Logically reversible measurements: Construction and application

JID:PLA AID:24710 /SCO Doctopic: Quantum physics

[m5G; v1.221; Prn:1/09/2017; 15:07] P.1 (1-5)

Physics Letters A ••• (••••) •••–•••

1

67

Contents lists available at ScienceDirect

2

68

3

69

Physics Letters A

4

70

5

71

6

72

www.elsevier.com/locate/pla

7

73

8

74

9

75

10

76

11 12 13 14 15 16 17 18

77

Logically reversible measurements: Construction and application

78 79

Sunho Kim a , Juncheng Wang a , Asutosh Kumar b,c , Akihito Soeda d , Junde Wu a,∗

80 81

a

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India c Homi Bhaba National Institute, Anushaktinagar, Mumbai 400094, India d Department of Physics, The University of Tokyo, Bunkyo Ku, Tokyo 113-0033, Japan

82

b

83 84

19

85

20 21

86

a r t i c l e

i n f o

a b s t r a c t

87

22 23 24 25 26 27 28 29 30 31

88

Article history: Received 14 July 2017 Received in revised form 29 August 2017 Accepted 29 August 2017 Available online xxxx Communicated by P.R. Holland

We show that for any von Neumann measurement, we can construct a logically reversible measurement such that Shannon entropies and quantum discords induced by the two measurements have compact connections. In particular, we prove that quantum discord for the logically reversible measurement is never less than that for the von Neumann measurement. © 2017 Published by Elsevier B.V.

89 90 91 92 93 94

Keywords: Quantum measurement Quantum information Quantum discord

95 96 97

32

98

33

99

34 35

100

1. Introduction

A

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Measurement, as envisaged, plays an inevitable role in quantum mechanics, and lies at the heart of “interpretational problem” of quantum mechanics. Nonetheless, different views of measurement almost universally agree on the measurement outcomes. A quantum measurement is described in terms of a complete set of positive operators for the system to be measured. A few examples of quantum measurement are von Neumann measurement [1] which consists of orthogonal projectors, positive-operator-valued measure (POVM) [2], unitarily reversible measurement [3,4], etc. The most general type of measurement that can be performed on a quantum system is known as a generalized measurement [5,6]. Any measurement on a quantum state is inherently associated with wave function collapse and probability distribution. We recollect the necessary preliminaries briefly below. Quantum measurements Let H be a finite dimensional complex Hilbert space, which represents some quantum system. The set of quantum states ρ on H is denoted by D(H). A quantum measurement on H is a set  ≡ { x }x∈ X ⊆ L (H) of positive operators indexed by x ∈ X and satisfies x x = 1H . Given a quantum state ρ ∈ D(H) and a quantum measurement  = {x }x∈ X , then a probability distribution p = { p (x)}x∈ X is induced where p (x) = T r (x ρ )

59 60 61 62 63 64 65 66

is the probability of the outcome x to occur. In this case,

*

Corresponding author. E-mail addresses: [email protected] (S. Kim), [email protected] (J. Wang), [email protected] (A. Kumar), [email protected] (A. Soeda), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.physleta.2017.08.062 0375-9601/© 2017 Published by Elsevier B.V.

ρ A†

ρ is trans†

101

formed into the quantum state ρx = px (x) x , where x = A x A x . If  = {x }x∈ X is a set of orthogonal projectors, then the measurement {x }x∈ X is said to be a von Neumann measurement [1]. The celebrated Neumark extension theorem [7,8] states that each quantum measurement can be seen as a von Neumann measurement on a larger Hilbert space [9]. We know that in a generalized measurement process, the input state ρ cannot always be retrieved with a nonzero success probability by a “reversing operation” on the state ρx . A measurement {x }x∈ X is called logically reversible [10] if the premeasurement state ρ of the measured system is uniquely determined from the postmeasurement state ρx and the outcome of the measurement. Ueda et al. in Ref. [10] have shown that the measurement {x }x∈ X is logically reversible if and only if each measurement operator x is a reversible operator. Moreover, if for each measurement operator x , there exists a unitary operator U x such that

102

† U x xU x

119

ρ

= ρ,

(1.1)

for each state ρ whose support lies on a subspace M of H, then {x }x∈ X is called the unitarily reversible measurement [4]. It is clear that any von Neumann measurement {x }x∈ X is not logically reversible except X has only a single element. Note that in a logically reversible measurement, the system’s information is preserved during the measurement process. Thus, the reversibility of a measurement is related to the information gained from that measurement. Quantum teleportation [11] can be seen as the problem of reversing a set of quantum operations [4].

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:PLA

AID:24710 /SCO Doctopic: Quantum physics

1 2 3

Suppose we are given a logically reversible measurement u = {u ,x }x∈ X . Since each measurement operator u ,x is a positive (reversible) operator, then, by the spectral decomposition theorem,



4 5

u , x =

8 9 10 11 12 13 14



i ∈x

The success probability p s of reversing, after the measurement with result x, has the upper bound [12,13]

ps ≤

mini ∈x {ax (i )} p u (x)

17 18 19

ptotal = s

24 25 26 27 28



p u (x) p s ,

(1.4)

x∈ X

22 23

(1.3)

,

where p u (x) = T r (u ,x ρ ). If we define the total success probability ptotal of reversing as s

20 21

(1.2)

x (i ) = 1H and ax (i ) > 0 for any i ∈ x . In particm ular, if for all x ∈ X there exist subsets {i s }s=x 1 ⊆ x such that mx s=1 x (i s ) are the same projector onto a subspace M and ax (i 1 ) = · · · = ax (imx ), then the measurement u is also a unitarily reversible on the subspace M [4]. where

15 16

ax (i )x (i ),

i ∈x

6 7

[m5G; v1.221; Prn:1/09/2017; 15:07] P.2 (1-5)

S. Kim et al. / Physics Letters A ••• (••••) •••–•••

2

then

ptotal s

≤

 x∈ X

min{ax (i )}. i ∈x

(1.5)

Note that the above bound is independent of the quantum state

ρ.

29 30 31 32 33 34 35 36 37

Shannon and von Neumann entropies A classical state is described by a probability distribution. Shannon entropy H ( p ), for the probability distribution p = { p (x)}x∈ X , is defined by [14]

H ( p) = −



p (x) log2 p (x).

(1.6)

x∈ X

For a quantum state ρ ∈ D(H), the quantum analog of Shannon entropy is von Neumann entropy, and is given by

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

S (ρ ) = − T r (ρ log2 ρ ).

(1.7)

An equivalent expression of S (ρ ) is [7],

S (ρ ) = min H ({ pa }), {|ψa , pa }

(1.8)

where the minimum is taken over all pure state convex decompositions of ρ . A decomposition minimizes { H ({ pa }) : {|ψa , pa }} if and only if it is a spectral decomposition of ρ . For an arbitrary ensemble {ρi , ηi }, which forms a convex decomposition of ρ , we have

S (ρ ) ≤ H ({ηi }) +



ηi S (ρi )

(1.9)

i

Given a von Neumann measurement  A = {xA }x∈ X on the quantum system H A , let us define a conditional  entropy on the quantum system H B by S B | A (ρ A B |{xA }) = x ηx S (ρ B |x ), where ρ B |x = ηx−1 T r A (xA ⊗ 1H B ρ A B ) and ηx = T r (xA ⊗ 1H B ρ A B ). Moreover, we denote by

J Bv|NA (ρ A B ) = S (ρ B ) − inf A



(1.11)

x

71 73 75

D vAN (ρ A B ) = I A : B (ρ A B ) − J Bv|NA (ρ A B )  = S (ρ A ) − S (ρ A B ) + inf ηx S (ρ B |x ),

79

A

(1.12)

MA



77 78

81 82

x

is called quantum discord, which is interpreted as a measure of quantum correlation [16–18]. It is an important informationtheoretic measure of quantum correlation [19], beyond entanglement measures [20]. Moreover, if we replace the von Neumann measurement in (1.12) with the generalized quantum measurement M A = { M zA }z∈ Z on H A (as described in the Introduction section), then the general quantum discord can be defined as follows:

D A (ρ A B ) = S (ρ A ) − S (ρ A B ) + inf

76

80

84 85 86 87 88 89 90 91 93

z

where ρ B |z = ηz−1 T r A (zA ⊗ 1H B ρ A B ) and ηz = T r ( M zA ⊗ HB ρ A B ). Clearly, D A (ρ A B ) ≤ D vAN (ρ A B ). Recall that, a purification of ρ ∈ D(H A ) is any pure state |φρ  φρ | ∈ D (H A ⊗ H B ) such that T r B (|φρ  φρ |) = ρ . It, then, follows from Neumark theorem and the additivity of von Neumann entropy with respect to tensor products, that

D A (ρ A B ) = D vANE (ρ A B ⊗ | 0  0 |).

83

92

ηz S (ρ B |z ),

(1.13)

This paper is organized as follows. Section 2 deals with the construction of a class of logically reversible measurements based on a von Neumann measurement, and provides a relationship between Shannon entropies of the two measurements. Section 3 presents an inequality between quantum discords induced by the two measurements. Conclusion is presented in Section 4.

94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

2. Logically reversible measurements

111

In this section, we show that it is possible to construct a logically reversible measurement from any given von Neumann measurement, and establish a compact relation between Shannon entropies induced by the two measurements. Let ρ ∈ D(H) and  = {x }x∈ X be a von Neumann measurement with | X | = n. Now, based on  and any a ∈ (0, n1 ), we can (a)

u ,x = {1 − (n − 1)a}x +

I A : B (ρ A B ) = S (ρ B ) − S (ρ B |ρ A ).

70

which is interpreted as a measure of classical correlation [16,17] between H A and H B . In general, I A : B (ρ A B ) and J Bv|NA (ρ A B ) are different, and the difference between them

Quantum discord Let H A and H B be (the Hilbert spaces of) two quantum systems, ρ A B ∈ D(H A ⊗ H B ) be a quantum state, ρ A and ρ B be the reduced states of ρ A B . In quantum information theory, quantum mutual information

is regarded as a measure of the total correlation [15] between H A and H B . With the quantum conditional entropy, S (ρ B |ρ A ) = S (ρ A B ) − S (ρ A ), quantum mutual information becomes

69

74

construct the following logically reversible measurement u = (a) {u ,x }x∈ X :

(1.10)

68

72

ηx S (ρ B |x ),

The equality is achieved if and only if {ρi } has mutual orthogonal supports.

I A : B (ρ A B ) = S (ρ A ) + S (ρ B ) − S (ρ A B ),

67

(a)



a y .

(2.1)

y =x

(a)

p u (x) =

113 114 115 116 117 118 119 120 121 122 123

(a)

The probability distribution p u = { p u (x)}x∈ X is induced, and the (a) probability p u (x) of the classical outcome x to occur is given by (a)

112

(a) T r (u ,x ρ ) = (1 − na) p (x) + a,

(2.2)

where p (x) = T r (x ρ ). It is easy to show that the measurement (ua) is not unitarily reversible on any subspace M with dim M = 1 of H. Note that the total success probability of reversing, after the original von Neumann measurement , is zero. However,

124 125 126 127 128 129 130 131 132

JID:PLA AID:24710 /SCO Doctopic: Quantum physics

[m5G; v1.221; Prn:1/09/2017; 15:07] P.3 (1-5)

S. Kim et al. / Physics Letters A ••• (••••) •••–•••

3

Case 2: If p (x) ≤ n1 , let

1

α3 = (1 − na) p (x) + a, β3 = p (x), and γ3 = a. α3 ≤ β3 + γ3 , and f (α3 ) ≤ f (β3 ) + f (γ3 )

Then 0 < α3 , β3 , γ3 ≤ n1 , (see Fig. 1(b)). Hence,

2 3 4 6 7 8

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

by inequality (1.5), the total success probability ptotal of reversing, s (a)

after the logically reversible measurement u , has the nonzero upper bound

ptotal ≤ na. s

(2.3)

Below, in Proposition 2.1, we give an important relationship between Shannon entropies induced by the two measurements. We will adopt the notation, H ( p ) := H ({ p (x)}). Proposition 2.1. For ρ ∈ D(H), and the logically reversible measure(a) (a) ment u = {u ,x }x∈ X which is induced by a von Neumann measurement  = {x }x∈ X where | X | = n and a ∈ (0, n1 ), we observe the relation (a)

H ( pu



 ) − n[max f (a), f (1 − na + a) ] ≤ H ( p ) ≤ H ( p (ua) ), (a)

(a) T r (u ,x ρ )

where p u (x) = f (x) = −x log2 x.

= (1 − na) p (x) + a, p (x) = T r (x ρ ) and

A = {x| p (x) ≤

n

}, B = {x| p (x) > }.

Note that for positive numbers p ≤

1 n

and q >

43 44 45 46 47

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Combining (2.4) and (2.7), the proposition is proved.

 Remark 2.2. Note that lima→0 max f (a), f (1 − na + a) = 0. So, it (a) follows from Proposition 2.1 that lima→0 H ( p u ) = H ( p ). This is (a) expected because when a → 0, u → . 

80 81 82

86

(a) (a) Du , A (ρ A B ) = I A : B (ρ A B ) − Ju , B | A (ρ A B ),

91

where

93



85 87 88 89 90 92 94

ηu,x S (ρu(a,)B |x ),

95

x

96

⊗ 1 B ρ A B ),

97

(a, A ) = ηu−,1x T r A (u ,x

98 99 100



S (ρ ) ≤ p 0 S (ρ1 ) + ( p 1 + p 2 ) S

− ( p1 + p2 ) H 0

≥ γ1 f  (α1 ) ≥ γ1 f  (β1 )  (a) (a) [− p (x) log2 p (x) + p u (x) log2 p u (x)]. ≥

102 103 104 105 106

Lemma 3.1. Let ρ , ρ1 , ρ2 ∈ D(H A ), p 0 + p 1 + p 2 = 1, ρ = ( p 0 + p 1 )ρ1 + p 2 ρ2 , and H 0 (r ) = −r log2 r − (1 − r ) log2 (1 − r ) for any r ∈ [0, 1]. Then,



x∈ A

p1 p1 + p2

p1 p1 + p2

ρ1 +





p2 p1 + p2

ρ2

107 108 109 110 111 112 113 114

+ H 0 ( p 2 ).

115 116

x∈ B

Thus, we obtain one-half of Proposition 2.1.

(2.4)

To prove the other half of Proposition 2.1, we consider two cases separately.

 (a) Case 1: If p (x) > n1 , let α2 = (1 − na) p (x) + a = p u (x) , β2 = p (x) 

and γ2 = 1 − na + a. Then n1 < α2 , β2 , γ2 ≤ 1, 0 < γ2 − α2 ≤ 1 − β2 , and f (α2 ) ≤ f (β2 ) + f (γ2 ) (see Fig. 1(a)). Hence,

Proof. Let us introduce two quantum systems H B and HC , and construct a quantum state ρ A BC ∈ D(H A ⊗ H B ⊗ HC ) as ρ A BC = p 0 ρ1 ⊗ |0 B 0| ⊗ |0C 0| + p 1 ρ1 ⊗ |0 B 0| ⊗ |1C 1| + p 2 ρ2 ⊗ |1 B 1| ⊗ |1C 1|. Then, we have S (ρ A ) = S (ρ ), S (ρ A B ) = H 0 ( p 2 ) + ( p0 + p 1 ) S (ρ1 ) + p 2 S (ρ2 ), S (ρ AC ) = H 0 ( p 0 ) + p 0 S (ρ1 ) + p1 p1 + p2

p2 p1 + p2

( p1 + p2 ) S ρ1 + ρ2 , and S (ρ A BC ) = H ( p ) + ( p 0 + p 1 ) S (ρ1 ) + p 2 S (ρ2 ), where the probability distribution p = ( p 0 , p 1 , p 2 ). Now, exploiting the strong subadditivity of von Neumann entropy [8], S (ρ A BC ) + S (ρ A ) ≤ S (ρ A B ) + S (ρ AC ), and simplifying we obtain the desired result. 2

ρ A B ∈ D(H A ⊗ H B ) with dim H A = n, a ∈ (0, n1 ),  

(2.5)

117 118 119 120 121 122 123 124 125 126 127 128

Theorem 3.2. For

n−1

(a)

− p u (x) log2 p u (x) ≤ − p (x) log2 p (x) − (1 − na + a) log2 (1 − na + a).

79

In this section, we study quantum discord with respect to log(a, A ) ically reversible measurement u on H A , where dim H A = n and a ∈ (0, n1 ). Quantum discord of state ρ A B ∈ D(H A ⊗ H B ) for the logically reversible measurement is defined by

this we need Lemma 3.1.

[− p u (x) log2 p u (x) + p (x) log2 p (x)]

(a)

78

84

(a)

(a)

77

83

(a) tum discord for logically reversible measurement, Du , A (ρ A B ). For

(a)

H ( p u ) ≥ H ( p ).

76

101

x∈ B

(a)

74

3. Quantum discord for logically reversible measurements

In the following, we establish an important relation between quantum discord for von Neumann measurement, D vAN (ρ A B ), and quan-

(a)



2

we have

Let us denote by α1 = maxx∈ A { p u (x)}, β1 = minx∈ B { p u (x)},   (a) (a) and γ1 = x∈ A ( p u (x) − p (x)) = x∈ B ( p (x) − p u (x)). Also, let f (x) = −x log2 x. Then, for any 0 < p ≤ q ≤ 1, we have f  ( p ) ≥ f  (q). Hence,

48 49

1 , n

 (a)  0≤ ( p u (x) − p (x)) = ( p (x) − p (ua) (x)). x∈ A

(2.7)

ηu,x = T r ((ua,,xA ) ⊗ 1 B ρ A B ).

n

Therefore,

42

75

ρu(a,)B |x

1

38

41

 (a) H ( p u ) − n max f (a), f (1 − na + a) ≤ H ( p ).

(a, A )

1

71 72



u

Proof. Let us consider the following two sets:

(2.6)

73

(a)

p ≤ (1 − na) p + a, q > (1 − na)q + a.

40

69

Now, summing (2.5) and (2.6) over allowed probabilities and adding them, we obtain

Ju , B | A (ρ A B ) = S (ρ B ) − inf

37 39

(a)



Fig. 1. f (α ) ≤ f (β) + f (γ ).

9 11

68 70

(a)

− p u (x) log2 p u (x) ≤ − p (x) log2 p (x) − a log2 a.

5

10

67

and the probability distribution pn,a = {(1 − (n − 1)a), a, · · · , a}, we have

129 130 131 132

JID:PLA

AID:24710 /SCO Doctopic: Quantum physics

[m5G; v1.221; Prn:1/09/2017; 15:07] P.4 (1-5)

S. Kim et al. / Physics Letters A ••• (••••) •••–•••

4

1 2

(a)

(a)

Du , A (ρ A B ) −

naJu , B | A (ρ A B ) 1 − na

3 4 5

(a) ≤ D vAN (ρ A B ) ≤ Du , A (ρ A B ) −

(a) naJu , B | A (ρ A B )

1 − na

6 7 8 9 10 11 12

(a, A )

= (1 − (k

− 1)a)xA

17 18 19 20

.

+



where k = | X | and a ∈ (0,

1 ). k



68

(a)

ηx S (ρ B |x ) ≤

Ju , B | A (ρ A B )

x∈ X

1 − na

+ H ( pn,a ).

(3.4)

a Ay ,

73

(a)

(a) Du , A (ρ A B ) −

naJu , B | A (ρ A B ) 1 − na

(a) Ju , B | A (ρ A B )

74

− H ( pn,a )

75 76

− H ( pn,a )

77

≤ I A : B (ρ A B ) − J Bv|NA (ρ A B ) = D vAN (ρ A B )

79

1 − na

78 80

(a)

≤ I A : B (ρ A B ) −

ρu(a,)B |x = ηu−,1x T r A ((ua,,xA ) ⊗ 1 B ρ A B ) a (1 − ka)ηx ρ B |x + ρB , = ηu , x ηu , x

Ju , B | A (ρ A B )

81 82

1 − na

83

(a)

(a) = Du , A (ρ A B ) −

naJu , B | A (ρ A B ) 1 − na

84

,

85

where ηu ,x = (1 − ka)ηx + a, ηx = ⊗ 1 B ρ A B ), and ρ B = T r A (ρ A B ). Thus, by the concavity of von Neumann entropy, we have

where the first inequality is due to (3.4), and the second inequality follows from (3.2). Hence, the proof. 2

24

(1 − ka)ηx

25

ηu , x

Remark 3.3. Note that Ju , B | A (ρ A B ) > 0, and from Theorem 3.2, we have

21 22 23

26 27 28 29

T r (xA

32 33

(1 − ka)

 x∈ X

42 43 44

ρu(a,)B |x =

47

and

52 53 54 55

58 59 60 61 62 63 64

(a)

S (ρu , B |x ) ≤

ρ B |x +

(1−na)ηx

ρ B −ηx ρ B |x , then 1−ηx

a(1 − ηx )

ηu , x

aη

ηu , x

(1 − na)ηx

ηu , x + H0(

S (ρ B |x ) +

a(1 − ηx )

ηu , x

a

ηu , x

 x∈ X

≤ (1 − na)

a(1−ηx )

ηu , x

S (ρ B ) −

,

ρ1 = ρ B | x , ρ2 = a

ηu , x

H 0 (ηx )

).

ηx S (ρ B |x ) + naS (ρ B ) + H ( pn,a ) − naH (ηu ),

x∈ X

x∈ X

ηu,x S (ρu(a,)B |x )

≤ (1 − na)

 x∈ X

and

≥

89 90 91 93

ρ

94 95 96

(a)

(a)

88

92

D vAN ( A B ),

where a ∈ (0, 1/n). Hence,

Du , A (ρ A B ) − D vAN (ρ A B ) ≥

87

naJu , B | A (ρ A B ) 1 − na

97

> 0.

Thus, we see that quantum discord for the logically reversible measurement exceeds that for the von Neumann measurement. See also [21].

98 99 100 101 102 103 104

In this paper, we have constructed the logically reversible measurement based on the von Neumann measurement. We then established relationships for Shannon entropies, and quantum discords with respect to these two measurements. In particular, we showed that quantum discord for the logically reversible measurement exceeds that for the von Neumann measurement.

106 107 108 109 110 111 112

Acknowledgements

113 114

AK acknowledges the research fellowship of Department of Atomic Energy, Government of India. This project is supported by National Natural Science Foundation of China (11171301, 11571307) and by the Doctoral Programs Foundation of the Ministry of Education of China (20120101110050).

115 116 117 118 119 120

References

121 122

where H ( pn,a ) =−(1 − (n − 1)a) log2 (1 − (n − 1)a) − (n − 1)a log2 a and H (ηu ) = − x∈ X ηu ,x log2 ηu ,x . Also, since a ≤ ηu ,x ≤ 1 − (n − 1)a for all x, we have H ( pn,a ) ≤ H (ηu ). Therefore,



1 − na

86

105

ρ B | X \{x} ,

ηu,x S (ρu(a,)B |x ) 

(a) Du , A (ρ A B ) −

(a) naJu , B | A (ρ A B )

4. Conclusion

After simple algebra, and using (2.4), we obtain

65 66

ρ B | X \{x} =

ρ B = ηx ρ B |x + (1 − ηx )ρ B | X \{x} .

56 57

(3.1)

(3.2)

, p1 = η x , p2 = u ,x ρ B | X \{x} . Then, by Lemma 3.1, we have

49 51

ηu , x

Let p 0 =

48 50

ρ

(1 − (n − 1)a)ηx

45 46

≤

J Bv|NA ( A B ).

Besides, if we denote by

38

41

ηu,x S (ρu(a,)B |x ).

(a)

1 − na

40

x∈ X



35

39



Because inf A x ηx S (ρ B |x ) is achieved on rank-one projectors, k = | X | = dim H A = n. Therefore, using (3.1), we have

Ju , B | A (ρ A B )

37

S (ρ B ) ≤

ηx S (ρ B |x ) + kaS (ρ B ) ≤

34 36

ηu , x

(a)

(a) S (ρu , B |x ),

implying

30 31

S (ρ B |x ) +

a

70 72

= I A : B (ρ A B ) −

Then the conditional state is

69 71

Now,

y =x

15 16

A

Proof. Let u = surement induced by von Neumann measurement  A = {xA }x∈ X . That is, (a, A ) u , x

67

= S (ρ B ) − inf

(a, A ) {u ,x }x∈ X be the logically reversible mea-

13 14

J Bv|NA (ρ A B )

− H ( pn,a )

ηx S (ρ B |x ) + naS (ρ B ) + (1 − na) H ( pn,a ), (3.3)

[1] A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, NorthHolland Publishing Company, Amsterdam, 1982. [2] A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic, Dordrecht, 1993. [3] H. Mabuchi, P. Zoller, Inversion of quantum jumps in quantum optical systems under continuous observation, Phys. Rev. Lett. 76 (1996) 3108. [4] M.A. Nielsen, C.M. Caves, Reversible quantum operations and their application to teleportation, Phys. Rev. A 55 (1997) 2547. [5] K. Kraus, States, Effects, and Operations, Springer-Verlag, Berlin, 1983. [6] C.W. Gardiner, Quantum Noise, Springer-Verlag, Berlin, 1991. [7] D. Spehner, Quantum correlations and distinguishability of quantum states, J. Math. Phys. 55 (2014) 075211.

123 124 125 126 127 128 129 130 131 132

JID:PLA AID:24710 /SCO Doctopic: Quantum physics

[m5G; v1.221; Prn:1/09/2017; 15:07] P.5 (1-5)

S. Kim et al. / Physics Letters A ••• (••••) •••–•••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

[8] J. Watrous, Theory of Quantum Information (Institute for Quantum Computing, University of Waterloo, 2008. [9] Neumark extension theorem [7,8]: Let  = {x = A 2x }x∈ X be a quantum measurement on H A with | X | = n. Then there exists a Hilbert space H E with dim H E = n, a pure state | 0  ∈ H E , a von Neumann measurement {xE } on H E , and a unitary operator U on H A ⊗ H E such that for each quantum state ρ ∈ D(H A ), A x ρ A x = T r E 1H A ⊗ πxE U ρ ⊗ | 0  0 |U † 1H A ⊗ πxE . It then follows that [7] x = A 2x = 0 |U † 1H A ⊗ πxE U | 0 , and the probability of the outcome x to occur is p (x) = Tr (x ρ ) = Tr (U † 1H A ⊗ πxE U ρ ⊗ | 0  0 |) [10] M. Ueda, N. Imoto, H. Nagaoka, Logical reversibility in quantum measurement: general theory and specific examples, Phys. Rev. A 53 (1996) 3808. [11] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein–Podolsky– Rosen channels, Phys. Rev. Lett. 70 (1993) 1895. [12] A.N. Korotkov, A.N. Jordan, Undoing a weak quantum measurement of a solidstate qubit, Phys. Rev. Lett. 97 (2006) 166805. [13] A.N. Jordan, A.N. Korotkov, Uncollapsing the wavefunction by undoing quantum measurements, Contemp. Phys. 51 (2010) 125. [14] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379–423, 623–656.

5

[15] B. Groisman, S. Popescu, A. Winter, Quantum, classical, and total amount of correlations in a quantum state, Phys. Rev. A 72 (2005) 032317. [16] L. Henderson, V. Vedral, Classical, quantum and total correlations, J. Phys. A 34 (2001) 6899. [17] V. Vedral, Classical correlations and entanglement in quantum measurements, Phys. Rev. Lett. 90 (2003) 050401. [18] H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett. 88 (2001) 017901. [19] K. Modi, A. Brodutch, H. Cable, T. Patrek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures, Rev. Mod. Phys. 84 (2012) 1655. [20] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865. [21] Moreover, by Eq. (1.13), we get the following important estimate of the general (a)

quantum discord D A (ρ A B ): Du , A E (ρ A B ⊗ | 0  0 |) − (a) H ( pn,a ) ≤ D A (ρ A B ) ≤ Du , A E (ρ A B

(a) naJu , B | A E (ρ A B ⊗| 0  0 |)

1−na

(a)

⊗ | 0  0 |) −

naJu , B | A E (ρ A B ⊗| 0  0 |) 1−na

.

−

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35

101

36

102

37

103

38

104

39

105

40

106

41

107

42

108

43

109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65

131

66

132