˆ

(5.31)

for trace class operators Aˆ on a Hilbert space H . Especially for qubits we have   1 ˆ 1 ˆ 1 + ρ · τˆ , ρˆ′ = 1 + ρ′ · τˆ ρˆ = 2 2 24 and hence q D(ˆ ρ, ρˆ′ ) = 14 trace + (ρ · τˆ − ρ′ · τˆ )2 

1 trace |ρ − ρ′ | = 4 1 |ρ − ρ′ | , = 2

q  +

ˆ1

i.e.: For qubit states ρˆ , ρˆ′ the trace distance is half the Euclidean distance of the corresponding Bloch vectors ρ , ρ′ . For general mixed states we have the following: Lemma 5.2.4 Let H be a finite-dimensional Hilbert space and ρˆ, ρˆ′ ∈ S(H) . Then25   D(ˆ ρ, ρˆ′ ) = max trace (Pˆ ρˆ) − trace (Pˆ ρˆ′ ) , (5.32) Pˆ ∈P

where P denotes the set of all projection operators on H . DRAFT, October 17, 2007

21

See also (Gilchrist et al., 2004). For infinite-dimensional systems, however, the trace distance is not physically adequate (Streater, 2003). 23 Note that  ∞ X  ρˆ = pν |Φν ihΦν | ,   ∞  1 X ′ ν=1 =⇒ D(ˆ ρ , ρ ˆ ) = |pν − p′ν | ∞ X  2 ν=1  ′ ′ ρˆ = pν |Φν ihΦν |   22

ν=1

if {Φν }ν∈IN is a MONS of H . 24 Recall that |e| = 1 25

2

=⇒ (e · τˆ ) = ˆ1

∀ e ∈ IR3 .

This formula holds also with P replaced by the set of all events; i.e. of all positive operators with trace ≤ 1 .

138

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION Outline of proof: The spectral theorem tells us that there are an orthonormal basis {φν }ν∈IN of H and real numbers λ1 , . . . , λn with ρˆ − ρˆ′ = and

n X

n X

λ ν φν

(5.33)

ν=1

λν = 0 .

(5.34)

ν=1

Then

q +

2

(ˆ ρ − ρˆ′ ) =

and, therefore,

n X

ν=1

|λν | φν

n

D(ˆ ρ, ρˆ′ )

= =

(5.34)

1X |λν | 2 ν=1 X λν

ν∈{1,...,n}



λν >0

=

trace

X

λ ν φν

ν∈{1,...,n}

≥

 trace Pˆ



λν >0

X

λ ν φν

ν∈{1,...,n}



 + trace Pˆ

Pˆ =

ν∈{1,...,n}

λ ν φν



∀ Pˆ ∈ P

λν ≤0

λν >0

with equality for

X

X

ν∈{1,...,n}

|φν ihφν | .

λν >0

By (5.33) and linearity of the trace, this implies (5.32).

Now it is obvious that the trace distance is a metric on the set of states:26 D(ˆ ρ1 , ρˆ2 ) ≥ 0 ,

D(ˆ ρ1 , ρˆ2 ) = 0

⇐⇒

ρˆ1 = ρˆ2 ,

D(ˆ ρ1 , ρˆ2 ) = D(ˆ ρ2 , ρˆ1 ) , D(ˆ ρ1 , ρˆ3 ) ≤ D(ˆ ρ1 , ρˆ2 ) + D(ˆ ρ2 , ρˆ3 ) . DRAFT, October 17, 2007

26

The inequality follows from (5.32) and     trace (ˆ ρ1 ) − trace (ˆ ρ3 ) = trace (ˆ ρ1 ) − trace (ˆ ρ2 ) + trace (ˆ ρ2 ) − trace (ˆ ρ3 ) .

139

5.2. ADAPTION TO QUANTUM COMMUNICATION

Corollary 5.2.5 Let H1 and H2 be finite-dimensional Hilbert spaces and let C ∈ Q(H1 , H1 ) be trace-preserving. Then Q(H1 , H1 ) is contractive w.r.t. the trace distance, i.e.:   D C(ˆ ρ), C(ˆ ρ′ ) ≤ D(ˆ ρ, ρˆ′ ) ∀ ρˆ, ρˆ′ ∈ S(H1 ) . Outline of proof: With φ1 , . . . , φn and λ1 , . . . , λn chosen as in the proof for (5.32), we have27   X D(ˆ ρ, ρˆ′ ) = trace λ ν φν ν∈{1,...,n} λν >0

¯ trace C

=



X

λ ν φν

ν∈{1,...,n}

¯ ≥ trace Pˆ C



!

λν >0

X

λ ν φν

ν∈{1,...,n}

¯ + trace Pˆ C



X

λ ν φν

ν∈{1,...,n}

!

λν ≤0

λν >0    ′ ˆ trace P C(ˆ ρ) − cˆ(ˆ ρ)

=

!

∀ Pˆ ∈ P .

This, together with (5.32), proves the theorem.

Corollary 5.2.6 Let H be a Hilbert space and N ∈ IN . Moreover, consider ρˆ1 , . . . , ρˆ′N ∈ S(H) and p1 , . . . , p′N ≥ 0 with N X

pν = 1 =

D

N X

pν ρˆν ,

ν=1

N X

p′ν ρˆ′ν

ν=1

!

p′ν .

(5.35)

ν=1

ν=1

Then

N X

≤

N N X 1X pν D(ˆ ρν , ρˆ′ν ) . |pν − p′ν | + 2 ν=1 ν=1

Outline of proof: By Lemma 5.2.4 there is a Pˆ ∈ P with ! ! N N N N X X X X ′ ′ ′ ′ = trace Pˆ D pν ρˆν pν ρˆν , pν ρˆν pν ρˆν − Pˆ ν=1

ν=1

 N    X pν trace Pˆ ρˆν − Pˆ ρˆ′ν + (pν − p′ν ) trace (Pˆ ρˆ′ν ) = | {z } | {z } ν=1 L.

≤

DRAFT, October 17, 2007

27

ν=1

ν=1

(5.36)

Recall Footnote 18 of Chapter 4.

N X

ν=1

≤

5.2.4

∈[0,1]

D(ρˆν ,ρˆ′ν )

pν D(ˆ ρν , ρˆ′ν ) +

X

ν∈{1,...,N } pν −p′ν >0

(pν − p′ν ) .

140

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION Together with N 1X |pν − p′ν | (pν − p′ν ) = (5.35) 2 ν=1 ν∈{1,...,N }

X

pν −p′ν >0

this implies (5.36).

A direct consequence of (5.36) is N X

D

pν ρˆν ,

N X

pν ρˆ′ν

ν=1

ν=1

!

≤

N X

pν D(ˆ ρν , ρˆ′ν ) .

(5.37)

ν=1

Moreover, setting ρˆ′ν ≡ ρˆ′ in (5.37) and recalling (5.35), we get D

N X

′

pν ρˆν , ρˆ

ν=1

!

≤

N X

pν D(ˆ ρν , ρˆ′ ) .

(5.38)

ν=1

Another important measure for the distance of states is the Bures fidelity:28 

Since

q +

def F (ˆ ρ, ρˆ′ ) = trace

s +

q +

ρˆ′

q †  q +

ρˆ

+

ρˆ′

q  +

2

ρˆ  .

|ψihψ| = |ψihψ| and hence rq +

+

|ψihψ| ρˆ′

q +

|ψihψ| = =

q +

|ψihψ| ρˆ′ |ψihψ|

q +

hψ | ρˆ′ ψi |ψihψ|

holds for all normalized ψ ∈ H , we have: ρˆ = |ψihψ| =⇒ F (ˆ ρ, ρˆ′ ) = hψ | ρˆ′ ψi

∀ ρˆ, ρˆ′ ∈ S(H) .

(5.39)

Symmetry of the fidelity in general is not evident from its definition but follows directly from Uhlmann’s theorem (Uhlman, 1976): DRAFT, October 17, 2007

28

For commuting ρˆ, ρˆ′ the Bures fidelity has a simple geometrical interpretation, since X  !2 ρˆ = pν |φν ihφν | ,   Xp ′ ν X =⇒ F (ˆ ρ, ρˆ ) = pν p′ν . ρˆ′ = p′ν |φν ihφν |   ν ν

See (Chen et al., 2002) for a geometrical interpretation of the Bures fidelity of general qubit states.

141

5.2. ADAPTION TO QUANTUM COMMUNICATION Theorem 5.2.7 Let H be a Hilbert-space and ρˆ1 , ρˆ2 ∈ S(H) . Then F (ˆ ρ1 , ρˆ2 ) = where29

def

Tρˆ =

max

Ψ1 ∈Tρˆ1 ,Ψ2 ∈Tρˆ2

n

|hΨ1 | Ψ2 i|2 ,



o

Ψ ∈ H ⊗ H : ρˆ = trace 2 |ΨihΨ|

∀ ρˆ ∈ S(H) .

Outline of proof:nLet j ∈ {1, 2}oand Ψj ∈ Tρˆj . By the spectral theorem, there is are (j)

(j)

orthonormal basis φ1 , . . . , φn with

(j)

(j)

of H , some n′ ∈ {1, . . . , n} , and s1 , . . . , sn′ > 0 ′

ρˆj =

n  X

s(j) ν

ν=1

2

φ(j) ν .

Therefore, the Schmidt-decomposition of Ψj has to be of the form ′

Ψj

= =

n X

(j) (j) s(j) ν φν ⊗ ψ ν

ν=1 n  X ν=1

p +

 ⊗ ψν(j) ρˆj φ(j) ν

o n (j) (j) of H . Considering j = 1 and j = 2 with some orthonormal basis ψ1 , . . . , ψn together we thus get hΨ2 | Ψ1 i =

n′ D p X +

ν,µ=1

p ED E + (2) (1) (1) ψ ρˆ2 φ(2) ρ ˆ φ . ψ 1 µ ν ν µ

(5.40)

In order to rewrite the r.h.s. of (5.40) as a trace we use the unitary operators Vˆ and Vˆ ′ characterized by (1) Vˆ φ(2) ∀ µ ∈ {1, . . . , n} µ = φµ and

Then

D E D E ˆ ′ (2) φ(2) = ψν(2) ψµ(1) µ V φν hΨ2 | Ψ1 i

= (5.40) = = =

n′ D p X +

ν,µ=1 n′ D X

∀ ν, µ ∈ {1, . . . , n} .

p ED E + ˆ ′ (2) ρˆ2 φ(2) ρˆ1 Vˆ φ(2) φ(2) ν µ µ V φν

p p E + ρˆ2 + ρˆ1 Vˆ Vˆ ′ φ(2) φ(2) ν ν ν=1   √ √ trace + ρˆ2 + ρˆ1 Vˆ Vˆ ′  √ √  trace Vˆ Vˆ ′ + ρˆ2 + ρˆ1

Applying Lemma 5.2.8, below, to the polar decomposition r  p p †  p p  p p + + + ′ ˆ + + ρˆ2 + ρˆ1 ρˆ2 ρˆ1 = U ρˆ2 + ρˆ1 , Vˆ Vˆ

ˆ unitary , U

DRAFT, October 17, 2007

29

Note that Tρˆ is the set of all purifications of ρˆ as introduced in Lemma A.4.8.

142

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION ucke, eine)) we always get (see, e.g., Lemma 7.3.20 of (L¨   q √ √
† √ √
2 + 2 + + |hΨ1 | Ψ2 i| ≤ trace ρˆ2 + ρˆ1 ρˆ2 + ρˆ1 = F (ˆ ρ1 , ρˆ1 ) .

(j)

Obviously, by appropriate choice of the ψν

we get equality.

Lemma 5.2.8 Let H be a finite-dimensional 30 Hilbert space. Then

ˆ ρˆ) ≤

B ˆ

trace (B

ˆ ∈ L(H) . ∀ ρˆ ∈ S(H) , B

Outline of proof: By the spectral theorem, there are a an orthonormal basis {φ1 , . . . , φn } of H and p1 , . . . , pn ≥ 0 with ρˆ =

n X

ν=1

Then

ˆ ρˆ) trace (B

pν |φν ihφν | ,

= = ≤ ≤

n X

pν = 1 .

ν=1

n X   ˆ |φν ihφν | pν trace B ν=1 n X E D ˆ φν pν φν | B ν=1 n D E X ˆ φν pν φν | B {z } | ν=1 ˆk ≤kB



ˆ

B .

Uhlmanns theorem and the definition of the Bures fidelity show: F (ˆ ρ1 , ρˆ2 ) = F (ˆ ρ2 , ρˆ1 ) , F (ˆ ρ1 , ρˆ2 ) ∈ [0, 1] ,  1 iff ρˆ1 = ρˆ2 F (ˆ ρ1 , ρˆ2 ) = 0 iff ρˆ1 ρˆ2 = 0 .

(5.41) (5.42) (5.43)

Using Uhlmanns theorem one may also show:31 q

def



ρ1 , ρˆ2 ) ∈ [0, π/2] is a metric , (5.44) A(ˆ ρ1 , ρˆ2 ) = arccos F (ˆ 

F C(ˆ ρ1 ), C(ˆ ρ2 ) F

X ν

pν ρˆν ,

X

p′ν ρˆ′ν

ν

!

≥ F (ˆ ρ1 , ρˆ2 ) , ≥

Xq

pν p′ν F (ˆ ρν , ρˆ′ν ) .

(5.45) (5.46)

ν

DRAFT, October 17, 2007

30

The given may be directly extended to the infinite-dimensional case. 31 Necessary and sufficient conditions for a given set of pure states to be transformable via a quantum operation into another given set of (not necessarily pure) states are given in (Chefles et al., 2003, Theorem 4).

143

5.2. ADAPTION TO QUANTUM COMMUNICATION For pure states φ, ψ : F (Pˆφ , Pˆψ ) = |hφ | ψi| =

q

1 − D(Pˆφ , Pˆψ )2 .

For general states: 1 − F (ˆ ρ1 , ρˆ2 ) ≤ D(ˆ ρ1 , ρˆ2 ) ≤

q

1 − F (ˆ ρ1 , ρˆ2 )2 .

Remark: In principle, fidelity and trace distance are of equal use to characterize the difference of states. However, usually, calculations are easier with fidelity. Therefore only the latter will be used in the following. Relevant for transmission of (unknown) states: def

Fmin (C)

=



min F ρˆ, C(ˆ ρ) ρˆ







= min F Pˆψ , C(Pˆψ ) . ψ (5.46) Relevant for the (approximate) realization of a gate Uˆ as the quantum operation C is the gate fidelity 

def F (Uˆ , C) = min F Uˆ ρˆ Uˆ −1 , C(ˆ ρ) ρˆ





= min F PˆUˆ ψ , C(PˆUˆ ψ ) ψ



for which we have, e.g.,32 q

q

q

arccos F (Uˆ1 Uˆ2 , C1 ◦ C2 ) ≤ arccos F (Uˆ1 , C1 ) + arccos F (Uˆ1 , C1 ) . Relevant for quantum sources producing ρˆj with probability pj and disturbed by C is the ensemble average fidelity def

F =

X j

DRAFT, October 17, 2007

32

Recall (5.44).





pj F ρˆj , C(ˆ ρj ) .

144

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

5.2.4

Schumacher Encoding

Lemma 5.2.9 Let H be a finite-dimensional Hilbert space, ρˆ0 ∈ S(H) , and δ > 0 . Then   ⊗n ˆ (n) lim trace ρ ˆ Λ 0 ρˆ0 ,δ = 1 n→∞ and33





ˆ (n) H⊗n ≥ 2n(S1 (ˆρ0 )−δ) 2n(S1 (ˆρ0 )+δ) ≥ dim Λ ρˆ0 ,δ

∀ n ∈ IN ,

ˆ (n) denotes the projector onto the subspace of all eigenvectors where, for n ∈ IN , Λ ρˆ0 ,δ h i n(S1 (ˆ ρ0 )−δ) n(S1 (ˆ ρ0 )+δ) of ρˆ⊗n with eigenvalues in 2 , 2 . 0 Lemma 5.2.10 Let H be a Hilbert space and Aˆ a self-adjoint operator on H . The for arbitrary ψ1 , . . . , ψN ∈ H and p1 , . . . , pN > 0 we have N X

ν=1

! N N D E 2 X X ˆ ˆ pν ψν A ψν ≥ 2 trace A pν . pν |ψν ihψν | − ν=1

Outline of proof: Apply the inequality D E to x = ψν Aˆ ψν .

x2 ≥ 2 x − 1 ∀ x ∈ IR

.. . DRAFT, October 17, 2007

33

See (Mitchison and Jozsa, 2003) in this connection.

ν=1

145

5.2. ADAPTION TO QUANTUM COMMUNICATION make a guess, Fano inequality

5.2.5

A la Nielsen/Chuang

Klein inequality (Nielsen and Chuang, 2001, Theorem 11.7):34 







trace ρˆ log(ˆ ρ) ≥ trace ρˆ log(ˆ ρ′ )

def



Hbin (p) = H {p, 1 − p}



∀ ρˆ, ρˆ′ ∈ S(H) .

(5.47)

∀ p ∈ [0, 1] .

.. . PPT criterion Cat states implemented for Josephson junctions or coherent states (entanglement laser)

5.2.6

Entropy35

Classically, every message can be encoded in a string of bits. But can quantum information always be encoded in a string of qubits?!

DRAFT, October 17, 2007

34

The r.h.s of (5.47) may become −∞ . Equality holds iff ρˆ = ρˆ′ . 35 See also (Ohya and Petz, 1993) and (Petz, 2001).

146

CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

Chapter 6 Handling Entanglement1 We have seen in 4.4 that perfect entanglement may be used for implementing noiseless quantum communication. Therefore, quantification and handling (e.g., distillation) of entanglement is important. Here, for simplicity, we consider only bipartite systems. One might expect, then, that separability is equivalent to the existence of corresponding (local) hidden variable models, hence to the validity of all (generalized) Bell inequalities.2 However, as shown in (Werner, 1989), that this is not the case.

6.1

Detecting Entanglement Detecting entanglement of pure states is very easy: ρˆ ∈ Spure (H1 ⊗ H2 ) ∪ Ssep (H1 ⊗ H2 )

⇐⇒

Lemma A.4.6 TheoremA.4.7

(

ρˆ ∈ Spure (H1 ⊗ H2 )  2 trace 2 (ˆ ρ) = trace 2 (ˆ ρ) .

It is mixedness which can make detection of entanglement a very hard problem (Gurvits, 2002).

6.1.1

Entanglement Witnesses and Non-Completely Positive Mappings

Lemma 6.1.1 A state ρˆ of the bipartite system S with state space H = H1 ⊗ H2 ˆ ∈ is non-separable 3 iff it possesses an entanglement witness, i.e. an operator W 4 L(H1 ⊗ H2 ) fulfilling the following two properties: DRAFT, October 17, 2007

1

See also (Horodecki et al., 2001) and references given there. 2 See (Werner and Wolf, 201; Collins and Gisin, 2003) in this connection. 3 Recall Definition A.4.4. 4 ˆ is Hermitian — thanks to the polarization identity The second property guarantees that W h(x1 , x1 ) =

3 1X α (−i) |x1 + iα x2 ihx1 + iα x2 | , 4 α=0

valid for every mapping h that is linear in the second and conjugate linear in the first argument; especially for h(φ, ψ) = |φihψ| .

147

148

CHAPTER 6. HANDLING ENTANGLEMENT

1.

ˆ ) < 0. trace (ˆ ρW

2.

ˆ)≥0 ρˆ′ separable =⇒ trace (ˆ ρW

∀ ρˆ′ ∈ S(H1 ⊗ H2 ) .

Outline of proof: The statement follows from the known fact5 that for every point X outside a convex set K there is a hyperplane separating X from K . n

o

(j)

Lemma 6.1.2 (Jamiolkowski) For j ∈ {1, 2} , let φ1 , . . . , φ(j) be a MONS of nj ˆ ∈ L(H1 ⊗ H2 ) there is a unique linear the Hilbert space Hj . Then for every W mapping LWˆ : L(H1 ) −→ L(H2 ) with ˆ = n1 (1 ⊗ L ˆ ) (PˆH+ ) , W W 1 where

n +* n 1 1 X X def 1 (1) (1) (1) (1) φ ⊗ φ φ ⊗ φ = µ µ ν ν . n1

PˆH+1



ED



φ(1) µ1 =

n2 X

ν2 ,µ2 =1

D



(2) (1) (2) ˆ φ(1) ν1 ⊗ φν2 | W φµ1 ⊗ φµ2

for all ν1 , ν2 ∈ {1, . . . , n1 } and: ⇐⇒

LWˆ positive where6

(6.2)

µ=1

ν=1

LWˆ fulfills LWˆ φ(1) ν1

(6.1)

E ED (2) φ(2) φν2 µ2

ˆ ρˆ) ≥ 0 ∀ ρˆ ∈ Ssep (H1 ⊗ H2 ) , trace (W def

Ssep (H1 ⊗ H2 ) = {ˆ ρ ∈ S(H1 ⊗ H2 ) : ρˆ separable} .

(6.3)

(6.4) (6.5)

Outline of proof: Defining LW ˆ by (6.3) plus linear continuation gives ˆ W

= = (6.3) =

lin. cont.

n1 X

ν1 ,µ1 =1 ν2 ,µ2 =1 n1  ED  X (1) φ(1) φν1 µ1 ν1 ,µ1 =1 n1 X

1 ⊗ LWˆ

DRAFT, October 17, 2007

5

n2 E ED D  X (1) (1) (2) (2) (1) (2) (2) ˆ φ ⊗ φ φ(1) ⊗ φ | W φ φ ⊗ φ ⊗ φ µ1 µ2 ν1 ν2 µ1 ν1 µ2 ν2




ν1 ,µ1 =1

 ED  (1) φ(1) ⊗ LW ˆ φν1 µ1

(1) φν1 ⊗ φ(1) ν1

ED (1) φ(1) µ1 ⊗ φµ1

!

See, e.g., (Neumark, 1959, $ 1, No. 9) and (Robertson and Robertson, 1967, Kap. 1, Satz 8). 6 Note that Ssep (H1 ⊗ H2 ) = S(H1 ) ⊗alg S(H2 ) . Theorem A.4.5

149

6.1. DETECTING ENTANGLEMENT and hence (6.1). Conversely, (6.1) gives   ED (1) φµ1 LWˆ φ(1) ν1

n1   X E ED  ED  D (1) (1) (1) (1) (1) φ ⊗ L φ φ(1) φ φ φ ′ ′ ′ ′ ˆ µ1 ν1 W µ1 ν1 µ1 ν1 ν1 ,µ1 =1 D E
ˆ + (1) φ(1) ˆ (PH1 ) φµ1 ν1 dim(H1 ) 1 ⊗ LW E D ˆ (1) , φ(1) ν1 W φµ1

= =

lin.

=

(6.1)

i.e. (6.3). Since ˆ trace W

N X

(1) ρˆk

k=1

⊗

!

(2) ρˆk

= dim(H1 )

Jami

N X

trace

k=1



(1) ρˆk

⊗ LW ˆ



(2) ρˆk



,

ˆ ˆ) for separable7 ρˆ . Conversely positivity of LW ˆ implies nonnegativity of trace (W ρ the latter implies positivity of LW ˆ , since  E ED E  D D (1) ˆ (2) (1) = φ ∀ φ(1) ∈ H1 , φ(2) ∈ H2 , φ W Ψ Ψ φ(2) LW (1) (2) (1) (2) φ ˆ φ ,φ φ ,φ (6.3) where

def



dim(H1 ) D

Ψφ(1) ,φ(2) = 

X

ν1 =1

 E  ⊗ φ(2) . φ(1) φ(1) φ(1) ν1 ν1

n

o

(j)

Corollary 6.1.3 For j ∈ {1, 2} , let φ1 , . . . , φ(j) be a MONS of the Hilbert nj 8 ˆ ∈ L(H1 ⊗ H2 ) we have space Hj . Then for every W ˆ ) < 0 =⇒ trace (ˆ ρW and





1 ⊗ L†Wˆ (ˆ ρ) 6≥ 0

ˆ ) ≥ 0 ∀ ρˆ ∈ Ssep (H1 ⊗ H2 ) trace (ˆ ρW

⇐⇒

∀ ρˆ ∈ S(H1 ⊗ H2 )

L†Wˆ (ˆ ρ2 ) ≥ 0 ∀ ρˆ2 ∈ S(H2 ) ,

(6.6)

(6.7)

where L†Wˆ denotes the Hilbert-Schmidt adjoint of the linear mapping LWˆ that is characterized by (6.3), i.e.: 







trace L†Wˆ (Aˆ2 ) Aˆ1 = trace Aˆ2 LWˆ (Aˆ1 )

∀ Aˆ1 ∈ L(H1 ) , Aˆ2 ∈ L(H2 ) .

DRAFT, October 17, 2007

7

Recall Footnote 6. 8 Positivity of Aˆ ∈ L(H) is easily to checked: Aˆ ≥ 0

⇐⇒

det(Aˆ − x ˆ1) = |{z} c ∈IR

dim(H)

X

ν=1

(−1)ν cν xν . |{z} ≥0

150

CHAPTER 6. HANDLING ENTANGLEMENT Outline of proof: (6.6) follows from 1 ˆ) trace (ˆ ρW n1

= (6.1) =

 
+ ˆ trace ρˆ 1 ⊗ LW ˆ (PH1 )    † + ˆ 1 ⊗ LWˆ (ˆρ) PH1 trace

and (6.7) from (6.4).

We may conclude: 



ˆ is an entanglement witness for ρˆ then 1 ⊗ L† (ˆ 1. If W ρ) 6≥ 0 and, thereˆ W fore, the positive map L†Wˆ cannot be completely positive.

2. For every positive mapping L′ : H2 −→ H1 we have:9 (1 ⊗ L′ )(ˆ ρ) 6≥ 0 =⇒ ρˆ ∈ / Ssep (H1 ⊗ H2 ) . 3. A state ρˆ ∈ S(H1 ⊗ H2 ) is separable if and only if (1 ⊗ L′ )(ˆ ρ) is positive ′ for all positive mappings L : H2 −→ H1 .

Remark: In general, for detecting entanglement, L†Wˆ is more useful ˆ since than W i.g. ˆ ρˆ) < 0 . L† (ˆ ρ) 6≥ 0 6=⇒ trace (W ˆ W

ˆ , on the other hand, allows to detect entanglement by local measureW uhne et al., 2002). ments (G¨

6.1.2

Examples

Lemma 6.1.4 The flip operator of the bipartite system S with state space H⊗H , i.e. the linear Operator Fˆ on H ⊗ H characterized by def Fˆ (ψ1 ⊗ ψ2 ) = ψ2 ⊗ ψ1

∀ ψ1 , ψ2 ∈ H ,

has the following properties:10 1. Fˆ = Fˆ † ,

Fˆ 2 = ˆ1 .

DRAFT, October 17, 2007

9

Of course this statement is relevant for non-completely positive L′ , only. 10 Recall Definition A.4.4.

(6.8)

151

6.1. DETECTING ENTANGLEMENT 2. Fˆ = Pˆ+ − Pˆ− ,

where the

  def 1 ˆ Pˆ± = 1 ± Fˆ 2 are the projectors onto the symmetric resp. anti-symmetric pure states of S : ˆ Ψ ∈ Pˆ± (H ⊗ H) ⇐⇒ Fˆ Ψ = ±Ψ ∀Ψ ∈ H ⊗ H.

3. For all β ∈ IR :

ˆ1 + β Fˆ ≥ 0 ⇐⇒ β ∈ [−1, +1] .

4. trace (Fˆ ) = − dim(H) . 5. For all ρˆ ∈ S(H ⊗ H) : ρˆ separable =⇒ trace (Fˆ ρˆ) ≥ 0 .

Outline of proof: The first four statements are more or less obvious and the last one follows from      ˆ trace F |ψ1 ihψ1 | ⊗ |ψ2 ihψ2 | = trace Fˆ |ψ1 ⊗ ψ2 ihψ1 ⊗ ψ2 |   = trace |ψ2 ⊗ ψ1 ihψ1 ⊗ ψ2 | = hψ1 ⊗ ψ2 | ψ2 ⊗ ψ1 i 2

= |hψ1 | ψ2 i| .

Consequence: The flip operator Fˆ is an entanglement witness for the mixed Werner states 11 ρˆW (β) with β ∈ [−1, −1/ dim(H)] , where ˆ1 + β Fˆ def ρˆW (β) = ∀ β ∈ [−1, +1] . (6.9) trace (ˆ1 + β Fˆ ) For H1 = H2 = H the positive linear mapping (6.3) associated with the n o flip operator (1) 12 (1) on H ⊗ H is the transposition w.r.t. {φ1 , . . . , φn } = φ1 , . . . , φn1 : 



LFˆ |φν1 ihφν2 |





= T |φν1 ihφν2 |

def

= |φν2 ihφν1 | 



= L′Fˆ |φν1 ihφν2 | DRAFT, October 17, 2007

11

Compare Footnote 27 of Chapter 4. 12 The transposition was considered at the end of 4.2.1.

(6.10) ∀ ν1 , ν2 ∈ {1, . . . , n} .

152

CHAPTER 6. HANDLING ENTANGLEMENT

While the flip operator cannot be an entanglement witness for, e.g., the Fˆ -invariant ρ) is pure states,13 a two-qubit state ρˆ is separable iff its partial transpose (1 ⊗ T) (ˆ positive. Slightly more generally we have: n

o

(j)

Theorem 6.1.5 For j ∈ {1, 2} , let φ1 , . . . , φ(j) be a MONS of the Hilbert nj space Hj . If n1 + n2 ∈ {4, 5} then an arbitrarily given state n1 X

ρˆ =

n2 X

ν1 ,µ=1 ν2 ,µ2 =1



(2) ρν1 ν2 ,µ1 µ2 φ(1) ν1 ⊗ φν2

ED



(2) φ(1) µ1 ⊗ φµ2 ∈ S(H1 ⊗ H2 )

is separable iff it has a positive partial transpose:14 n1 X

n2 X

ν1 ,µ=1 ν2 ,µ2 =1



(2) ρν1 ν2 ,µ1 µ2 φ(1) ν1 ⊗ φµ2

ED



(2) φ(1) µ1 ⊗ φν2 ≥ 0 .

Proof:15 See (Horodecki et al., 1996, Theorem 3).

n

Another example for H1 = H2 = H and {φ1 , . . . , φn } = o

(2)

is φ1 , . . . , φ(2) n1

n

(1)

φ1 , . . . , φ(1) n1

o

=

ˆ = ˆ1H⊗H − n PˆH+ W =

n  X

ν, µ=1









ˆ+ = m 1 ⊗ LD H (PH ) , where ˆ LWˆ (B)

=

ˆ LD H (B)

def

ˆ −B ˆ trace (B)

(6.1)

=



|φν ihφµ | ⊗ δνµ ˆ1 − |φν ihφµ |

(6.11) ˆ ∈ L(H) , ∀B

(6.12)

Obviously, LD H is positive and Therefore,16





1 ⊗ LD ρ) = trace 2 (ˆ ρ) ⊗ ˆ1 − ρˆ ∀ ρˆ ∈ L(H) . H (ˆ trace 2 (ˆ ρ) ⊗ ˆ1 6≥ ρˆ =⇒ ρˆ ∈ / Ssep (H ⊗ H) .

(6.13)

DRAFT, October 17, 2007

13

See, however, (Horodecki and Horodecki, 1996). 14 Positivity of the partial to the partial transpose itself, does not depend on o n transpose, contrary (2) (2) the choice for the MONS φ1 , . . . , φn2 . 15 The essential point is that — provided n1 + n2 ∈ {4, 5} — every positive mapping L′ : L(H2 ) −→ L(H1 ) is decomposable; see (Labuschagne et al., 2003) and references given there. For entangled PPT states in higher dimensions see (Ha et al., 2003). 16 See (Hiroshima, 2003) in this connection.

153

6.1. DETECTING ENTANGLEMENT

6.1.3

Other Criteria17

Up to now, in view of Corollary 6.1.3, we considered the entanglement criterion (1 ⊗ |{z} L )(ˆ ρ) 6≥ 0 =⇒ ρˆ ∈ / Ss (H1 ⊗ H1 ) .

(6.14)

≥0

only for positive maps L : L(H2 ) −→ L(H1 ) . But, of course, (6.14) also holds for positive maps L : L(H2 ) −→ L(H2 ) . Typical examples are 1. L = T : Every separable state ρˆ ∈ S(H1 ⊗ H2 ) is PPT, i.e. has a positive partial transpose (1 ⊗ T)(ˆ ρ) — for every choice of MONS. 2. L = LD H :

ρˆ ∈ Ssep (H1 ⊗ H2 ) =⇒ trace 2 (ˆ ρ) ⊗ ˆ1 ≥ ρˆ .

Remarks: 1. Since18

and



(1 ⊗ T)(ˆ ρ)

†



= (1 ⊗ T)(ˆ ρ) ∀ ρˆ ∈ S(H1 ⊗ H2 ) 

trace (1 ⊗ T)(ˆ ρ) = 1 ∀ ρˆ ∈ S(H1 ⊗ H2 ) , we have19

(1 ⊗ T)(ˆ ρ) 6≥ 0 ⇐⇒ k(1 ⊗ T)(ˆ ρ)k1 > 1 ∀ ρˆ ∈ S(H1 ⊗ H2 ) , as exploited in (Vidal and Werner, 2002). 2. Since20

T (ˆ τ ν ) = (−1)δν2 τˆν

we have for two-qubit states ρˆ :21 (1 ⊗ T)(ˆ ρ) = ρˆ

⇐⇒



∀ ν ∈ {0, . . . , 3} , 

trace (ˆ τ µ ⊗ τˆ2 ) ρˆ = 0 ∀ µ ∈ {0, . . . , 3} .

DRAFT, October 17, 2007

17

See also (Doherty et al., 2003) 18 Note that  † ˆ (1 ⊗ T)(Aˆ† ) = (1 ⊗ T)(A)

19

Recall (5.31). Recall (A.22) 21 Compare (Altafini, 2003, Sect. II.B., Corollary 1). 20

∀ Aˆ ∈ L(H1 ⊗ H2 ) .

(6.15)

154

CHAPTER 6. HANDLING ENTANGLEMENT 3. For all ρˆ ∈ S(H1 ⊗ H2 ) : (1 ⊗ T)(ˆ ρ) Ψ = |{z} λ Ψ 6= 0 =⇒

(

 0 fulfilling (6.30). Then, if we define ˆ j def M =

q +

ˆ† p′j ρˆ′1 U j

q +

ρˆ1 /\ ρˆ H


−1

ˆ1 − Pˆρˆ H Pˆρˆ H + q p′j

∀ j ∈ {1, . . . , N ′ }

159

6.2. LOCAL OPERATIONS AND CLASSICAL COMMUNICATION (6.29) and (6.30) imply ′

N X

ˆ †M ˆ j = ˆ1 M j

(6.31)

   ˆj ⊗ ˆ ˆ † ⊗ ˆ1 = p′j trace 2 (ˆ M 1 ρˆ M ρ′ ) ∀ j ∈ {1, . . . , N ′ } , j

(6.32)

j=1

and trace 2 Since



trace 2 (ˆ ρ′′ ) = trace 2 (ˆ ρ′ ) ′ ′′ ρˆ , ρˆ ∈ Spure (H ⊗ H)



=⇒

TheoremA.4.7

(

   ˆ † = ρˆ′ ˆ ρˆ′′ ˆ1 ⊗ U ˆ1 ⊗ U ˆ ∈ L(H) , for some unitary U 

(6.32) shows that there are unitary Vˆ1 , . . . VˆN ′ ∈ L(H) with     ˆ † ⊗ Vˆ † = p′j ρˆ′ ∀ j ∈ {1, . . . , N ′ } . ˆ j ⊗ Vˆj ρˆ M M j j

This, together with (6.30) and (6.31), shows that ρˆ can be transformed into ρˆ′ by LOCC.

ˆ H ˆ ′ be Hermitian operators on the Hilbert space H with Lemma 6.2.5 Let H, spectral decompositions ˆ = H

n X

ν=1

Eν |ψν ihψν | ,

ˆ H ˆ ′ iff the conditions Then H

n X

Eν =

max π∈Sn

are fulfilled.

ν=1

ν=1

n X

Eν′ |ψν′ ihψν′ | .

Eν′

(6.33)

∀ n′ ∈ {1, . . . , n − 1}

(6.34)

′

′

n X

n X

ν=1

ν=1

and

ˆ′ = H

Eπ(ν) ≤ max π∈Sn

n X

ν=1

′ Eπ(ν)

Proof: See (Nielsen and Chuang, 2001, Proposition 12.11).

Remarks: 1. Obviously,

1 ˆ 1  ρˆ1 ∀ ρˆ1 ∈ S(H) . dim H 2. According to Theorem 6.2.4, therefore,31 



Spure (H ⊗ H) ⊂ SLOCC PˆH+ . DRAFT, October 17, 2007

31

Recall (6.2).

160

CHAPTER 6. HANDLING ENTANGLEMENT 3. For qubit states ρˆ, ρˆ′ we always have either ρˆ  ρˆ′ or ρˆ′  ρˆ or both.

4. However, for higher dimensional H neither ρˆ  ρˆ′ nor ρˆ′  ρˆ need be true for ρˆ, ρˆ′ ∈ S(H) as application of Lemma 6.2.5 to, e.g., the case ρˆ =

1 (7 φ1 + 7 φ2 + φ3 ) , 15

1 (11 φ1 + 2 φ2 + 2 φ3 ) 15

ρˆ =

shows if {φ1 , φ2 , φ3 } is a MONS of H . Note that LOCC would be much more powerful if intermediate use of nonlocally entangled ancillary pairs, restoring their original states, could be made: n

o

(j)

For j = 1 resp. j = 2 let φ1 , . . . , φ(j) be a MONS of the Hilbert nj space Hj , let Ψ=

n1 X

νµ=1

(1) λνµ φ(1) ν ⊗ φµ ,

Ψ′ =

n1 X

ν,µ1

(1) λ′νµ φ(1) ν ⊗ φµ

be pure states of the bipartite system with state space H1 ⊗ H1 and let n1 X n2 X

Φ= Φ′ =

ν1 α,β=1 n1 X n2 X







(2)

(2) (1) λνµ λanc ⊗ φ(1) αβ φν ⊗ φα µ ⊗ φβ

ν1 α,β=1









(2)

(2) (1) λ′νµ λanc ⊗ φ(1) αβ φν ⊗ φα µ ⊗ φβ

,



be pure states of the bipartite system with state space (H1 ⊗ H2 ) ⊗ (H1 ⊗ H2 ). Then it may happen that32 |Ψi 6 |Ψ′ i

but

|Φi  |Φ′ i

— an effect of the ancillary system in the state

n2 X

α,β=1

is called entanglement catalysis.

(2)

(2) λanc αβ φα ⊗ φβ that

Finally, given ρˆ, ρˆ′ ∈ Spure (H ⊗ H) , let us note that ρˆ can be transformed into ρˆ′ by local operations without communication iff ˆ trace 2 (ˆ ˆ† , trace 2 (ˆ ρ′ ) ∝ K ρ) K

ˆ †K ˆ ≤ ˆ1 K

(6.35)

ˆ ∈ L(H) . Let holds for some K trace 2 (ˆ ρ) =

n X

ν=1

pν |ψν ihψν | ,

trace 2 (ˆ ρ′ ) =

n X

ν=1

p′ν |ψν′ ihψν′ |

DRAFT, October 17, 2007

32

See (Nielsen and Chuang, 2001, Exercise 12.21) for an explicit example with (n1 , n2 ) = (4, 2) .

161

6.3. QUANTIFICATION OF ENTANGLEMENT

be spectral decompositions of ρˆ, ρˆ′ . Then, exploiting unitary transformations and ˆ one can show that (6.35) is equivalent to existence of the polar decomposition of K real numbers k1 , . . . , kn and a permutation π ∈ Sn with: n X

ν=1

6.2.2

p′ν = (kν )2 pπ(ν)

(kν )2 ≤ 1 ,

∀ ν ∈ {1, . . . , n} .

Entanglement Dilution

.. .

6.2.3

Entanglement Distillation

See, e.g., (Bowmeester et al., 2000, Section 8.4) and (Devetak and Winter, 2005).

6.3

Quantification of Entanglement33

For pure states |ΨihΨ| of a bipartite system S with state space H = H1 ⊗ H2 there is a generally accepted measure of entanglement, namely the entropy of entanglement 34 





def

Epure |ΨihΨ|



S1 trace 2 |ΨihΨ|

=

Th.









= S1 trace 1 |ΨihΨ| A.4.7

(6.36) ,

i.e. the von Neumann entropy of the partial states. Since 

Epure 



′

n X √

ν=1

pν φ(1) ν

⊗

 φ(2) ν

′

=−



n X

ν=1



p log2 (pν )

ν |{z} >0

holds for Schmidt decompositions, Epure |ΨihΨ| becomes maximal35 for n

o

n′ = min dim(H1 ), dim(H2 ) ,

pν =

DRAFT, October 17, 2007

ˇ aˇcek and Hradil, 2002). See also (Reh´ The unit of entanglement is one ebit. 35 Recall Remark 3 in the beginning of Section 5.1.

33

34

1 n′

∀ ν ∈ {1, . . . , n′ } .

162

CHAPTER 6. HANDLING ENTANGLEMENT

Hence







o

n

max E |ΨihΨ| = log2 min dim(H1 ), dim(H2 )

Ψ∈H1 ⊗H2 kΨk=1

(6.37)

holds for E = Epure . Strict requirements for every entanglement measure E on S(H1 ⊗ H2 ) : 1. E(ˆ ρ) ≥ 0 ∀ ρˆ ∈ S(H1 ⊗ H2 ) . 2. E(ˆ ρ) = 0 ∀ ρˆ ∈ Ssep (H1 ⊗ H2 ) . 3. E(ˆ ρ) = 0 =⇒ ρˆ ∈ Ssep (H1 ⊗ H2 )

∀ ρˆ ∈ S(H1 ⊗ H2 )

(could be relexed by cosideration of additional entanglement measures). 4. ρˆ′ ∈ SLOCC (ˆ ρ) =⇒ E(ˆ ρ′ ) ≤ E(ˆ ρ)

∀ ρˆ, ρ′ ∈ S(H1 ⊗ H2 ) .

5. For all pure states |Ψ1 ihΨ1 | , |Ψ2 ihΨ2 | ∈ S(H1 ⊗ H2 ) : 







Epure |Ψ1 ihΨ1 | < Epure |Ψ2 ihΨ2 |







Desirable for entanglement measures E on S(H1 ⊗ H2 ) : 1. E(ˆ ρ) ≥ E(ˆ ρ′ ) =⇒ ρˆ′ ∈ SLOCC (ˆ ρ)

∀ˆ ρ, ρˆ′ ∈ S(H1 ⊗ H2 )

(not possible). 2. Continuity 3. Additivity 4. Subadditivity 5. Convexity Standard entanglement measures:36 DRAFT, October 17, 2007

36



=⇒ E |Ψ1 ihΨ1 | < E |Ψ2 ihΨ2 | .

There cannot be a unique entanglement measure (Morikoshi et al., 2003).

163

6.3. QUANTIFICATION OF ENTANGLEMENT 1. Entanglement of formation37 is the convex continuation def

EF (ˆ ρ) =

ρˆ=

P inf ν

p

ρˆ

ν ν |{z} |{z} 0

pure

P





ρν ) ν pν S1 trace 2 (ˆ

of the entropy of entanglement (6.36) to all of S(H1 ⊗ H2 ) . Additivity of the entanglement of formation is generally conjectured but not yet proved. 2. Squashed Entanglement (Christandl and Winter, 2004) • In general, Werner states are not maximally entangled, i.e. their entanglement of formation (tangle) is not maximal for given a fixed degree of mixedness (linear entropy) (White et al., 2001). • Compare with (Eckert et al., 2002). (see also quant-ph/0210107)

• What about local superselection rules? (Verstraete and Cirac, 2003; Bartlett and Wiseman, 2003)

• Exploit the notion of truncated expectation values. (Lee et al., 2003) • What is the generalization of the latter for mixed states? • Existiert ein Abstandsmaß a la (Lee et al., 2003)?

• Warum verwendet man nicht den Hilbert-Schmidt-Abstand von Zust¨anden, der sich leicht mithilfe von Erwartungswerten von (Produkten von) Pauli-Operatoren ausdr¨ ucken l¨aßt? • Besteht ein Zusammenhang mit (Lee et al., 2003)? Der Abstand zur Menge der separablen Zust¨ande sollte doch ein Maß f¨ ur Verschr¨anktheit sein... Remarks: 1. For (dim(H1 ), dim(H1 )) either (2,2) or (3,2) PPT w.r.t. the second factor is necessary and sufficient for separability (Horodecki et al., 1996). 2. Otherwise states with bound entanglement, i.e. entangled PPT states, exist (Horodecki et al., 1996, Appendix). 3. In the 2-qubit case every entangled mixed state can be represented as a convex combination of a separable (in general mixed) state with a pure entangled state (Lewenstein and Sanpera, 1998). The representation with minimal norm of the pure state is unique. 4. Also this shows that, for the 2-qubit case, separability is equivalent to PPT. DRAFT, October 17, 2007

37

For pairs of qubits the entanglement of formation coincides with the concurrence (Wootters, 2001, Section 3.1).

164

CHAPTER 6. HANDLING ENTANGLEMENT

Appendix A A.1

Turing’s Halting Problem

The halting problem is the following: There is no algorithm by which one may decide for every program and every finite input to the program whether the program will halt or loop forever. The proof given by Turing is essentially as follows: Since every program may be encoded into a finite sequence (b1 , . . . , bn ) of bits the programs may be indexed by the corresponding numbers Pn n−ν . The same holds for all finite inputs. Now assume that ν=1 bν 2 there is an algorithm telling us for all (j, k) ∈ ZZ2+ whether program j will halt on input k . Then this algorithm may be used to write a program P with the following property: For every j ∈ ZZ+ , program P will hold on input j iff program j will not. Obviously, program P is different from program j for every j ∈ ZZ+ — a contradiction. A heuristic explanation is the following: There are uncountably many possibilities for infinite loops which, therefore, cannot be checked in a systematic way. But we cannot be sure whether a given program will halt on a given input or not unless • an infinite loop is found by chance or

• the program was actually tested on the input and found to halt. Let us finally note that the halting problem is a solution to Hilbert’s 23rd problem (see http://aleph0.clarku.edu/ djoyce/hilbert/).

165

166

APPENDIX A.

A.2

Some Remarks on Quantum Teleportation

Even though quantum teleportation, described in 1.2.2, seems to indicate some kind of quantum nonlocality, there is a naive ‘explanation’ relying on locality and some kind of realism: There is a set of four compatible relations, corresponding to the Bell states, between a pair of qubits. These correlations are so strong that the state (predicting ensemble averages) of the second qubit is fixed by that of the first qubit and the Bell relation (considered as an element of reality). Therefore, Alice need only inform Bob about the Bell relation of qubit 1 to some qubit 2 with known Bell relation to Bob’s qubit 3 in order to enable Bob to transform qubit 3 into the unknown state of qubit 1. If qubits 1 and 2 are accessible to Alice and far apart from Bob, then Alice can access this information without influencing Bob’s qubit (thanks to locality) , although disturbing qubits 1 and 2 in an uncontrollable way. Strictly speaking, of course, this picture is inconsistent: Two qubits may be in a factorized 2-qubit state with factors meeting none of the Bell relations. Nevertheless we claim that one of the Bell relations is an element of reality which we find by measuring w.r.t. to the Bell basis. This inconsistency, however, is typical for our talking about quantum systems: Even when we know the state Φ of a system (since its preparation is well specified), we may ask for the probability |hΨ | Φi|2 to find it in another state Ψ . Concerning the Bell relation the situation is even less disturbing: The 1-qubit states give no information about the actual relation between the partners of the individual pairs. Selection into subensembles corresponding to the 4 Bell relations has to be expected to change the partial 1-qubit states – even from a classical point of view. In order to test for the Bell relations it seems necessary to get the qubits into contact1 — not necessarily into interaction (Resch et al., 2002; Hofmann and Takeuchi, 2002). This way they loose their identity – a natural reason for the change of the total (internal) state through measurement. It seems that the Bell relations may be taken as elements of reality, but they can be applied to only one (freely chosen) set of 1-particle ‘properties’.2 DRAFT, October 17, 2007

1

See (Lloyd, 2000), however. 2 More generally, see (Griffiths, 2002).

167

A.3. QUANTUM PHASE ESTIMATION AND ORDER FINDING

A.3

Quantum Phase Estimation and Order Finding

˘ ϕ is Exercise 25 Show that the following quantum network acts as indicated if Ψ a n-qubit eigenstate of the unitary Operator Uˆ with eigenvalue ei ϕ : |0i |0i |0i ˘ϕ Ψ

  

H .. . H

.. .

.. . ···

s s

H .. .

ˆ 20 U

√1 2

s

···

.. .

.. . √1 2 √1 2

··· ···

.. .

.. . ···

ˆ 21 U

 

.. .

ˆ 2m U



|0i + ei 2  

m

|0i + ei 2 |0i + ei 2

ϕ

|1i

1

ϕ

0

ϕ

|1i |1i




 

˘ϕ . Ψ

First, let us consider the case ϕ = 2π

I(b) 2m+1

for some b ∈ {0, 1}m+1 .

Since (A.1) =⇒ e

i 2(m+1)−ν ϕ

(2.22) tells us that

= exp i 2π

ˆ −1

(A.1) =⇒ |bi = F



− m+1 2

2

m+1 X

ν=1

|0i + e



(m+1)−ν−µ 

bµ 2

µ=(m+1)−ν+1

m+1 O

(A.1)

i 2(m+1)−ν ϕ

|1i



!

.

,

(A.2)

Therefore, the phase ϕ considered in Exercise 25 can be determined by applying the inverse quantum Fourier transform to the state m+1 ˘ ϕ def Φ = 2− 2

m+1 O ν=1

− m+1 2

= 2

2m+1 X−1 j=0

(m+1)−ν ϕ

|0i + ei 2

|1i



(A.3)

ei ϕ j |jim+1

of the first m + 1 qubits of the output produced by the described network and measuring the result — if (A.1) holds exactly and everything works perfectly.

168

APPENDIX A.

If ϕ ∈ [0, 2π) is not of the form (A.1), we have ˘ϕ Fˆ −1 Φ

1

= m+1 (A.3),(2.18) 2 1

=

2m+1 1

=

2m+1 1

=

2m+1

2m+1 X−1 j,k=0

2π

ei ϕ j e−i k 2m+1 j |kim+1

2m+1 X−1 

ϕ i2π ( 2π −

e

k 2m+1

)

j,k=0

j

|kim+1

2m+1 X−1

1 − ei2π(2 2π −k) |kim+1 ϕ k 1 − ei2π( 2π − 2m+1 )

2m+1 X−1

1 − ei 2 ϕ |kim+1 . ϕ k 1 − ei2π( 2π − 2m+1 )

k=0

k=0

m+1 ϕ

m+1

(A.4)

˘ϕ Then (A.4) implies that the probability p(k) for finding |kim+1 when testing Fˆ −1 Φ fulfills the inequality p(k) ≤ Let us define



1

1−e 22m



ϕ i2π ( 2π −

k 2m+1







−2

)

.

(A.5)

ϕ def D(k) = min 2m+1 − ν − k ∀ k ∈ ZZ . ν∈ZZ 2π m Then, given d ∈ {3, . . . , 2 − 1} , the probability pd for getting any state |kim+1 ˘ ϕ fulfills the inequality with D(k) > d when testing Fˆ −1 Φ

pd ≤

1 . 2 (d − 2)

 Proof: Choosing k0 ∈ 0, . . . , 2m+1 − 1 such that def

∆ = 2m+1

ϕ − k0 ∈ (0, 1) 2π

we get pd

X

=

p(k)

k∈{0,...,2m+1 −1} D(k)>d

≤ (A.5) =

≤

1 22m 1 22m 1 22m

X

k∈{0,...,2m+1 −1} minν∈ZZ |k0 −k−ν 2m+1 |≥d

X

j∈{−2m +1,...,2m }

minν∈ZZ |j−ν 2m+1 |≥d

X

j∈{−2m +1,...,2m } j ∈{−d+1,...,d−1} /

k−k0 −2 −i2π m+1 i2π ∆ 2 1 − e 2m+1 e

−2 j −i2π m+1 i2π ∆ 2 1 − e 2m+1 e

−2 j −i2π m+1 i2π ∆ 2 . 1 − e 2m+1 e

(A.6)

A.3. QUANTUM PHASE ESTIMATION AND ORDER FINDING

169

By (A.6), therefore3 ≤

pd

=

≤ ≤

X

1 22m

j∈{−2m +1,...,2m } j ∈{−d+1,...,d−1} /

−2 2 2π ∆ − j π m+1 2

 2m −d X 1 X −2 −2 (∆ − j)  (∆ − j) + 4 j=−2m +1 j=d   m 2 −d X 1 X −2 j −2 + (1 − j)  4 j=−2m +1 

j=d

1 2

m 2X −1

j −2

j=d−1 Z ∞

dx x2

≤

1 2

=

1 . 2 (d − 2)

d−2

˘ ϕ w.r.t. the (A.6) tells us that, with probability ≥ 1 − (d − 2)−1 /2 , testing Fˆ −1 Φ computational (m + 1)-qubit base gives a state |bi for which I(b) d min ϕ − 2π ν − 2π m+1 ≤ 2π m+1 . ν∈ZZ 2 2

Now, let x and N be arbitrarily given coprime positive integers. Then the order finding problem is to determine n

def

o

′

r = min r′ ∈ IN : xr = 1 mod N , the order of x modulo N . Defining

n

def

L = min l ∈ IN : N ≤ 2l and

def

o

[j mod N ] = min {k ∈ ZZ+ : k = j mod N }

∀ j ∈ ZZ ,

DRAFT, October 17, 2007

3

Note that

since

d dθ

|θ| |θ| 1 θ iθ 1 − e = sin ≥ √ ≥ 2 2 π 2 2 

θ θ sin − √ 2 2 2



=

∀ θ ∈ [−π, +π] ,

1 θ 1 cos − √ ≥ 0 ∀ θ ∈ [0, +π] . 2 2 2 2

170

APPENDIX A.

we have4

and

Dh

( ) h iE j r = x mod N : j ∈ IN L

i h iE = δjk xj mod N xk mod N L

∀ j, k ∈ {0, 1, . . . , r − 1} .

Therefore the states

r−1 h iE def 1 X −i 2π s j j r e x mod N Ψs = √ L r j=0

∀ s ∈ {0, . . . , r − 1}

(A.7)

are normalized and, thanks to (2.17), fulfill the equation h iE X s 1 r−1 √ e+i 2π r j Ψs = xj mod N L r s=0

∀ j ∈ {0, . . . , r − 1} .

Especially for j = 0 the latter gives

X 1 r−1 Ψs . |1iL = √ r s=0

(A.8)

We do not yet know the states Ψs explicitly since we do not yet know r . But we know that these states exist and have the nice property that they are eigenstates of the unitary5 L-qubit operator Uˆ characterized by def Uˆ |yiL =



|[x y mod N ]iL |yi

if y ∈ {0, . . . , N − 1} else

n

o

∀y ∈ 0, . . . , 2L − 1 .

More precisely, we have s Uˆ Ψs = ei 2π r Ψs

∀ s ∈ {0, . . . , r − 1} .

˘ ϕ by |1i , in Exercise 25 we get the total output Therefore, replacing n by L and Ψ L 6 state X 1 r−1 ˘ 2π s ⊗ Ψs . √ Φ r r s=0

Since the Ψs form an orthonormal subset of the L-qubit state space, we may assume that the partial state of the system of the first m + 1 qubits is one of the vector ˘ 2π s with equal probability and the results concerning phase estimation show: states Φ r DRAFT, October 17, 2007

4

Recall (1.12). 5 Note that, because gcd(x, N ) = 1 , y1 6= y2 mod N

6

Recall (A.3).

=⇒ x y1 6= x y2 mod N .

171

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS ˘ 2π s is tested w.r.t. the computational (m + 1)-qubit basis When Fˆ −1 Φ r the probability for getting a state |kim+1 with s d k − m+1 ≤ m+1 , r 2 2

for d ≥ 3 , is not less than 1 −

1 . 2 (d − 2)

Choosing d and 2m+1 /d sufficiently large we obtain, this way, an excellent approximation k/2m+1 of s/r for some random s ∈ {0, . . . , r − 1} .

A.4 A.4.1

Finite-Dimensional Quantum Kinematics General Description

• The state space of a finite-dimensional quantum system is a finite-dimensional complex Euclidean space H the inner product of which we denote by h. | .i . • The vectors ψ ∈ H with norm 1 correspond to pure states.7 • In the interaction picture, used here, the state vectors ψ do not depend on time as long as the states are not disturbed by additional interaction (e.g. with some ‘measurement’ apparatus). • The state vectors label equivalence classes of preparation procedures for ensembles of individual systems of the considered type. • Preparation procedures are called equivalent if the ensembles they provide cannot be distinguished by the statistical outcome of measurements. • Important measurements performable (in principle) on individual elements of an ensemble are projective measurements: The individual drawn from an ensemble with state vector ψ=

n X

ν=1

hφν | ψi φν

will be forced to a transition (if necessary) into one of the φν ensembles.8 According to the rules of quantum theory we have9 |hφν | ψi|2 = probability for the transition ψ 7→ φν

(A.9)

for all ν ∈ {1, . . . , n} . DRAFT, October 17, 2007

7

We assume that there are no superselection rules. 8 I.e., an ensemble formed by a (sufficiently ) large number of individuals for which φν is ‘measured’ actually corresponds (sufficiently well) to φν , unless additional perturbations have appeared. 9 Therefore, hφn u | ψi is called probability amplitude for the transition ψ 7→ φν .

172

APPENDIX A.

• Observables are operators Aˆ ∈ L(H) of the form Aˆ =

n X

ν=1

a |φν ihφν | ,

{φ1 , . . . , φn } an orthonormal basis of H ,

ν |{z} ∈IR

(A.10)

with the following interpretation: In a state with state vector φν the physical entity A (corresponding ˆ has the definite value aν . to A) This together with (A.9) implies:10 



trace |ψihψ| Aˆ =

(

expectation value for A in a state with state vector ψ .

(A.11)

• Individuals which are only known to be members of an ensemble with state vector ψj with probability λj for j ∈ {1, . . . , N } , to be described by the density matrix 11 ρˆ =

N X

j=1

in the sense that

λj

|{z}

ψj

|{z}

,

N X

λj = 1 , form an ensemble

j=1

trace (ˆ ρ) = 1 ,

(A.12)

≥0 normalized

ˆ = expectation value for Aˆ . trace (ˆ ρ A) • In this sense, the set of all states corresponds to def

S(H) =

(A.13)

o

n

ˆ =1 . Aˆ ∈ L(H) : Aˆ ≥ 0 , trace (A)

States12 with ρˆ2 6= ρˆ are called mixed. DRAFT, October 17, 2007

10

Note that

n E D   X 2 aν |hφν | ψi| = ψ Aˆ ψ . trace |ψihψ| Aˆ = ν=1

11

Such ensembles arise, e.g., from projective measurements on pure states if the individuals are not selected according to the ‘measurement’ results. 12 From now on we identify states with their density matrices. Note that ρˆ = |ψihψ|

for some ψ ∈ H

⇐⇒

ρˆ2 = ρˆ .

173

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS Lemma A.4.1 Let ρˆ1 , ρˆ2 ∈ S(H) . Then 0 ≤ trace (ˆ ρ1 ρˆ2 ) ≤ 1 and ⇐⇒

trace (ˆ ρ1 ρˆ2 ) = 1

∃ ψ ∈ H : ρˆ1 = ρˆ2 = |ψihψ| .

Outline of proof: Thanks to the spectral theorem there are an orthonormal basis {φ1 , . . . , φn } of H and λ1 , . . . , λn ≥ 0 with ρˆ1 =

n X

ν=1

and hence

λν |φν ihφν |

n X

trace (ˆ ρ1 ρˆ2 ) =

ν,µ=1 n X

=

ν=1

λν hφµ | φν ihφν | ρˆ2 φµ i

λν hφν | ρˆ2 φν i .

Therefore, 0 ≤ trace (ˆ ρ1 ρˆ2 ) ≤ 1 and   trace (ˆ ρ1 ρˆ2 ) = 1 ⇐⇒ 0 < λν < 1 =⇒ hφν | ρˆ2 φν i = 1

∀ µ ∈ {1, . . . , n}

⇐⇒ ∃ ν0 ∈ {1, . . . , n} : ρˆ1 = ρˆ2 = |φν0 ihφν0 | .

Lemma A.4.2 Let13 ψ1 , . . . , ψN ∈ H . Then N X

k=1

!

|ψk ihψk | H = span

N [

k=1

!

{ψk } .

(A.14)

Outline of proof: Thanks to the spectral theorem there is an ONS {φ1 , . . . , φn′ } ⊂ H with n′ N X X λν |φν ihφν | (A.15) |ψk ihψk | = ν=1

k=1

for suitable λ1 , . . . , λn′ > 0 and, consequently, ! N X |ψk ihψk | H = span k=1

Then

N X

k=1

2

|hχ | ψk i|

= =

(A.15)

=

N X

k=1 n′ X

ν=1 n′ X

ν=1

N [

k=1

{φk }

!

.

(A.16)

hχ | ψk ihψk | χi λν hχ | φν ihφν | χi λν |hχ | φν i|

2

∀χ ∈ H

DRAFT, October 17, 2007

13

If not stated otherwise, by H , H1 , , H2 we denote arbitrarily given finite-dimensional complex Euclidean vector spaces.

174

APPENDIX A. and, consequently, χ ⊥ ψk

∀ k ∈ {1, . . . , N }

⇐⇒

i.e. N [

span

k=1

{ψk }

!

χ ⊥ φν 

= span 



′

n [

ν=1

The latter together with (A.16) implies (A.14).

∀ ν ∈ {1, . . . , n} ,

{φν } .

′ Corollary A.4.3 Let ψ1 , . . . , ψN , ψ1′ , . . . , ψN ∈ H . Then N X

k=1

|ψk ihψk | =

N X

k=1

|ψk′ ihψk′ | .

(A.17)





holds iff there is a unitary N × N -matrix Uj k with14 ψk =

N X

Uk j ψj′

∀ k ∈ {1, . . . , N } .

j=1

(A.18)

Proof: Assume that (A.17) holds and, as in the proof of Lemma A.4.2, let us choose an orthonormal basis {φ1 , . . . , φn } ⊂ H and λ1 , . . . , λn′ > 0 for which (A.15) holds. Then Lemma A.4.2 implies   ′ ! n N [ [ span {φν } . {ψk } = span  ν=1

k=1

Especially, the ψk ’s can be written as linear combinations ′

ψk =

n X

ν=1

ckν

p +

λ ν φν

of φν ’s. Then we have ′

n X

ν=1

λν |φν ihφν |

= (A.15)

n′  N X X

k=1 ν,µ=1

ckµ

∗

ckν λν |φν ihφµ | .

Since the |φν ihφµ | form an ONS w.r.t. the Hilbert-Schmidt scalar product D E ˆ def ˆ ∀ A, ˆ B ˆ ∈ S(H) , Aˆ B = trace (Aˆ† B) (A.19) this implies

N X

k=1 DRAFT, October 17, 2007

14

ckν


∗

ckµ = δνµ

∀ µ, ν ∈ {1, . . . , n′ } ,

  In general, of course, the Uj k are not fixed by (A.18)

175

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS i.e. the

 c1ν   cν =  ...  , cN ν 

ν ∈ {1, . . . , n′ } ,

form an orthonormal system in CN .
Extending this to an orthonormal basis of CN we get a unitary N × N -matrix ckν with ψk =

N X

p +

ckν

ν=1

where

λ ν φν

∀ k ∈ {1, . . . , N } ,

(A.20)

def

λν = 0 for ν > n′ . ν
Similarly, we get a unitary N × N -matrix c′k with ψk′ =

N X

c′k

ν

ν=1

and hence p +

λ ν φν =

N X

p + c′l

l=1

λ ν φν

ν
∗

ψl′

∀ k ∈ {1, . . . , N }

∀ ν ∈ {1, . . . , N } .

(A.21)

Combining (A.20) with (A.21) we get (A.18) for the unitary matrix with components def

Uk l =

N X

ckν c′l

ν=1

ν
∗

∀ k, l ∈ {1, . . . , N } .

Conversely, (A.17) follows from (A.18) by straightforward calculation.

A.4.2

Qubits

Qubits are 2-dimensional quantum systems for which some orthonormal computational basis {|0i , |1i} of their state space H is chosen. According to the standard convention      α  ψ= w.r.t. |0i , |1i  β ∀ α, β ∈ C  def  ⇐⇒ ψ = α |0i + β |1i we have

|0i =

 

1 0

,

|1i =

 

All Aˆ ∈ L(H) may be identified with their matrix

tional basis:

ψ=



α β



=⇒

Aˆ ψ =



A11 A21

A12 A22

0 1

.



A11 A21



α β



A12 A22



w.r.t. the computa



w.r.t. |0i , |1i .

176

APPENDIX A.

n √ √ o Here an orthonormal basis w.r.t. the scalar product (A.19) is τˆ0 / 2, . . . , τˆ3 / 2 , where 0

τˆ =



1 0 0 1



1

τˆ =



0 1 1 0



2

τˆ =



0 i

−i 0



3

τˆ =



1 0 0 −1



(A.22)

are the well-known Pauli matrices. Therefore we have 3 1X ˆτ ν ) τˆν ˆ trace (Aˆ A= 2 ν=0

especially ρˆ = where the vector15

 1 ˆ 1 + ρ · τˆ 2

∀ Aˆ ∈ L(H) ,

∀ ρˆ ∈ S(H) ,

def

ρ = trace (ˆ ρ τˆ ) fulfills16 |ρ| ≤ 1 ,

ρˆ2 = ρˆ ⇐⇒ |ρ| = 1 .

Note that the components of ρ are just expectation values of observables which are sufficient for quantum state tomography. Remark: Since (eϑ,ϕ · τˆ ) χϑ,ϕ = χϑ,ϕ

holds for

 ϑ cos  e def  1 eϑ,ϕ χϑ,ϕ =  ϕ ϑ, +i 2 e sin 1 every pure state corresponds to a definite spin orientation in the spin- 12 case — where h ¯ ˆ 2 τ is the spin vector observable .

A.4.3



 sin ϑ cos ϕ def =  sin ϑ sin ϕ  , cos ϑ



−i

ϕ 2

Bipartite Systems n

(j)

o

For j = 1, 2 let φ1 , . . . , φ(j) be an orthonormal basis of the state space Hj nj of a quantum mechanical system Sj . If S1 and S2 are distinguishable then the bipartite system S composed of these two has the state space H = H1 ⊗ H2 with orthonormal basis n

o

(2) φ(1) ν ⊗ φµ : ν ∈ {1, . . . , n1 } , µ ∈ {1, . . . , n2 } .

DRAFT, October 17, 2007

15

Usually, the vector of coherence ρ is called Bloch vector when associated with electron spin and Stokes vector when associated with photon polarization. More generally see (Altafini, 2003). 16 Note that  1 2 1 = trace (ˆ ρ) ≥ trace (ˆ ρ2 ) = 1 + |ρ| . 2

177

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS Extending ⊗ to a bilinear mapping of H1 × H2 into H1 ⊗ H2 we get 



ˆ1 ⊗ A ˆ2 : A ˆ 1 ∈ L(H1 ) , A ˆ 2 ∈ L(H2 ) , L(H1 ⊗ H2 ) = span A where the linear operators Aˆ1 ⊗ Aˆ2 are fixed by 

Aˆ1 ⊗ Aˆ2











def ψ (1) ⊗ ψ (2) = Aˆ1 ψ (1) ⊗ Aˆ2 ψ (2)







∀ ψ (1) , ψ (2) ∈ H1 ⊗ H2 . (A.23) If Aˆ1 resp. Aˆ2 is the observable of S1 resp. S1 corresponding to the physical entity A1 resp A2 then Aˆ1 ⊗ Aˆ2 is the observable corresponding to the physical entity A1 A2 . If one is only interested in the subsystem S1 of the total system S in state ρˆ then it is sufficient to know its partial state 17 def

def

ρˆ1 = trace 2 (ˆ ρ) =

n2 D X

µ=1

since:

 

trace ρˆ Aˆ1 ⊗ ˆ1 Note that







E

(2) ˆ φµ , φ(2) µ ρ

(A.24)

= trace (ˆ ρ1 Aˆ1 ) ∀ Aˆ1 ∈ L(H1 ) . i.g.

ρˆ pure

6=⇒

ρˆ1 pure .

Example: If S1 and S2 are qubit systems then for the the Bell state 1 def Ψ− = √ 2

 

1 0

⊗

 

0 1

−

 

0 1

⊗

 

1 0

we have

  ED 1 trace 2 Ψ− Ψ− = ˆ1 , 2 i.e. the partial states give no information at all: 



  ED 1 trace Ψ− Ψ− |ψihψ| ⊗ ˆ1 = kψk2 2

∀ ψ ∈ C2 .

But there are strong (non-classical) correlations between the subsystems, since 1 Ψ− = √ (ψ ⊗ ψ⊥ − ψ⊥ ⊗ ψ) 2 DRAFT, October 17, 2007

17

The so-called partial trace trace 2 w.r.t. the second factor is the linear mapping of L(H1 ⊗H2 ) into L(H1 ) characterized by   |ψ1 ihψ1′ | ∀ ψ1 , ψ1′ ∈ H1 , ψ2 , ψ2′ ∈ H2 . hψ2′ | ψ2 i trace 2 |ψ1 ⊗ ψ2 ihψ1′ ⊗ ψ2′ | = | {z }   =trace |ψ2 ihψ2′ | The partial trace w.r.t. the first factor, written trace 1 , is defined similarly.

178

APPENDIX A. for all normalized ψ ∈ C2 , where 

α β



def

=

⊥



−β ∗ α∗



for all α, β ∈ C ,

and hence   ED    1 − − trace Ψ Ψ |ψihψ| ⊗ |φihφ| = |hφ | ψ⊥ i|2

2

holds for all normalized ψ, φ ∈ C2 .

Definition A.4.4 A state ρˆ of the bipartite system S with state space H1 ⊗ H2 is called separable iff there is a sequence {ˆ ρN }N ∈IN of states of the form ρˆN = with 18

N X

λ

ρˆ(1) ⊗ ρˆ(2) ν

ν ν |{z} |{z} ν=1

≥0 ∈S(H1 )

|{z}

(A.25)

∈S(H2 )

lim kˆ ρ − ρˆN k1 = 0 .

N →∞

Theorem A.4.5 (Horodecki) Let S be a bipartite system with state space H1 ⊗ H2 . Then the separable states of S are exactly those of the form (A.25) with N ≤ (dim(H1 ⊗ H2 ))2 . Proof: See (Horodecki, 1997).

Lemma A.4.6 Let Ψ be a normalized vector in H1 ⊗ H2 . Then ρˆ = |ΨihΨ| is separable iff Ψ = φ(1) ⊗ φ(2) (A.26)

holds for some normed φ(1) ∈ H1 and φ(2) ∈ H2 .

Outline of proof: Let ρˆ = ρˆ2 be separable, hence of the form ρˆ =

N X

λν ρˆ(1) , ⊗ ρˆ(2) ν ν |{z} |{z} |{z}

ν=1 >0 ∈S(H) DRAFT, October 17, 2007

18

Recall (5.31) and Footnote 22 of Chapter 5.

∈S(H)

N X

ν=1

λν = 1 .

179

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS Then, by Lemma A.4.1, trace (ˆ ρ2 ) N X λν λµ trace H1 (ˆ ρ(1) ˆ(1) = ρ(2) ˆ(2) ν ρ µ ) trace H2 (ˆ ν ρ µ ) | {z } {z } | ν,µ=1

1 =

∈[0,1]

and, therefore,

∈[0,1]

ρ(2) ˆ(2) ρ(1) ˆ(1) trace H1 (ˆ ν ρ µ ) = 1 ∀ ν, µ ∈ {1, . . . , N } . ν ρ µ ) = trace H2 (ˆ By by Lemma A.4.1, again, the latter is equivalent to the existence of normed φ(1) ∈ H1 and φ(2) ∈ H2 with ED (j) ρˆ(j) φ(j) ∀ ν ∈ {1, . . . , N } , j ∈ {1, 2} ν = φ and hence (A.26).

Conversely, (A.26) together with ED def ρˆ(j) = φ(j) φ(j)

∀ j ∈ {1, 2}

implies

|ΨihΨ| = ρˆ(1) ⊗ ρˆ(2)

and hence separability of |ΨihΨ| .

Theorem A.4.7 Every vector state on H1 ⊗ H2 allows a Schmidt decomposition,19 i.e. for every Ψ ∈n H1 ⊗ H2 there number n′ ∈ IN o are a n unique Schmidt o (1) (1) (2) (2) and orthonormal subsets φ1 , . . . , φn′ resp. φ1 , . . . , φn′ of H1 resp. H2 with ′

Ψ=

n X

s

ν |{z}

ν=1 >0

(2) φ(1) ν ⊗ φν

(A.27)

for suitable Schmidt  coefficients s1 , . . . , sn′ . If the eigenvalues of the partial state trace 2 |ΨihΨ| are non-degenerated then the Schmidt decomposition of Ψ is unique. Outline of proof: Consider any o 2 . Thanks to the spectral theorem n Ψ ∈ H1 ⊗ H (1) (1) there are an orthonormal basis φ1 , . . . , φn1 of H1 , a positive integer n′ ≤ n1 , and s1 , . . . , sn′ > 0 with n1   X ED 2 (1) φ trace 2 |ΨihΨ| = (sν ) φ(1) ν , ν ν=1

def

sν = 0

∀ ν > n′ .

(A.28)

Then there are ψ1 , . . . , ψn1 ∈ H2 with Ψ=

n1 X

ν=1

φ(1) ν ⊗ ψν

DRAFT, October 17, 2007

19

For the generalization to n-partite systems see (Carteret et al., 2000)

(A.29)

180

APPENDIX A. and, therefore, n1 ED   X φ(1) hψµ | ψν i φ(1) trace 2 |ΨihΨ| = µ . ν ν,µ=1

The latter together with (A.28) implies 2

hψµ | ψν i = (sν ) δνµ

∀ ν, µ ∈ {1, . . . , n1 } .

Thus, with

ψν ∀ ν ∈ {1, . . . , n′ } , sν (A.29) becomes equivalent to (A.27). Since, conversely, (A.27) implies (A.28) the stated uniqueness properties are obvious. def

φ(2) = ν

Remarks: 1. Note that the Bell states (4.55) have Schmidt number 2 and hence, by Lemma A.4.6, be separable. 2. The vector state  1 |0, 0i + |0, 1i + |1, 0i + |1, 1i , 2 







ˆ |0i , where ˆ |0i ⊗ H however, is separable since equal to H  1  ˆ |0i def H = √ |0i + |1i , 2

 1  ˆ |1i def H = √ |0i − |1i 2

characterizes the unitary Hadamard operator, strongly used in quantum computing. 3. For mixed states ρˆ ∈ S(H1 ⊗ H2 ) there is a Schmidt-like decomposition of the Form ′

ρˆ =

n X

ν=1

s

ν |{z} >0

ˆ(2) Aˆ(1) ν ⊗ Aν

with n′ ≤ (dim H1 )2 , (dim H2 )2 . According to (Herbut, 2002, Corollary 1) the operators Aˆ(j) ν may be chosen Hermitian and such that 



ˆ(j) = 0 . ν 6= µ =⇒ trace Aˆ(j) ν Aµ However, in general, they cannot be positive. Pure states are non-separable iff their partial states are mixed.20 Therefore, the correlations in non-separable pure states are non-classical.21 Obviously, the partial DRAFT, October 17, 2007

20

Usually, non-separable states are called entangled; see (Verstraete and Cirac, 2003), however. 21 In a way, also the mixed non-separable states are non-classically correlated (Werner, 1989).

181

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS transpose 





¯  1⊗T

= (4.13)

n2 X

n1 X

λ ν 1 µ1 ν 2 µ2

ν1 ,µ1 =1 ν2 ,µ2 =1

n2 X

ν2 ,µ2 =1





ED (2) (2) φ(1) ⊗ φ λν1 µ1 ν2 µ2 φ(1) ν1 µ2 µ1 ⊗ φν2 n

(2)

w.r.t. the orthonormal basis φ1 , . . . , φ(2) n2 states: ! (1 ⊗ T)

N X

 ED (1) (2)  φ(1) φν1 ⊗ φ(2) µ1 ⊗ φµ2 ν2

T2

λν ρˆ(1) ν

ν=1

⊗

ρˆ(2) ν

o

=

of H2 must be positive for separable N X

ν=1

ρ(2) λν ρˆ(1) ν ⊗ T(ˆ ν ) ≥ 0.

ˆ λ with λ > 1/3 , considered at the end of Therefore, the mixed Werner states W 22 Section 4.2.1, are non-separable. Lemma A.4.8 For every state ρˆ ∈ H there is a purification in H ⊗ H , i.e. a normalized vector Ψ ∈ H ⊗ H with 



ρˆ = trace 2 |ΨihΨ| .

(A.30)

Outline of proof: Thanks to the spectral theorem there are an orthonormal basis o n (1) (1) of H and λ1 , . . . , λn ≥ 0 with φ1 , . . . , φn ρˆ =

n X

ν=1

For the normalized vector23

ED (1) φ λν φ(1) ν , ν def

Ψ =

n p X +

ν=1

n X

λν = 1 .

(A.31)

ν=1

(1) λν φ(1) ν ⊗ φν

then, we get (A.30).

DRAFT, October 17, 2007

22

A decomposition of the 2-qubit Werner states, taking the form (A.25) for λ ≤ 1/3 , is presented on page 6 of: http://www.physik.uni-augsburg.de/weh-school/bruss.pdf 23 For the set of all suitable Ψ see (Kuah and Sudarshan, 2003, Lemma 1).

182

APPENDIX A.

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Index adder, 16

word, 108 coding quantum dense, 24 coherence vector of, 176 coherent superposition, 18, 23 complexity computational, 31 computation -al basis, 18–20 classical, 15, 17, 22 reversible, 15, 16 complexity, 31 concurrence, 163 CSS codes, 114

balanced, 22 Bell measurement, 25, 26 network, 24, 26 states, 24 Bernstein-Vazirani oracle, 23 binary digit, 11 bit, 11 qu-, 18 Bloch vector, 176 Campbell-Hausdorff formula, 64 circuit classical logic, 12, 15 computational, 11 equivalent, 11 graph, 12 logic, 14 classical computation, 15, 17, 22 network, 18 logic circuit, 12, 15 reversible computation, 15, 16 reversible network, 15 closed quantum system, 103 code Calderbank-Shor-Steane, 114 linear classical dual, 109 quantum, 110 words, 110 stabilizer, 113 Steane, 116

decoherence, 104 dense coding, 24 Deutsch-Jozsa oracle, 22, 23 problem, 22 dual code, 109 ebit, 161 entangled, 24 pure state, 24, 26, 103 entanglemen catalysis, 160 entanglement of formation, 163 swapping, 26 witness, 147, 151 entropic inequalities, 154 entropy `i, 155 Reny equivalent classical network, 11 199

200 preparation procedures, 171 quantum network, 21 error correction, 18 Euler angles, 29 fast Fourier, 43 flip operator, 100 Fourier transform fast, 43 Fredkin gate, 14 function balanced, 22 decision, 14 gate, 11 Fredkin, 14 non-deterministic, 19 quantum, 19, 20 universal, 27 reversible, 13 Toffoli, 14 universal, 15, 16 graph, 12 halting problem, 11, 165 Hamming distance, 108 interaction picture, 171 LOCC, 156 logic circuit, 14 measurement, 18, 19 Bell, 25, 26 projective, 171 mixture, 104 network adder, 16 Bell, 24, 26 classical 2-bit, 17 computational, 18 reversible, 15

INDEX optimization, 16 quantum equivalent, 21 quantum computational, 18 reversible, 15 teleportation, 25 open quantum system, 104 operator positive support, 136 oracle Bernstein-Vazirani, 23 Deutsch-Jozsa, 22, 23 order of an integer modulo N , 169 Partial transpose, 152 Pauli group, 111 Pauli matrices, 104 polarization identity, 147 post selection, 26 problem Deutsch-Jozsa, 22 halting, 165 projective measurement, 171 quantum closed system, 103 code, 110 code words, 110 computation network, 18 result, 19 dense coding, 24 gate, 19, 20 measurement, 18 network equivalent, 21 open system, 104 parallelism, 18 state entangled, 24, 26, 103 tomography, 176

201

INDEX state collapse, 18 teleportation, 25, 26 wire, 19 qubit, 18

norm, 137 partial, 177 transposition, 100, 151 partial, 152

Range criterion, 154 register, 11 n-qubit, 19 state space, 18 `i entropies, 155 Reny result of a quantum computation, 19 reversible classical network, 15

universal gate, 15, 16 quantum gate, 27

Sleator-Weinfurter construction, 31 stabilizer, 113 codes, 113 state Bell, 24 mixed, 104 pure, 171 quantum entangled, 24, 26, 103 separable, 151, 178 Werner, 151 state space, 171 Steane code, 116 Stokes vector, 176 superoperator, 96 superposition coherent, 23 teleportation, 26 network, 25 of entanglement, 26 tensor product formalism of quantum mechanics, 19 Toffoli gate, 14 tomography quantum state, 176 trace

Werner states, 151 wire, 15 quantum, 19