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Duan Jing, Luo Yu, Li Yong-Ming. A new class of states of reversible entanglement manipulation under positive partial transpose operations. Chinese Physics B, 2018, 27(11): 110305  
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A new class of states of reversible entanglement manipulation under positive partial transpose operations
Duan Jing1, Luo Yu2, Li Yong-Ming1, 2, †
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China
College of Computer Science, Shaanxi Normal University, Xi’an 710119, China

 

† Corresponding author. E-mail: liyongm@snnu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11671244).

Abstract

We first study the reversibility for a class of states under the operations which completely preserve the positivity of partial transpose (PPT) and show that the entanglement cost is equal to the distillable entanglement for a rank-two mixed state on the 4⊗4 antisymmetric subspace under PPT operations. By using a similar method in finding the irreversibility, we also find that the value of a new efficiently computable additive lower bound Eη(ρ) for the asymptotic PPT-relative entropy of entanglement presented in [Phys. Rev. Lett. 119, 180506 (2017)] is equal to the regularized Rains’ bound and an upper bound EN(ρ) for distillable entanglement for these states. Furthermore, we find states on the symmetric subspace satisfying the relation mentioned above, generalize the antisymmetric states and symmetric states in higher dimensions, and give a specific value for distillable entanglement and entanglement cost for these states under the PPT operations.

PACS: 03.67.-a;03.67.Mn;03.67.Bg
Keyword:distillable entanglement;entanglement cost;semidefinite programming
1. Introduction

Quantum entanglement is one of the most important physical resources in quantum mechanics and can be used in quantum information, especially in quantum communication and quantum coding.[1] Based on these results, much work on entanglement has been carried out recently.[2–7]

We study two kinds of fundamental measures for quantum entanglement, including the entanglement cost (EC) and the distillable entanglement (ED). The distillable entanglement[8–12] portrays the maximum efficiency with which one can obtain the maximally entangled states from a bipartite state ρ by local operations and classical communication (LOCC), which means that the distillable entanglement quantifies the optimal efficiency r of transforming ρ⊗n to rn maximally entangled states. The cost of quantum entanglement[13–15] describes the optimal rate r with which one can obtain the given bipartite state ρ from the maximally entangled states by LOCC alone, which means that the entanglement cost quantifies the efficiency r of transforming rn maximally entangled states to ρ⊗n. Since the LOCC operations are difficult to capture and lead to more difficulties in calculating ED and EC, Rains proposed the positive partial transpose (PPT) operations including LOCC operations,[16] which are more powerful than LOCC operations, such as the creation of any bound entangled state and distilling a non-positive partial transpose (NPT) state, i.e, any state that cannot be created by PPT operations.[17] At the same time, PPT operations can give a more simple classification of states. How to calculate ED remains unknown, and calculating EC for general quantum states is an NP-hard (non-deterministic polynomial-time hard) problem.[14]

A natural idea is to find some bounds[14,15] to estimate the distillable entanglement and the entanglement cost. For distillable entanglement, there are many well-known upper bounds, such as Rains’ bound[9] and the relative entropy of entanglement (REE).[18,19] Unfortunately, these well-known bounds are difficult to calculate, and are only easy for some special states.[20,21] Since the distillable entanglement and the entanglement cost are important but difficult to calculate, people attach more importance to finding a method to evaluate them. In Refs. [12], [22, and [23], the semidefinite programming (SDP) bounds for ED,PPT and EC,PPT have been proposed, which can be computable and have a lot of interesting characteristics.

For Werner states and orthogonally invariant states, Rains’ bound is proved to be equal to the asymptotic relative entropy of entanglement with respect to PPT states.[24,25] It has been shown that ED is equal to EC in the asymptotic limit of many infinitely identical copies of a pure state,[26] i.e.,

On one hand, the reversibility under PPT operations means that ED,PPT is equal to EC,PPT and the entanglement manipulation between the given states and the maximally entangled states is reversible. On the other hand, it is proved that reversibility no longer holds for mixed states.[27] In fact, the authors gave an example of reversibility under PPT operations in Ref. [17]. In Ref. [28], ED is proved to be strictly less than EC and the regularized Rains’ bound is strictly less than a new SDP lower bound Eη(ρ) for entanglement cost for a class of rank-two 3⊗3 antisymmetric states under PPT operations. There are some states[28] that satisfy Eη(ρ) > R∞(ρ), while the states we find satisfy Eη(ρ) = R∞(ρ). At the same time, Eη(ρ) > EN(ρ) for the states in Ref. [28], while we find some states that satisfy Eη(ρ) = EN(ρ), where EN(ρ) is the logarithmic negativity of a state ρ.

In this paper, we achieve our aims through finding a similar method to that of Ref. [28] to demonstrate a class of 4⊗4 antisymmetric states which are reversible under PPT operations. We use the method in constructing the irreversibility[28] while we also obtain the reversibility and show that Eη(ρ) is equal to the regularized Rains’ bound and EN(ρ) for this class of states. By using an additive SDP lower bound for the asymptotic REE, we prove that

For a class of 4⊗4 antisymmetric states which we construct, they also satisfy
Consequently,
We also generalize this class of states to higher dimensions. Indeed, it is very important that the method we use helps us obtain the specific value of ED and EC for a new class of states under PPT operations while ED and EC are difficult to calculate. At the same time, the asymptotic reversibility under PPT operations for these states will lead to a unique measure of entanglement which plays a role similar to entropy in thermodynamics and the unique ordering is provided.

This paper is organized as follows. In Section 2, we briefly review some notions related to distillable entanglement and entanglement cost. In Section 3, we give a new class of states which are reversible under PPT operations. In Section 4, we give the conclusion of this paper.

2. Preliminaries

In this section, let us review some notions related to entanglement. We use the symbols A and B to denote the Hilbert spaces. Let ℓ(A) denote the set of linear operators over A. A quantum state ρ must satisfy ρ ≥ 0 and Tr(ρ) = 1. The support of ρ, denoted by supp(ρ), is a subspace spanned by the eigenvectors of ρ with positive eigenvalues.

For a linear operator P on the Hilbert spaces, we define and the trace norm is ∥P∥1 = Tr|P|, where P† is the conjugate of P. The operator norm ∥P∥∞ is defined as the maximum eigenvalue of |P|. A positive semidefinite operator XAB ∈ ℓ(A⊗B) is said to completely preserve positivity of partial transpose if , i.e.,

The partial transpose can also be given by Eq. (5).

Semidefinite programming is a very important and useful tool in entanglement theory with a lot of applications.[29–35] The computable SDP upper bound EW(ρ)[22] for distillable entanglement and lower bound EM(ρ)[12] for entanglement cost under PPT operations help us to estimate ED,PPT and EC,PPT better.

The distillable entanglement[36] is defined as follows:

where , and Λ ranges over operations including LOCC operations, separable operations (SEP), and operations completely preserving positivity of partial transpose. If Λ ranges over PPT operations, the entanglement distillation is noted as ED,PPT.

The entanglement cost[36] is defined by

where and Λ ranges over operations including LOCC operations, PPT operations, and SEP operations.

It is clear that

The PPT-relative entropy of entanglement[24,37] is given as a convex optimization

The PPT-relative entropy of entanglement is very important in quantum theory; it is a well-known upper bound for distillable entanglement which describes the minimum distance between the given states and the PPT states.[38–40] The asymptotic PPT-relative entropy of entanglement is defined as follows:

An improved bound for distillable entanglement is Rains’ bound which was introduced in Ref. [16] and defined as follows:

where S(ρ∥τ) = Tr(ρ log ρ – ρ log τ) is the relative entropy. The regularized Rains’ bound was introduced in Ref. [41] which is given as follows:

Rains’ bound is very important in quantum theory; it describes the minimum distance between the given states and all the possible separable states.

In Ref. [41], it is shown that a lower bound for the PPT-entanglement cost is the asymptotic PPT-relative entropy of entanglement, and the following always holds:

In Ref. [28], a new important SDP lower bound for entanglement cost is given as follows:

where P is the projector of ρ, YTB is the partial transpose of Y. That work also indicates that for any quantum state ρ and as a consequence
It has been proved that R∞(ρ) < Eη(ρ) for a class of states.[28] The relation between R∞(ρ) and the new SDP lower bound Eη(ρ) remains a question.

In Ref. [42], the logarithmic negativity of a state ρ is given as follows:

Then the following conclusion is drawn:
Eη(ρ) is an SDP lower bound for the PPT-entanglement cost and EN(ρ) is an SDP upper bound for the PPT-distillable entanglement. The relation between Eη(ρ) and EN(ρ) is unknown.

In Refs. [12] and [22], the one-copy deterministic PPT-distillable entanglement of ρ is defined by

where
where PAB is the projector of ρ, IAB is an identity matrix. The strong dual SDP is given by

3. Main results

In this section, we first consider a class of bipartite states supported on the 4⊗4 antisymmetric subspace.

Because a 4 ⊗ 4 maximally entangled state can efficiently prepare an exact copy of ρ by LOCC operations, we will find that EC,LOCC (ρ) ≤ 1 and

Then we will have

Applying Eη(ρ), we obtain that and it is very significant that we use the bounds for the entanglement cost to get EC,PPT(ρ) = 1 for the states we mentioned above. Whether all the states satisfy remains a question.

We can also evaluate the PPT-distillable entanglement of ρ by using the upper bound EN(ρ) for the regularized Rains’ bound and the one-copy deterministic PPT-distillable entanglement

Firstly, we can get the one-copy deterministic PPT-distillable entanglement of ρ from Eq. (18):

Here P is the projection onto sup p(ρ).

Now, we choose

It is easy to check P ≤ R ≤ I, which means P is a feasible solution to Eq. (34).

We can calculate that

Therefore,

Secondly, we find that

Then
As a result, we have

Finally, we obtain that ED,PPT(ρ ) = R∞(ρ) = EN(ρ) = log2 = 1.

Applying the upper bound EN(ρ) and the lower bound for the distillable entanglement, we obtain that ED,PPT(ρ) = R∞(ρ) = EN(ρ) = log2 = 1, while it is difficult to calculate the specific value of ED,PPT(ρ). It remains a question whether all the states satisfy ED,PPT(ρ) = R∞(ρ), and the relation between R∞(ρ) and the new SDP lower bound Eη(ρ) also remains a problem. It is more important that R∞(ρ) = Eη(ρ) and EN(ρ) = Eη(ρ) for the states we find.

By Proposition 1 and Proposition 2, we have shown a class of special states which are reversible under PPT operations and satisfy EN(ρ) = R∞(ρ) = Eη(ρ).

Firstly, choosing we find that and Noting that Y ± PTB ≥ 0, we can find that Y is a feasible solution to Eη(ρ). It is clear that Thus, It is obvious that a maximally entangled state can efficiently prepare an exact copy of ρ by LOCC. By using Eqs. (15) and (13), we can obtain that Finally, it is easy to obtain that and As a result, ED,PPT(ρ) = EC,PPT(ρ) = R∞(ρ) = Eη(ρ) = EN(ρ) = 1.

Next, we find a class of states supported on the N ⊗ N antisymmetric subspace satisfying reversibility and EN(ρ) = R∞(ρ) = Eη(ρ).

4. Conclusion

In this paper, we study the entanglement cost and distillable entanglement to obtain reversibility and EN(ρ) = Eη(ρ) = R∞(ρ) on some conditions. As a result, we find a class of rank-two 4 ⊗ 4 antisymmetric states and symmetric states under PPT operations and extend them to a class of states in high dimensions satisfying reversibility and EN(ρ) = Eη(ρ) = R∞(ρ). Indeed, we also calculate ED,PPT(ρ) and EC,PPT(ρ) for these states. It also remains unknown whether all the states satisfy Eη(ρ) ≥ R∞(ρ) and Eη(ρ) ≥ EN(ρ). The asymptotic reversibility of a class of operations will lead to a unique measure of entanglement which plays a role similar to the entropy in thermodynamics and the unique ordering is provided for some entangled states. In future work, we want to find a way to classify the states with respect to reversibility and irreversibility under PPT operations, and we will attach more importance to the measure of entanglement.

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