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Multipartite distribution property of one way discord beyond measurement
Annals of Physics 354 (2015) 157–164
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Multipartite distribution property of one way discord beyond measurement Si-Yuan Liu a,b , Yu-Ran Zhang b,∗ , Wen-Li Yang a , Heng Fan b,c a
Institute of Modern Physics, Northwest University, Xian 710069, PR China
b
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, PR China c
Collaborative Innovation Center of Quantum Matter, Beijing 100190, PR China
article
info
Article history: Received 19 July 2014 Accepted 17 December 2014 Available online 26 December 2014 Keywords: One way discord Multipartite distribution property
abstract We investigate the distribution property of one way discord in the multipartite system by introducing the concept of polygamy deficit for one way discord. The difference between one way discord and quantum discord is analogue to the one between entanglement of assistance and entanglement of formation. For tripartite pure states, two kinds of polygamy deficits are presented with the equivalent expressions and physical interpretations regardless of measurement. For four-partite pure states, we provide a condition which makes one way discord polygamy satisfied. In addition, we generalize these results to the case for N-partite pure states. Those results can be applicable to multipartite quantum systems and are complementary to our understanding of the shareability of quantum correlations. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Quantum correlations, such as entanglement and quantum discord, are considered as valuable resources for a variety of quantum information tasks [1–7]. On the other hand, in general, entanglement and discord are quite different from each other. Since quantum entanglement does not seem to capture all quantum features of quantum correlations, other measures of quantum correlations are proposed and investigated. Quantum discord is a widely accepted one among them [8–15]. Quantum
∗
Corresponding author. E-mail addresses: [email protected] (Y.-R. Zhang), [email protected] (W.-L. Yang), [email protected] (H. Fan).
http://dx.doi.org/10.1016/j.aop.2014.12.013 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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discord plays an important role in the research of quantum correlations due to its potential applications in a number of quantum processes, such as quantum critical phenomena [16–19], quantum evolution under decoherence [20–22] and the deterministic quantum computation with one qubit (DCQ1) protocol [23]. Since quantum discord, a significant resource in quantum-information processing, quantifies the quantum correlation in a bipartite state, it is interesting to study its distribution property in the multipartite system. The monogamy property which characterizes the restriction for sharing a resource or a quantity is helpful to provide significant information for this issue and deserves systematic investigations. In general, the limits on the shareability of quantum correlations are described by monogamy inequalities [24–29]. Recently, the monogamy relation for quantum discord has been studied in Refs. [30–36]. It is found that the monogamy of quantum discord does not always hold for any tripartite pure state [33], that is to say, the polygamy relation for quantum discord can hold for some states. Recently, a quantum correlation called one-way unlocalizable quantum discord has been presented which is similar as quantum discord [37]. The one way discord has an operational interpretation—for any tripartite pure state, the polygamy relation always holds. It is interesting to study the differences and connections between one way discord and quantum discord. The distribution property of one way discord in multipartite system is also worth considering. In this paper, we present the concept of polygamy deficit of one way discord. For any tripartite pure state, using the equivalent expression of polygamy deficit, we can control the polygamy degree of one way discord. For 4-partite pure states, we provide a condition for the case that one way discord is polygamy. In addition, the above results can be generalized to N-partite pure states. We believe that our results provide a useful method in understanding the distribution property of one way discord. Our results get rid of the optimal measurement problem which is difficult or even impossible to overcome in most researches on one way discord and quantum discord, and give important relations to simplify the calculation, Therefore, we believe that our results may have great applications in quantum information processing and can be applied to physical models of many-body quantum systems. This paper is organized as follows. In Section 2, we give a brief review of the definition of one way discord and corresponding correlations. In Section 3, we study the differences and connections between one way discord, quantum discord and corresponding quantum correlations. In Section 4, we define the polygamy deficit of one way discord. For any tripartite pure state, the polygamy degree of one way discord is considered. For 4-partite pure states, we provide a condition that makes one way discord polygamy. In Section 5, we generalize our results in Section 4 to N-partite pure states. In Section 6, we summarize our results. 2. The definition of one way discord and corresponding correlations In order to study the distribution property of one way discord, we give a brief review of one way discord and corresponding correlations. For a bipartite state ρAB , the one-way unlocalizable quantum discord is defined as the difference between the mutual information and the one way unlocalizable entanglement (UE) [37], namely,
δu← (ρAB ) = I (ρAB ) − Eu← (ρAB ) ,
(1)
where I (ρAB ) = S (ρA ) + S (ρB ) − S (ρAB ) is mutual information and S (ρ) = −Tr(ρ log2 ρ) is the von Neumann entropy [38]. The one way UE is defined as:
Eu (ρAB ) = min S (ρA ) − ←
MkB
pk S (ρ ) , A k
(2)
k
where the minimum is taken over all possible rank-1 measurements {MkB } applied on the subsystem B, pk = Tr[(I A ⊗ MkB )ρ AB ] is the probability of the outcome k, and ρkA = TrB [(I A ⊗ MkB )ρAB ]/pk is the state of system A when the outcome is k [39]. The definition of quantum discord is similar as one way discord, D← (ρAB ) = I (ρAB ) − J ← (ρAB ) ,
(3)
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where J ← (ρAB ) is the classical correlation which is defined as
J ← (ρAB ) = max S (ρA ) − MkB
pk S (ρkA ) .
(4)
k
For any tripartite pure state, we have the Koashi–Winter relation [27] J ← (ρAB ) + Ef (ρAC ) = S (ρA ) ,
(5)
where Ef (ρAC ) is the entanglement of formation (EOF) of ρAC . Similarly, we have the Buscemi–Gour– Kim equality [39]: Eu← (ρAB ) = S (ρA ) − Ea (ρAC ) ,
(6)
where Ea (ρAC ) is the entanglement of assistance (EOA) of ρAC , which is defined by the maximum average entanglement of ρAC [40,41], Ea (ρAC ) =
max
{px , |φx
⟩AC
}
px S (ρxA ),
(7)
x
where the maximum is taken over all possible pure-state decompositions of ρAC , satisfying ρAC = AC A AC x px |φx ⟩ ⟨φx | and ρx = TrC (|φx ⟩ ⟨φx |). 3. Differences and connections between one way discord and quantum discord One way discord and quantum discord are two similar quantum correlations. The differences and connections between them are interesting and significant. In this section, we study this issue carefully. First of all, let us consider the difference between one way discord and quantum discord. According to the definitions, we have D← (ρAB ) + J ← (ρAB ) = δu← (ρAB ) + Eu← (ρAB ) = I (ρAB ) . Using Eqs. (5) and (6), we have
δu← (ρAB ) − D← (ρAB ) = Ea (ρAC ) − Ef (ρAC ) ≥ 0.
(8)
This formula tells an interesting fact that the difference between one way discord and quantum discord for ρAB is equivalent to the difference between EOA and EOF for ρAC . We know that the one way discord is greater than or equal to the quantum discord in general, but how to measure the difference between them seems unclear. For an arbitrary tripartite pure state, this equation shows that the difference can be measured by the difference between EOA and EOF for another two parties. Since EOF and EOA do not involve measurements, it is much easier to calculate the difference between them, which provides a simple method for measuring the difference between one way discord and quantum discord. If ρABC changes from a pure state to another pure state, the changes of both sides of this equation are equivalent. In other words, we can control the difference between this two kinds of quantum discord by adjusting the corresponding entanglement measure. In particular, when EOA and EOF are equal, the two kinds of quantum discord are equivalent. Here, we give a simple example. In Fig. 1(a), for a family of GHZ states |ψ⟩ = cos θ |000⟩+sin θ |111⟩ with θ ∈ [0, π /2] , δu← (ρAB ) − D← (ρAB ) is plotted as a function of 2θ /π . It shows that the difference between one way discord and quantum discord for ρAB first increases and then decreases with increasing θ . In particular, when θ = 0 or π /2, the states are separable and we have δu← (ρAB ) = D← (ρAB ). When θ = π /4, the state is the maximally entangled state, the difference between one way discord and quantum discord for ρAB reaches the maximum value. Consider the distribution property of one way discord, we provide an interesting relationship as follows:
δu← (ρAB ) − δu← (ρCB ) = D← (ρAB ) − D← (ρCB ) = S (B|C ) .
(9)
Here, we give a simple proof. Using the definition and Buscemi–Gour–Kim equality (6), we have δu← (ρAB ) = I (ρAB ) − Eu← (ρAB ) with Eu← (ρAB ) = S (ρA ) − Ea (ρAC ). Similarly, we have δu← (ρAB ) = Ea (ρAC )− S (A|B) and δu← (ρCB ) = Ea (ρCA )− S (C |B). Combined these two equations together, we have
δu← (ρAB ) − δu← (ρCB ) = S (ρA ) − S (ρC ) = S (B|C ) .
(10)
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a
b
Fig. 1. (Color online). (a) δu← (ρAB ) − D← (ρAB ) vs. 2θ/π for the GHZ state |ψ⟩ = cos θ |000⟩ + sin θ|111⟩. (b) The difference between one way discord and quantum discord of ρAB and ρCB is plotted as a function of 2θ/π for a state |ψ⟩ = cos θ |Φ + ⟩AB |0⟩C + sin θ |0⟩A |Ψ + ⟩BC .
For quantum discord, using the Koashi–Winter equality (5), we have D← (ρAB ) = Ef (ρAC ) − S (A|B) and D← (ρCB ) = Ef (ρCA ) − S (C |B). The difference between D← (ρAB ) and D← (ρCB ) is equivalent to S (ρA ) − S (ρC ), which completes the proof. This equation gives us another interesting fact that the difference between one way discord of ρAB and ρCB is equivalent to the difference between quantum discord of the same states. Both differences equal to the conditional entropy S (B|C ) or −S (B|A), that is, the difference is independent of measure and quantum correlations. For any tripartite pure states, if S (ρA ) is greater than or equal to S (ρC ), we have the conditional entropy S (B|C ) ≥ 0, which means the one way discord or quantum discord of ρAB is always greater than or equal to ρCB , and vice versa. This formula also gives a new physical meaning for the conditional entropy: it reflects the distribution property of one way discord or quantum discord for relevant states. To control the distribution of one way discord or quantum discord between ρAB and ρCB , we only need to adjust the corresponding conditional entropy. For example, we consider a series of states, |ψ⟩ = cos θ |Φ + ⟩AB |0⟩C + sin θ |0⟩A |Ψ + ⟩BC with |Ψ + ⟩ and |Φ + ⟩ the Bell states. In Fig. 1(b), the difference between one way discord and quantum discord of ρAB and ρCB is plotted as a function of 2θ /π with θ ∈ [0, π /2]. This figure shows that the difference between one way discord and quantum discord of ρAB and ρCB decreases from 1 to −1 with increasing θ . In particular, when θ = π /4, the conditional entropy S (B|C ) equals zero, the one way discord or quantum discord of ρAB and ρCB are equal. 4. The polygamy deficit of one way discord In order to study the distribution property of one way discord carefully, similar as the quantum discord, we give two kinds of polygamy deficits of one way discord [35],
← △← ρA(BC ) − δu← (ρAB ) − δu← (ρAC ) , δu(A) = δu → △→ ρA(BC ) − δu→ (ρAB ) − δu→ (ρAC ) . δu(A) = δu
(11) (12)
→ △← δu(A) involves the local measurements on B, C and a coherent measurement on BC , while the △δu(A)
only involves local measurements on A. The first kind of polygamy deficit can be rewritten as ← ← △← δu(A) = Eu (ρAB ) − Ea (ρAB ) = Eu (ρAC ) − Ea (ρAC ) .
Here, we give a simple proof. For tripartite pure states, we have △← δu(A)
(13)
= S (ρA ) − δu← (ρAB ) − δu← (ρAC ).
Using the formulas we have proved, the polygamy deficit can be re-expressed as ← △← δu(A) = S (ρA ) − I (ρAB ) + Eu (ρAB ) − Ea (ρAB ) + S (A|C )
= Eu← (ρAB ) − Ea (ρAB ) .
(14)
Using the Buscemi–Gour–Kim equality (6) Eu← (ρXY ) + Ea (ρXZ ) = S (ρX ) (X , Y , Z ∈ {A, B, C }), we have Eu← (ρAB ) − Ea (ρAB ) = Eu← (ρAC ) − Ea (ρAC ).
S.-Y. Liu et al. / Annals of Physics 354 (2015) 157–164
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According to Ref. [37], the one way discord is polygamy for tripartite pure states, but we do not know the degree of polygamy for a particular state. It is shown in Eq. (13) that the degree of polygamy for one way discord is determined by the difference of UE and EOA. That is to say, we can control the degree of polygamy by adjusting the difference of UE and EOA for corresponding reduced state. It is worth noting that the right hand side of this equation only involves one local measurement on B or C , so the experiment and calculations can be greatly simplified. The difference of UE and EOA for ρAB and ρAC changes in the same step so that we only need to consider one of them. Then, we consider the equivalent expression of the second polygamy deficit of one way discord. For → → → ← tripartite pure states, we have △→ δu(A) = S (ρA ) − δu (ρAB ) − δu (ρAC ), where δu (ρAB ) = δu (ρBA ) and
δu→ (ρAC ) = δu← (ρCA ). Since we have proved in the previous section that δu← (ρXY ) + S (X |Y ) = Ea (ρXZ ) with (X , Y , Z ∈ {A, B, C }). It can be rewritten as △→ δu(A) = S (ρA ) + S (B|A) + S (C |A) − 2Ea (ρBC ) .
(15)
For tripartite pure states, we have
△→ δu(A) = I (ρBC ) − 2Ea (ρBC ) .
(16)
This equation provides an interesting fact that the second polygamy inequality also holds for one way discord, since Ea (ρBC ) ≥ 12 I (ρBC ) always holds for tripartite pure states. Thus, both the first polygamy inequality (contains local and coherent measurements) and the second polygamy inequality (only contains local measurement) hold for one way discord. The polygamy degree of the second polygamy inequality is decided by the difference of I (ρBC ) and 2Ea (ρBC ). In other words, we can control the polygamy degree by adjusting the mutual information and EOA for ρBC . It is worth noting that the right hand side of this equation does not include any measurement, so the experiment and calculations can be greatly simplified. We can also prove that ← ← ← ← △→ δu(A) = Eu (ρBA ) − δu (ρBA ) = Eu (ρCA ) − δu (ρCA ) .
(17)
It shows that the second polygamy deficit is equivalent to the difference between UE and one way ← ← discord for ρBA or ρCA . In particular, when Ea (ρBC ) = 12 I (ρBC ), we have △→ δu(A) = 0, Eu (ρBA ) = δu (ρBA )
and Eu← (ρCA ) = δu← (ρCA ). Then, a simple example is given. In Fig. 2(a), for a group of GHZ states |ψ⟩ = cos θ|000⟩+ sin θ |111⟩ with θ ∈ [0, π /2], the △→ δu(A) = I (ρBC ) − 2Ea (ρBC ) is plotted as a function of 2θ /π . This figure shows that the second polygamy deficit of one way discord first decreases then increases with increasing θ . In particular, when θ = 0 or π /2, the states are separable states, we have △→ δu(A) = 0. When θ = π /4, the state is maximally entangled state, the polygamy degree of the second polygamy inequality reaches its maximum, △→ δu(A) = −1. Similarly, for 4-partite pure state, we define the two kinds of polygamy deficits as follows:
△(δ4u()← = δu← ρA(BCD) − δu← (ρAB ) − δu← (ρAC ) − δu← (ρAD ) , A) △(δ4u()→ = δu→ ρA(BCD) − δu→ (ρAB ) − δu→ (ρAC ) − δu→ (ρAD ) . A)
(18) (19)
(4)← We can provide an upper bound of these two polygamy deficits: △δu(A) ≤ S (ρA )− 21 I (ρAB )− 12 I (ρAC )− (3) 1 I (ρAD ) = 21 I (ρABC ). Since I (ρABC ) = I (ρA(BC ) ) − I (ρAB ) − I (ρAC ) = △I , we have 2 (A)
△(δ4u()← ≤ A) △(δ4u()→ ≤ A)
1 2 1 2
I (ρABC ) = I (ρABC ) =
1 2 1 2
△(I(4A)) = △(I(3A)) ,
(20)
△(I(4A)) = △(I(3A)) .
(21) (4)←
When mutual information is polygamy I (ρABC ) ≤ 0, we must have △δu(A) ≤ 0.
So far, similar as the polygamy deficit for tripartite states, we have defined two kinds of polygamy deficits for 4-partite pure states. We have provided an upper bound for these two polygamy deficits
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a
b
Fig. 2. (Color online). (a) For state |ψ⟩ = cos θ|000⟩ + sin θ |111⟩ (θ ∈ [0, π/2]), △→ δu(A) against 2θ/π . (b) The interaction
information I (ρABC ) is plotted as a function of 2θ/π for |ψ⟩ = cos θ|0000⟩ + sin θ |1111⟩.
which is equivalent to the interaction information for its tripartite reduced state ρABC . In general, I (ρABC ) can be positive or negative. When I (ρABC ) ≤ 0, two kinds of polygamy inequalities hold for (3) one way discord. It is worth noting that I (ρABC )/2 = △I A holds for any tripartite reduced state ρABC . ( )
That is to say, when the mutual information is polygamy for ρABC , we must have I (ρABC ) ≤ 0, then both polygamy inequalities of one way discord hold for ρABCD . We can therefore make the polygamy inequalities hold by adjusting the tripartite interaction information I (ρABC ) or the corresponding mutual information. Since I (ρABC ) does not involve any measurement, the experiments and calculations can be very simple. Then, we consider a family of states, |ψ⟩ = cos θ|0000⟩ + sin θ |1111⟩ with θ ∈ [0, π /2]. In Fig. 2(b), the half of the interaction information is plotted as a function of 2θ /π . This figure shows that I (ρABC )/2 first decreases then increases with increasing θ . In particular, when θ = 0 or π /2, the state is separable and the polygamy deficits are upper bounded by 0. When θ = π /4, the state is the maximally entangled state, the polygamy deficits are upper bounded by −1/2. That is to say, the one way discord is always polygamy for this state. 5. The N -partite polygamy deficit of one way discord In this section, we define two kinds of N-partite monogamy deficits of one way discord. For N-partite pure states, we can provide a unified upper bound for these two polygamy deficits. For an arbitrary N-partite state, two kinds of polygamy deficits are defined as N ← △(δNu )← = δ ρ − δu← ρA1 Ai , A A ··· A ( ) u N 1 2 A
( 1)
(22)
i=2
N → △(δNu )→ = δ ρ δu→ ρA1 Ai . A1 (A2 ···AN ) − u A
( 1)
(23)
i=2
Similar as the tripartite or four partite case, we can provide a unified upper bound for these two polygamy deficits. Since δu→ (ρAB ) ≥ I (ρAB ) /2, we have
△(δNu )← (A )
≤
1
△(δNu )→ (A ) 1
(N )
where △I
(A1 )
≤
1 2 1 2
2S ρA1 −
N
I ρA1 Ai
=
i=2
2S ρA1 −
= I (ρA1 (A2 ···AN ) )−
N
I ρA1 Ai
i=2
N
i =2
=
1 2 1 2
△I(NA) ,
(24)
△I(NA) ,
(25)
( 1)
( 1)
I (ρA1 Ai ) is the N-partite polygamy deficit of mutual information. It (N )
shows that when mutual information is polygamy △I
(A1 )
≤ 0, the one way discord must be polygamy.
S.-Y. Liu et al. / Annals of Physics 354 (2015) 157–164
a
163
b
(N )→ u(A1 )
Fig. 3. (Color online). (a) The △δ
is plotted as a function of 2θ/π for |ψ⟩ = cos θ|0⟩⊗N + sin θ |1⟩⊗N . (b) For state |ψ⟩ =
cos θ |0⟩⊗N + sin θ |1⟩⊗N (θ ∈ [0, π/2]), the
1 2
△(I (NA)1 ) against 2θ/π .
So far, similar as the polygamy deficit for tripartite and 4-partite states, we have provided a unified upper bound for these two polygamy deficits, which is equivalent to half of the polygamy deficit (N ) ≤ 0, these two kinds of polygamy inequalities hold for one way of mutual information. When △I (A1 )
discord. So we can make the polygamy inequalities hold by adjusting the corresponding mutual information. As before the mutual information does not involve any measurement, so the experiment and calculations can be greatly simplified. In particular, when N = 3, the bound equals zero. When N = 4, the bound returns to the result in last section. In addition, similar as Eq. (16), we can provide an equivalent expression of the N-partite polygamy deficit of one way discord. Using Eqs. (22) and (23), we have N △(δNu )← = NS ρA1 − I ρA1 Ai + Ea ρA1 (A2 ···Ai−1 Ai+1 ···AN ) , A
( 1)
△(δNu )→ = A ( 1)
(26)
i=2 N N S ρ Ai − I ρA1 Ai + Ea ρAi (A2 ···Ai−1 Ai+1 ···AN ) . i=1
(27)
i=2
For example, we consider a family of states, |ψ⟩ = cos θ|0⟩⊗N + sin θ|1⟩⊗N with θ ∈ [0, π /2]. After (N ) 3 some calculations, we get that 12 △I (A ) = N − = (cos2 θ log2 cos2 θ + sin2 θ log2 sin2 θ ) and △(δNu()→ 2 A ) 1
1
(N − 2)(cos2 θ log2 cos2 θ + sin2 θ log2 sin2 θ ). (N )→ In Fig. 3(a), the N-partite polygamy deficit (N ≥ 3) of one way discord △δu(A ) is plotted as a func1 tion of 2θ /π . In Fig. 3(b), its upper bound, the half of the polygamy deficit of mutual information, (N )→ is plotted as a function of 2θ /π . When θ = π /4, the state is the maximally entangled state, △δu(A ) 1 reaches its minimum 2 − N and its upper bound also reaches its minimum (3 − N )/2. We find that the one way discord is always polygamy for this state and the polygamy degree grows linearly with increasing N. 6. Conclusions and discussion In summary, we have considered the distribution property of one way discord for multipartite quantum systems. We have showed that the difference between one way discord and quantum discord equals the difference between EOA and EOF. The distribution property of one way discord for tripartite pure state has also been investigated. Moreover, we have introduced the concepts of two polygamy deficits for one way discord of which we have found the equivalent expressions. Using this result, we can obtain physical interpretations for these two polygamy deficits and control the polygamy degree of one way discord regardless of the optimal measurement problem. For 4-partite
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pure states, we have provided a condition for the case that one way discord is polygamy. That is to say, if this condition is satisfied, the one way discord must be polygamy. The above results can also be generalized to the N-partite pure states. For the N-partite GHZ state, we provide an equivalent expression of the polygamy deficit of one way discord and the corresponding upper bound. We believe that our results provide a powerful but computationally simplified method in understanding the distribution property of one way discord in multipartite quantum systems. Our results may have applications in quantum information processing and can be applied to physical models of many-body quantum systems. Acknowledgments We thank Yu Zeng and Xian-Xin Wu for valuable discussions. This work is supported by ‘‘973’’ program (2010CB922904), NSFC (11075126, 11031005, 11175248) and NWU graduate student innovation funded YZZ12083. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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Annals of Physics journal homepage: www.elsevier.com/locate/aop
Multipartite distribution property of one way discord beyond measurement Si-Yuan Liu a,b , Yu-Ran Zhang b,∗ , Wen-Li Yang a , Heng Fan b,c a
Institute of Modern Physics, Northwest University, Xian 710069, PR China
b
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, PR China c
Collaborative Innovation Center of Quantum Matter, Beijing 100190, PR China
article
info
Article history: Received 19 July 2014 Accepted 17 December 2014 Available online 26 December 2014 Keywords: One way discord Multipartite distribution property
abstract We investigate the distribution property of one way discord in the multipartite system by introducing the concept of polygamy deficit for one way discord. The difference between one way discord and quantum discord is analogue to the one between entanglement of assistance and entanglement of formation. For tripartite pure states, two kinds of polygamy deficits are presented with the equivalent expressions and physical interpretations regardless of measurement. For four-partite pure states, we provide a condition which makes one way discord polygamy satisfied. In addition, we generalize these results to the case for N-partite pure states. Those results can be applicable to multipartite quantum systems and are complementary to our understanding of the shareability of quantum correlations. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Quantum correlations, such as entanglement and quantum discord, are considered as valuable resources for a variety of quantum information tasks [1–7]. On the other hand, in general, entanglement and discord are quite different from each other. Since quantum entanglement does not seem to capture all quantum features of quantum correlations, other measures of quantum correlations are proposed and investigated. Quantum discord is a widely accepted one among them [8–15]. Quantum
∗
Corresponding author. E-mail addresses: [email protected] (Y.-R. Zhang), [email protected] (W.-L. Yang), [email protected] (H. Fan).
http://dx.doi.org/10.1016/j.aop.2014.12.013 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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discord plays an important role in the research of quantum correlations due to its potential applications in a number of quantum processes, such as quantum critical phenomena [16–19], quantum evolution under decoherence [20–22] and the deterministic quantum computation with one qubit (DCQ1) protocol [23]. Since quantum discord, a significant resource in quantum-information processing, quantifies the quantum correlation in a bipartite state, it is interesting to study its distribution property in the multipartite system. The monogamy property which characterizes the restriction for sharing a resource or a quantity is helpful to provide significant information for this issue and deserves systematic investigations. In general, the limits on the shareability of quantum correlations are described by monogamy inequalities [24–29]. Recently, the monogamy relation for quantum discord has been studied in Refs. [30–36]. It is found that the monogamy of quantum discord does not always hold for any tripartite pure state [33], that is to say, the polygamy relation for quantum discord can hold for some states. Recently, a quantum correlation called one-way unlocalizable quantum discord has been presented which is similar as quantum discord [37]. The one way discord has an operational interpretation—for any tripartite pure state, the polygamy relation always holds. It is interesting to study the differences and connections between one way discord and quantum discord. The distribution property of one way discord in multipartite system is also worth considering. In this paper, we present the concept of polygamy deficit of one way discord. For any tripartite pure state, using the equivalent expression of polygamy deficit, we can control the polygamy degree of one way discord. For 4-partite pure states, we provide a condition for the case that one way discord is polygamy. In addition, the above results can be generalized to N-partite pure states. We believe that our results provide a useful method in understanding the distribution property of one way discord. Our results get rid of the optimal measurement problem which is difficult or even impossible to overcome in most researches on one way discord and quantum discord, and give important relations to simplify the calculation, Therefore, we believe that our results may have great applications in quantum information processing and can be applied to physical models of many-body quantum systems. This paper is organized as follows. In Section 2, we give a brief review of the definition of one way discord and corresponding correlations. In Section 3, we study the differences and connections between one way discord, quantum discord and corresponding quantum correlations. In Section 4, we define the polygamy deficit of one way discord. For any tripartite pure state, the polygamy degree of one way discord is considered. For 4-partite pure states, we provide a condition that makes one way discord polygamy. In Section 5, we generalize our results in Section 4 to N-partite pure states. In Section 6, we summarize our results. 2. The definition of one way discord and corresponding correlations In order to study the distribution property of one way discord, we give a brief review of one way discord and corresponding correlations. For a bipartite state ρAB , the one-way unlocalizable quantum discord is defined as the difference between the mutual information and the one way unlocalizable entanglement (UE) [37], namely,
δu← (ρAB ) = I (ρAB ) − Eu← (ρAB ) ,
(1)
where I (ρAB ) = S (ρA ) + S (ρB ) − S (ρAB ) is mutual information and S (ρ) = −Tr(ρ log2 ρ) is the von Neumann entropy [38]. The one way UE is defined as:
Eu (ρAB ) = min S (ρA ) − ←
MkB
pk S (ρ ) , A k
(2)
k
where the minimum is taken over all possible rank-1 measurements {MkB } applied on the subsystem B, pk = Tr[(I A ⊗ MkB )ρ AB ] is the probability of the outcome k, and ρkA = TrB [(I A ⊗ MkB )ρAB ]/pk is the state of system A when the outcome is k [39]. The definition of quantum discord is similar as one way discord, D← (ρAB ) = I (ρAB ) − J ← (ρAB ) ,
(3)
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159
where J ← (ρAB ) is the classical correlation which is defined as
J ← (ρAB ) = max S (ρA ) − MkB
pk S (ρkA ) .
(4)
k
For any tripartite pure state, we have the Koashi–Winter relation [27] J ← (ρAB ) + Ef (ρAC ) = S (ρA ) ,
(5)
where Ef (ρAC ) is the entanglement of formation (EOF) of ρAC . Similarly, we have the Buscemi–Gour– Kim equality [39]: Eu← (ρAB ) = S (ρA ) − Ea (ρAC ) ,
(6)
where Ea (ρAC ) is the entanglement of assistance (EOA) of ρAC , which is defined by the maximum average entanglement of ρAC [40,41], Ea (ρAC ) =
max
{px , |φx
⟩AC
}
px S (ρxA ),
(7)
x
where the maximum is taken over all possible pure-state decompositions of ρAC , satisfying ρAC = AC A AC x px |φx ⟩ ⟨φx | and ρx = TrC (|φx ⟩ ⟨φx |). 3. Differences and connections between one way discord and quantum discord One way discord and quantum discord are two similar quantum correlations. The differences and connections between them are interesting and significant. In this section, we study this issue carefully. First of all, let us consider the difference between one way discord and quantum discord. According to the definitions, we have D← (ρAB ) + J ← (ρAB ) = δu← (ρAB ) + Eu← (ρAB ) = I (ρAB ) . Using Eqs. (5) and (6), we have
δu← (ρAB ) − D← (ρAB ) = Ea (ρAC ) − Ef (ρAC ) ≥ 0.
(8)
This formula tells an interesting fact that the difference between one way discord and quantum discord for ρAB is equivalent to the difference between EOA and EOF for ρAC . We know that the one way discord is greater than or equal to the quantum discord in general, but how to measure the difference between them seems unclear. For an arbitrary tripartite pure state, this equation shows that the difference can be measured by the difference between EOA and EOF for another two parties. Since EOF and EOA do not involve measurements, it is much easier to calculate the difference between them, which provides a simple method for measuring the difference between one way discord and quantum discord. If ρABC changes from a pure state to another pure state, the changes of both sides of this equation are equivalent. In other words, we can control the difference between this two kinds of quantum discord by adjusting the corresponding entanglement measure. In particular, when EOA and EOF are equal, the two kinds of quantum discord are equivalent. Here, we give a simple example. In Fig. 1(a), for a family of GHZ states |ψ⟩ = cos θ |000⟩+sin θ |111⟩ with θ ∈ [0, π /2] , δu← (ρAB ) − D← (ρAB ) is plotted as a function of 2θ /π . It shows that the difference between one way discord and quantum discord for ρAB first increases and then decreases with increasing θ . In particular, when θ = 0 or π /2, the states are separable and we have δu← (ρAB ) = D← (ρAB ). When θ = π /4, the state is the maximally entangled state, the difference between one way discord and quantum discord for ρAB reaches the maximum value. Consider the distribution property of one way discord, we provide an interesting relationship as follows:
δu← (ρAB ) − δu← (ρCB ) = D← (ρAB ) − D← (ρCB ) = S (B|C ) .
(9)
Here, we give a simple proof. Using the definition and Buscemi–Gour–Kim equality (6), we have δu← (ρAB ) = I (ρAB ) − Eu← (ρAB ) with Eu← (ρAB ) = S (ρA ) − Ea (ρAC ). Similarly, we have δu← (ρAB ) = Ea (ρAC )− S (A|B) and δu← (ρCB ) = Ea (ρCA )− S (C |B). Combined these two equations together, we have
δu← (ρAB ) − δu← (ρCB ) = S (ρA ) − S (ρC ) = S (B|C ) .
(10)
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a
b
Fig. 1. (Color online). (a) δu← (ρAB ) − D← (ρAB ) vs. 2θ/π for the GHZ state |ψ⟩ = cos θ |000⟩ + sin θ|111⟩. (b) The difference between one way discord and quantum discord of ρAB and ρCB is plotted as a function of 2θ/π for a state |ψ⟩ = cos θ |Φ + ⟩AB |0⟩C + sin θ |0⟩A |Ψ + ⟩BC .
For quantum discord, using the Koashi–Winter equality (5), we have D← (ρAB ) = Ef (ρAC ) − S (A|B) and D← (ρCB ) = Ef (ρCA ) − S (C |B). The difference between D← (ρAB ) and D← (ρCB ) is equivalent to S (ρA ) − S (ρC ), which completes the proof. This equation gives us another interesting fact that the difference between one way discord of ρAB and ρCB is equivalent to the difference between quantum discord of the same states. Both differences equal to the conditional entropy S (B|C ) or −S (B|A), that is, the difference is independent of measure and quantum correlations. For any tripartite pure states, if S (ρA ) is greater than or equal to S (ρC ), we have the conditional entropy S (B|C ) ≥ 0, which means the one way discord or quantum discord of ρAB is always greater than or equal to ρCB , and vice versa. This formula also gives a new physical meaning for the conditional entropy: it reflects the distribution property of one way discord or quantum discord for relevant states. To control the distribution of one way discord or quantum discord between ρAB and ρCB , we only need to adjust the corresponding conditional entropy. For example, we consider a series of states, |ψ⟩ = cos θ |Φ + ⟩AB |0⟩C + sin θ |0⟩A |Ψ + ⟩BC with |Ψ + ⟩ and |Φ + ⟩ the Bell states. In Fig. 1(b), the difference between one way discord and quantum discord of ρAB and ρCB is plotted as a function of 2θ /π with θ ∈ [0, π /2]. This figure shows that the difference between one way discord and quantum discord of ρAB and ρCB decreases from 1 to −1 with increasing θ . In particular, when θ = π /4, the conditional entropy S (B|C ) equals zero, the one way discord or quantum discord of ρAB and ρCB are equal. 4. The polygamy deficit of one way discord In order to study the distribution property of one way discord carefully, similar as the quantum discord, we give two kinds of polygamy deficits of one way discord [35],
← △← ρA(BC ) − δu← (ρAB ) − δu← (ρAC ) , δu(A) = δu → △→ ρA(BC ) − δu→ (ρAB ) − δu→ (ρAC ) . δu(A) = δu
(11) (12)
→ △← δu(A) involves the local measurements on B, C and a coherent measurement on BC , while the △δu(A)
only involves local measurements on A. The first kind of polygamy deficit can be rewritten as ← ← △← δu(A) = Eu (ρAB ) − Ea (ρAB ) = Eu (ρAC ) − Ea (ρAC ) .
Here, we give a simple proof. For tripartite pure states, we have △← δu(A)
(13)
= S (ρA ) − δu← (ρAB ) − δu← (ρAC ).
Using the formulas we have proved, the polygamy deficit can be re-expressed as ← △← δu(A) = S (ρA ) − I (ρAB ) + Eu (ρAB ) − Ea (ρAB ) + S (A|C )
= Eu← (ρAB ) − Ea (ρAB ) .
(14)
Using the Buscemi–Gour–Kim equality (6) Eu← (ρXY ) + Ea (ρXZ ) = S (ρX ) (X , Y , Z ∈ {A, B, C }), we have Eu← (ρAB ) − Ea (ρAB ) = Eu← (ρAC ) − Ea (ρAC ).
S.-Y. Liu et al. / Annals of Physics 354 (2015) 157–164
161
According to Ref. [37], the one way discord is polygamy for tripartite pure states, but we do not know the degree of polygamy for a particular state. It is shown in Eq. (13) that the degree of polygamy for one way discord is determined by the difference of UE and EOA. That is to say, we can control the degree of polygamy by adjusting the difference of UE and EOA for corresponding reduced state. It is worth noting that the right hand side of this equation only involves one local measurement on B or C , so the experiment and calculations can be greatly simplified. The difference of UE and EOA for ρAB and ρAC changes in the same step so that we only need to consider one of them. Then, we consider the equivalent expression of the second polygamy deficit of one way discord. For → → → ← tripartite pure states, we have △→ δu(A) = S (ρA ) − δu (ρAB ) − δu (ρAC ), where δu (ρAB ) = δu (ρBA ) and
δu→ (ρAC ) = δu← (ρCA ). Since we have proved in the previous section that δu← (ρXY ) + S (X |Y ) = Ea (ρXZ ) with (X , Y , Z ∈ {A, B, C }). It can be rewritten as △→ δu(A) = S (ρA ) + S (B|A) + S (C |A) − 2Ea (ρBC ) .
(15)
For tripartite pure states, we have
△→ δu(A) = I (ρBC ) − 2Ea (ρBC ) .
(16)
This equation provides an interesting fact that the second polygamy inequality also holds for one way discord, since Ea (ρBC ) ≥ 12 I (ρBC ) always holds for tripartite pure states. Thus, both the first polygamy inequality (contains local and coherent measurements) and the second polygamy inequality (only contains local measurement) hold for one way discord. The polygamy degree of the second polygamy inequality is decided by the difference of I (ρBC ) and 2Ea (ρBC ). In other words, we can control the polygamy degree by adjusting the mutual information and EOA for ρBC . It is worth noting that the right hand side of this equation does not include any measurement, so the experiment and calculations can be greatly simplified. We can also prove that ← ← ← ← △→ δu(A) = Eu (ρBA ) − δu (ρBA ) = Eu (ρCA ) − δu (ρCA ) .
(17)
It shows that the second polygamy deficit is equivalent to the difference between UE and one way ← ← discord for ρBA or ρCA . In particular, when Ea (ρBC ) = 12 I (ρBC ), we have △→ δu(A) = 0, Eu (ρBA ) = δu (ρBA )
and Eu← (ρCA ) = δu← (ρCA ). Then, a simple example is given. In Fig. 2(a), for a group of GHZ states |ψ⟩ = cos θ|000⟩+ sin θ |111⟩ with θ ∈ [0, π /2], the △→ δu(A) = I (ρBC ) − 2Ea (ρBC ) is plotted as a function of 2θ /π . This figure shows that the second polygamy deficit of one way discord first decreases then increases with increasing θ . In particular, when θ = 0 or π /2, the states are separable states, we have △→ δu(A) = 0. When θ = π /4, the state is maximally entangled state, the polygamy degree of the second polygamy inequality reaches its maximum, △→ δu(A) = −1. Similarly, for 4-partite pure state, we define the two kinds of polygamy deficits as follows:
△(δ4u()← = δu← ρA(BCD) − δu← (ρAB ) − δu← (ρAC ) − δu← (ρAD ) , A) △(δ4u()→ = δu→ ρA(BCD) − δu→ (ρAB ) − δu→ (ρAC ) − δu→ (ρAD ) . A)
(18) (19)
(4)← We can provide an upper bound of these two polygamy deficits: △δu(A) ≤ S (ρA )− 21 I (ρAB )− 12 I (ρAC )− (3) 1 I (ρAD ) = 21 I (ρABC ). Since I (ρABC ) = I (ρA(BC ) ) − I (ρAB ) − I (ρAC ) = △I , we have 2 (A)
△(δ4u()← ≤ A) △(δ4u()→ ≤ A)
1 2 1 2
I (ρABC ) = I (ρABC ) =
1 2 1 2
△(I(4A)) = △(I(3A)) ,
(20)
△(I(4A)) = △(I(3A)) .
(21) (4)←
When mutual information is polygamy I (ρABC ) ≤ 0, we must have △δu(A) ≤ 0.
So far, similar as the polygamy deficit for tripartite states, we have defined two kinds of polygamy deficits for 4-partite pure states. We have provided an upper bound for these two polygamy deficits
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S.-Y. Liu et al. / Annals of Physics 354 (2015) 157–164
a
b
Fig. 2. (Color online). (a) For state |ψ⟩ = cos θ|000⟩ + sin θ |111⟩ (θ ∈ [0, π/2]), △→ δu(A) against 2θ/π . (b) The interaction
information I (ρABC ) is plotted as a function of 2θ/π for |ψ⟩ = cos θ|0000⟩ + sin θ |1111⟩.
which is equivalent to the interaction information for its tripartite reduced state ρABC . In general, I (ρABC ) can be positive or negative. When I (ρABC ) ≤ 0, two kinds of polygamy inequalities hold for (3) one way discord. It is worth noting that I (ρABC )/2 = △I A holds for any tripartite reduced state ρABC . ( )
That is to say, when the mutual information is polygamy for ρABC , we must have I (ρABC ) ≤ 0, then both polygamy inequalities of one way discord hold for ρABCD . We can therefore make the polygamy inequalities hold by adjusting the tripartite interaction information I (ρABC ) or the corresponding mutual information. Since I (ρABC ) does not involve any measurement, the experiments and calculations can be very simple. Then, we consider a family of states, |ψ⟩ = cos θ|0000⟩ + sin θ |1111⟩ with θ ∈ [0, π /2]. In Fig. 2(b), the half of the interaction information is plotted as a function of 2θ /π . This figure shows that I (ρABC )/2 first decreases then increases with increasing θ . In particular, when θ = 0 or π /2, the state is separable and the polygamy deficits are upper bounded by 0. When θ = π /4, the state is the maximally entangled state, the polygamy deficits are upper bounded by −1/2. That is to say, the one way discord is always polygamy for this state. 5. The N -partite polygamy deficit of one way discord In this section, we define two kinds of N-partite monogamy deficits of one way discord. For N-partite pure states, we can provide a unified upper bound for these two polygamy deficits. For an arbitrary N-partite state, two kinds of polygamy deficits are defined as N ← △(δNu )← = δ ρ − δu← ρA1 Ai , A A ··· A ( ) u N 1 2 A
( 1)
(22)
i=2
N → △(δNu )→ = δ ρ δu→ ρA1 Ai . A1 (A2 ···AN ) − u A
( 1)
(23)
i=2
Similar as the tripartite or four partite case, we can provide a unified upper bound for these two polygamy deficits. Since δu→ (ρAB ) ≥ I (ρAB ) /2, we have
△(δNu )← (A )
≤
1
△(δNu )→ (A ) 1
(N )
where △I
(A1 )
≤
1 2 1 2
2S ρA1 −
N
I ρA1 Ai
=
i=2
2S ρA1 −
= I (ρA1 (A2 ···AN ) )−
N
I ρA1 Ai
i=2
N
i =2
=
1 2 1 2
△I(NA) ,
(24)
△I(NA) ,
(25)
( 1)
( 1)
I (ρA1 Ai ) is the N-partite polygamy deficit of mutual information. It (N )
shows that when mutual information is polygamy △I
(A1 )
≤ 0, the one way discord must be polygamy.
S.-Y. Liu et al. / Annals of Physics 354 (2015) 157–164
a
163
b
(N )→ u(A1 )
Fig. 3. (Color online). (a) The △δ
is plotted as a function of 2θ/π for |ψ⟩ = cos θ|0⟩⊗N + sin θ |1⟩⊗N . (b) For state |ψ⟩ =
cos θ |0⟩⊗N + sin θ |1⟩⊗N (θ ∈ [0, π/2]), the
1 2
△(I (NA)1 ) against 2θ/π .
So far, similar as the polygamy deficit for tripartite and 4-partite states, we have provided a unified upper bound for these two polygamy deficits, which is equivalent to half of the polygamy deficit (N ) ≤ 0, these two kinds of polygamy inequalities hold for one way of mutual information. When △I (A1 )
discord. So we can make the polygamy inequalities hold by adjusting the corresponding mutual information. As before the mutual information does not involve any measurement, so the experiment and calculations can be greatly simplified. In particular, when N = 3, the bound equals zero. When N = 4, the bound returns to the result in last section. In addition, similar as Eq. (16), we can provide an equivalent expression of the N-partite polygamy deficit of one way discord. Using Eqs. (22) and (23), we have N △(δNu )← = NS ρA1 − I ρA1 Ai + Ea ρA1 (A2 ···Ai−1 Ai+1 ···AN ) , A
( 1)
△(δNu )→ = A ( 1)
(26)
i=2 N N S ρ Ai − I ρA1 Ai + Ea ρAi (A2 ···Ai−1 Ai+1 ···AN ) . i=1
(27)
i=2
For example, we consider a family of states, |ψ⟩ = cos θ|0⟩⊗N + sin θ|1⟩⊗N with θ ∈ [0, π /2]. After (N ) 3 some calculations, we get that 12 △I (A ) = N − = (cos2 θ log2 cos2 θ + sin2 θ log2 sin2 θ ) and △(δNu()→ 2 A ) 1
1
(N − 2)(cos2 θ log2 cos2 θ + sin2 θ log2 sin2 θ ). (N )→ In Fig. 3(a), the N-partite polygamy deficit (N ≥ 3) of one way discord △δu(A ) is plotted as a func1 tion of 2θ /π . In Fig. 3(b), its upper bound, the half of the polygamy deficit of mutual information, (N )→ is plotted as a function of 2θ /π . When θ = π /4, the state is the maximally entangled state, △δu(A ) 1 reaches its minimum 2 − N and its upper bound also reaches its minimum (3 − N )/2. We find that the one way discord is always polygamy for this state and the polygamy degree grows linearly with increasing N. 6. Conclusions and discussion In summary, we have considered the distribution property of one way discord for multipartite quantum systems. We have showed that the difference between one way discord and quantum discord equals the difference between EOA and EOF. The distribution property of one way discord for tripartite pure state has also been investigated. Moreover, we have introduced the concepts of two polygamy deficits for one way discord of which we have found the equivalent expressions. Using this result, we can obtain physical interpretations for these two polygamy deficits and control the polygamy degree of one way discord regardless of the optimal measurement problem. For 4-partite
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pure states, we have provided a condition for the case that one way discord is polygamy. That is to say, if this condition is satisfied, the one way discord must be polygamy. The above results can also be generalized to the N-partite pure states. For the N-partite GHZ state, we provide an equivalent expression of the polygamy deficit of one way discord and the corresponding upper bound. We believe that our results provide a powerful but computationally simplified method in understanding the distribution property of one way discord in multipartite quantum systems. Our results may have applications in quantum information processing and can be applied to physical models of many-body quantum systems. Acknowledgments We thank Yu Zeng and Xian-Xin Wu for valuable discussions. This work is supported by ‘‘973’’ program (2010CB922904), NSFC (11075126, 11031005, 11175248) and NWU graduate student innovation funded YZZ12083. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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