Irreversibility of Asymptotic Entanglement Manipulation Under ...

Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose Xin Wang1∗ and Runyao Duan1,2† 1

arXiv:1606.09421v1 [quant-ph] 30 Jun 2016

Centre for Quantum Computation and Intelligent Systems (QCIS), Faculty of Engineering and Information Technology, University of Technology Sydney (UTS), NSW 2007, Australia and 2 UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve positivity of partial transpose (PPT), which resolves a major open problem in quantum information theory. To be more specific, we show that for any rank-two mixed state supporting on the 3 ⊗ 3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a byproduct, we find that for this class of quantum states, both the Rains’ bound and its regularization, are strictly less than the asymptotic relative entropy of entanglement with respect to PPT states. So in general there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further present a feasible sufficient condition for the irreversibility of entanglement manipulation under PPT operations. PACS numbers:

Introduction: A basic feature of many physical processes is the appearance of irreversibility. In the area of quantum information science, entanglement plays a central role and the irreversibility in entanglement manipulations has been studied in the last two decades. The manipulation of entanglement under local operations and classical communication (LOCC) is generally irreversible in the finite-copy regime. More precisely, the amount of pure entanglement that can be distilled from a finite number of copies of a state ρ is usually strictly smaller than the amount of pure entanglement needed to prepare the same number of copies of ρ [1]. Surprisingly, in the asymptotic settings where the number of copies tend to infinite, this process of entanglement manipulation for bipartite pure states is shown to be reversible [2]. In contrast, for mixed states, this asymptotic reversibility under LOCC operations does not hold anymore [3–7]. In particular, one requires a positive rate of pure state to generate the so-called bound entanglement by LOCC [3, 9], while it is well known that no pure state can be distilled from it [8]. Various approaches have been considered to enlarge the class of operations to ensure reversible interconversion of entanglement in the asymptotic setting. A natural candidate is the class of quantum operations that completely preserve positivity of partial transpose (PPT) [10]. A remarkable result is that any state with a nonpositive partial transpose (NPT) is distillable under this class of operations [11]. This suggests the possibility of reversibility under PPT operations and there are examples of mixed states which can be reversibly converted into pure states in the asymptotic setting, e.g. the class of antisymmetric states of arbitrary dimension [12]. However, the reversibility under PPT operations remained

unknown so far since there were no further examples. Recently, a reversible theory of entanglement considering all asymptotically non-entangling transformations was studied in Refs. [13, 14] and the unique entanglement measure is identified as the asymptotic (regularized) relative entropy of entanglement. A more general reversible framework for quantum resource theories was introduced in Ref. [15]. When the pure entanglement is set to be the standard 2⊗2 maximally entangled state, two fundamental ways of entanglement manipulation are well known, namely, entanglement distillation and entanglement dilution [1, 2]. These two tasks also naturally raise two fundamental entanglement measures of distillable entanglement ED and entanglement cost EC [1]. To be specific, ED is the highest rate at which one can obtain 2 ⊗ 2 maximally entangled states from the given state by LOCC [1] while EC is the optimal rate for converting 2⊗2 maximally entangled states to the given state by LOCC. Then the problem of the reversibility in entanglement manipulation under PPT operations is to determine whether distillable entanglement always coincides with entanglement cost under PPT operations. This problem is very difficult since for general mixed states it is highly nontrivial to evaluate these two measures both of which are given by a limiting procedure. Note that PPT bound entanglement is not a candidate for irreversibility anymore since it can be used as free resources under PPT operations. In this paper, we demonstrate that irreversibility still exists in the asymptotic entanglement manipulation under PPT operations, which resolves a major open problem in quantum information theory [16, 17]. Our approach is to show a gap between the regularized Rains’ bound and the asymptotic relative entropy of entangle-

2 ment with respect to PPT states, which also resolves another open problem in Ref. [17]. To be specific, we introduce an additive semidefinite programming (SDP) [18] lower bound for the asymptotic relative entropy of entanglement with respect to PPT states and compare it to the Rains’ bound. With this new lower bound, we are able to show that the PPT-assisted entanglement cost of any rank-two state supporting on the 3 ⊗ 3 antisymmetric subspace is exactly one entanglement bit (ebit) while its PPT distillable entanglement is strictly smaller than one ebit. As an immediate corollary, we show that there is no unique entanglement measure under PPT operations. Finally, we give a sufficient condition which can be used to efficiently verify the irreversibility of asymptotic entanglement manipulation under PPT operations. Then a more general class of sates can be constructed to further illustrate the irreversibility. Before we present our main results, let us review some notations and preliminaries. In the following we will frequently use symbols such as A (or A0 ) and B (or B 0 ) to denote (finite-dimensional) Hilbert spaces associated with Alice and Bob, respectively. The set of linear operators over A is denoted by L(A). Note that for a linear√ operator R over a Hilbert space, we define |R| = R† R, and the trace norm of R is given by kRk1 = Tr |R|, where R† is the complex conjugate of R. The operator norm kRk∞ is defined as the maximum eigenvalue of |R|. A deterministic quantum operation (quantum channel) N from A0 to B is simply a completely positive and trace-preserving (CPTP) linear map from L(A0 ) to L(B). P The Choi-Jamiolkowski matrix of N is given by JAB = ij |iA ihjA | ⊗ N (|iA0 ihjA0 |), where {|iA i} and {|iA0 i} are orthonormal basis on isomorphic Hilbert spaces A and A0 , respectively. A positive semidefinite operator EAB ∈ L(A ⊗ B) is said to be PPT if TB EAB ≥ 0, where TB means the partial transpose over the system B, i.e., (|iA jB ihkA lB |)TB = |iA lB ihkA jB |. A bipartite operation is said to be a PPT operation if its Choi-Jamiolkowski matrix is PPT. A well known fact is that the classes of PPT operations, Separable operations (SEP) [10] and LOCC obey the following strict inclusions [19], LOCC ( SEP ( PPT.

In Ref. [20], Hayden, Horodecki and Terhal proved that EC equals to the regularised entanglement of formation EF [1], the convex roof extension of the entropy of entanglement. The Rains’ bound is introduced in Ref. [10] to upper bound ED,P P T and is reformulated in Ref. [21] as the following convex optimization problem : R(ρ) = min S(ρ||σ) s.t. σ ≥ 0, Tr |σ TB | ≤ 1.

(1)

In this formula, S(ρ||σ) = Tr(ρ log ρ − ρ log σ) denotes the quantum relative entropy, where we take log ≡ log2 throughout the paper. Since Rains’ bound is not additive [22], it is necessary to consider the regularized Rains’ bound, i.e., R∞ (ρ) = lim

n→∞

1 R(ρ⊗n ). n

(2)

The relative entropy of entanglement (REE) [23–25] with respect to the PPT states is given by ER,P P T (ρ) = min S(ρ||σ) s.t. σ, σ TB ≥ 0, Tr σ = 1. And the asymptotic relative entropy of entanglement with respect to the PPT states is given by ∞ ER,P P T (ρ) = lim

n→∞

1 ER,P P T (ρ⊗n ). n

(3)

The logarithmic negativity [26, 27] is an upper bound on PPT-assisted distillable entanglement, i.e., EN (ρ) = log kρTB k1 . An improved version of logarithmic negativity is introduced in Ref. [28] as TB EW (ρ) = min log kXAB k1 , s.t. XAB ≥ ρ.

It was shown in Ref. [28] that ED,P P T (ρ) ≤ EW (ρ) ≤ EN (ρ). The following one-copy PPT-assisted deterministic distillable entanglement was also obtained in Ref. [28], (1)

TB E0,D,P P T (ρ) = max − log2 kRAB k∞ ,

s.t. PAB ≤ RAB ≤ 1AB .

(4)

Let Ω represent one of LOCC, SEP or PPT. Then the concise definitions of distillable entanglement and entanglement cost by the class of Ω operations can be given as follows [17]:

where PAB is the projection onto supp(ρ), the support of ρ. Note that supp(ρ) is defined to be the subspace spanned by the eigenvectors of ρ with positive eigenval(1) ues. Clearly E0,D,P P T (ρ) is efficiently computable by ⊗n rn ED,Ω (ρAB ) = sup{r : lim ( inf kΛ(ρAB ) − Φ(2 )k1 ) = 0}, SDP, and we have n→∞ Λ∈Ω rn EC,Ω (ρAB ) = sup{r : lim ( inf kρ⊗n AB − Λ(Φ(2 ))k1 ) = 0}, n→∞ Λ∈Ω

Pd where Φ(d) = 1/d i,j=1 |iiihjj| represents the standard d ⊗ d maximally entangled state. For simplicity, we denote ED,LOCC and EC,LOCC as ED and EC , respectively.

(1)

E0,D,P P T (ρ) ≤ ED,P P T (ρ) ≤ R∞ (ρ) ≤ R(ρ), which is very helpful to determine the exact values of PPT-assisted distillable entanglement and the Rains’ bound for some states.

3 The entanglement distillation and dilution of bipartite states under LOCC/SEP/PPT operations satisfy the following chain of inequalities: ED (ρ) ≤ ED,SEP (ρ) ≤ ED,P P T (ρ) ≤ R∞ (ρ) ∞ ≤ ER,P P T (ρ) ≤ EC,P P T (ρ) ≤ EC,SEP (ρ) ≤ EC (ρ),

(5)

so we can overcome the difficulty of estimating the regularised relative entropy of entanglement. The additivity of Eη (·) can be proved by utilizing the duality theory of SDP. A detailed proof of the additivity of Eη (·) is presented in the appendix. More details about the applications of SDP in quantum information theory can be found in Ref. [31].

where the fifth inequality is proved in Ref. [29]. Irreversibility under PPT operations: We are going to show the irreversibility under PPT operations by using a 3 ⊗ 3 state, namely, ρv = 12 (|v1 ihv1 | + |v2 ihv2 |) with 1 1 |v1 i = √ (|01i − |10i), |v2 i = √ (|02i − |20i). 2 2 The projection onto the support of ρv is given by PAB = |v1 ihv1 |+|v2 ihv2 |. It is clear that the state ρv is a rank-two state supporting on the 3 ⊗ 3 antisymmetric subspace. In Ref. [30], Chitambar and one of us showed that this state can be transformed into a 2 ⊗ 2 pure entangled state by a suitable separable operation while no finite-round LOCC protocol can do that. Theorem 1 For the state ρv , ∞ ∞ EC,P P T (ρv ) = ER,P P T (ρv ) > R (ρv ) = ED,P P T (ρv ).

Proof From the Propositions 3 and 4 below, the gap between the regularized Rains’ bound and the asymptotic relative entropy of entanglement is clear to see. In particular, the irreversibility under PPT operations is directly proved by

FIG. 1: The relative entropy of entanglement with respect to PPT states is defined as the smallest quantum relative entropy from the state ρ to the state σ taken from the set of PPT states Γ. Assume that ρ0 and σ0 give the smallest quantum relative entropy from D(ρ) to Γ. It is clear that ER,P P T (ρ) = S(ρ||σ) ≥ S(ρ0 ||σ0 ) and we show S(ρ0 ||σ0 ) ≥ − log Tr P σ0 ≥ Eη (ρ) in Proposition 2, where P is the projection onto the support of ρ. This lower bound Eη (ρ) is powerful since it still works in the asymptotic setting due to its additivity under tensor product.

1 EC,P P T (ρv ) = 1 > log(1 + √ ) = ED,P P T (ρv ). 2 t u ∞ SDP lower bound for ER,P (ρ): Our aim is PT to obtain an efficiently computable lower bound for ∞ ER,P P T (ρ). In the one-copy case, we clearly need to do some relaxations of the minimization of S(ρ||σ) with respect to PPT states. Let us define D(ρ) = {ρ0 : supp(ρ0 ) ⊆ supp(ρ)} to be the set of quantum states supporting on supp(ρ), and denote the set of PPT states by Γ. We can first relax the minimization of S(ρ||σ) to the smallest relative entropy distance between D(ρ) and the set Γ. See FIG. 1 for the idea. Then applying some properties of quantum relative entropy, the problem can be relaxed to minimizing − log Tr P σ over all PPT states σ, where P is the projection onto supp(ρ). Noting that this is SDP computable, we can further use SDP techniques to obtain the following bound

Proposition 2 For any state ρ, ∞ ER,P P T (ρ) ≥ Eη (ρ).

(7)

Proof Firstly, let us introduce a CPTP map by N (τ ) = P τ P + (1 − P )τ (1 − P ). Then for ρ0 ∈ D(ρ) and σ0 ∈ Γ, we have that S(ρ0 ||σ0 ) ≥ S(N (ρ)||N (σ0 )) = S(ρ0 ||P σ0 P /Tr P σ0 P ) − log Tr P σ0

(8)

≥ − log Tr P σ0 , where the first inequality is from the monotonicity of quantum relative entropy [32, 33] and the second inequality is due to the non-negativity of quantum relative entropy. Therefore,

Eη (ρ) = max − log kY TB k∞ , s.t. − Y ≤ P TB ≤ Y. (6) min S(ρ||σ) ≥ Interestingly, Eη (·) is additive under tensor product, i.e., Eη (ρ1 ⊗ ρ2 ) = Eη (ρ1 ) + Eη (ρ2 ),

σ∈Γ

min

ρ0 ∈D(ρ),σ0 ∈Γ

S(ρ0 ||σ0 ) ≥ min − log Tr P σ0 . σ0 ∈Γ

Secondly, utilizing the weak duality theory of SDP, we

4 can easily see that max Tr P σ0 ≤ min t s.t. Y TB ≤ t1, P TB ≤ Y

σ0 ∈Γ

≤ min t s.t. − t1 ≤ Y TB ≤ t1, −Y ≤ P TB ≤ Y = min kY TB k∞ , s.t. − Y ≤ P TB ≤ Y. Thus, ER,P P T (ρ) ≥ − log max Tr P σ0 ≥ Eη (ρ). σ0 ∈Γ

Finally, noting that Eη (ρ) is additive, we have that 1 ER,P P T (ρ⊗n ) n 1 ≥ lim Eη (ρ⊗n ) = Eη (ρ). n→∞ n

∞ ER,P P T (ρ) = lim

n→∞

∞ ER,P P T (ρv ) ≤ EC,P P T (ρv ) ≤ EC (ρv ) ≤ EF (ρv ), we have that EC,P P T (ρv ) = EC (ρv ) = 1. t u Remark Our approach to evaluate the PPT-assisted entanglement cost is to combine the lower bound Eη and upper bound EF . This result provides a new proof of the entanglement cost of the rank-two 3 ⊗ 3 antisymmetric state in Ref. [34]. Moreover, our result gives a stronger argument that the entanglement cost under PPT operations of this state is still one ebit. t u PPT-assisted distillable entanglement of ρv : We can evaluate the PPT-assisted distillable entanglement of ρv by the upper bound of Rains’ bound and the SDP characterization of the one-copy PPT-assisted deterministic distillable entanglement [28].

Proposition 4 For the state ρv ,

t u PPT-assisted entanglement cost of ρv : Applying the lower bound Eη (ρ), we can now prove that the PPTassisted entanglement cost of ρv is still one ebit.

√ ED,P P T (ρv ) = R∞ (ρv ) = log(1 + 1/ 2).

Proof On one √ hand, it is easy to calculate that kρTv B k1 = 1 + 1/ 2. Then, √ R∞ (ρv ) ≤ EN (ρv ) ≤ log(1 + 1/ 2).

Proposition 3 For the state ρv , ∞ EC,P P T (ρv ) = ER,P P T (ρv ) = 1.

(9)

Proof Firstly, suppose that Q = |01ih01| + |10ih10| + |02ih02| + |20ih20| and we can prove that Eη (ρv ) ≤ ∞ ER,P P T (ρv ) ≤ 1 by choosing a PPT state τ = Q/4 such that S(ρv ||τ ) = 1. Secondly, we are going to prove Eη (ρv ) ≥ 1. To see this, suppose that 1 Y = [Q + |00ih00| + (|11i + |22i)(h11| + h22|)]. 2 Noting that TB Y − PAB =

1 (|00i + |11i + |22i)(h00| + h11| + h22|), 2

TB it is clear that PAB ≤ YAB . Moreover,

1 TB Y + PAB = Q + (|00i − |11i − |22i)(h00| − h11| − h22|), 2 TB which means that PAB ≥ −Y . Then YAB is a feasible solution to the SDP (6) of Eη (ρv ). Thus,

Eη (ρv ) ≥ − log kY TB k∞ = − log 1/2 = 1, and we can conclude that ∞ Eη (ρv ) = ER,P P T (ρv ) = 1.

Finally, noting that EF (ρv ) = 1 (from Ref. [5], or simply applying the fact that any state in the support of ρv is a 2 ⊗ 2 maximally entangled state) and

(10)

(11)

On the other hand, suppose that √ RAB = (3 − 2 2)(|r1 ihr1 | + |r2 ihr2 |) + PAB √ √ with |r1 i = (|01i + |10i)/ 2 and |r2 i = (|02i + |20i)/ 2. It is easy to check that PAB ≤ RAB ≤ 1, which means that RAB is a feasible solution to SDP (4) of (1) E0,D,P P T (ρv ). Therefore, √ (1) TB k∞ = log(1 + 1/ 2). (12) E0,D,P P T (ρv ) ≥ − log kRAB Finally, combining Eq. (11) and Eq. √(12), we have that ED,P P T (ρv ) = R∞ (ρv ) = log(1 + 1/ 2). t u General irreversibility under PPT operations: We have shown the irreversibility of the asymptotic entanglement manipulation of ρv under PPT operations. One can use similar technique to prove the irreversibility for any ρ with spectral decomposition ρ = p|u1 ihu1 | + (1 − p)|u2 ihu2 | (0 < p < 1), where √ √ |u1 i = (|01i − |10i)/ 2, |u2 i = (|abi − |bai)/ 2. Interestingly, it holds that ED,P P T (ρ) < 1 = EC,P P T (ρ). (See appendix). More generally, we can provide a sufficient condition for the irreversibility under PPT operations and construct a general class of such states. Theorem 5 If EW (ρ) < Eη (ρ), then ED,P P T (ρ) < EC,P P T (ρ).

5 Proof

This is clear to see from

1

∞ ED,P P T (ρ) ≤ EW (ρ) < Eη (ρ) ≤ ER,P P T (ρ) ≤ EC,P P T (ρ).

0.95

E η (ρ

Indeed, we can obtain a more specific condition if we use logarithmic negativity EN instead of EW . That is, for a bipartite state ρ, if there exists a Hermitian matrix Y TB such that PAB ± Y ≥ 0 and kρTB k1 < kY TB k−1 ∞ , then ED,P P T (ρ) < EC,P P T (ρ). t u We further show the irreversibility in asymptotic manipulations of entanglement under PPT operations by a class of 3 ⊗ 3 states in defined by ρ(α) = (|ψ1 ihψ1 | + |ψ2 ihψ2 |)/2, √ √ √ where |ψ1 i = α|01i − 1 − α|10i and |ψ2 i = α|02i − √ 1 − α|20i with 0.42 ≤ α ≤ 0.5. Then the projection onto the range of ρ(α) is PAB = |ψ1 ihψ1 | + |ψ2 ihψ2 |. One can easily calculate that p EN (ρ(α) ) = log k(ρ(α) )TB k1 = log(1 + 2α(1 − α)). We then construct a feasible solution to the dual SDP (6) of Eη (ρ(α) ), i.e., Y p = α(|01ih01| + |02ih02|) + (1 − α)(|10ih10| + |20ih20|) + α(1 − α)(|00ih00| + |11ih11| + |22ih22|+|11ih22|+|22ih11|). It can be checked that −Y ≤ TB ≤ Y and kY TB k∞ ≤ 1 − α. Thus, PAB Eη (ρ(α) ) ≥ − log(1 − α). When 0.42 p ≤ α ≤ 0.5, it is easy to check that − log(1− α) > log(1 + 2α(1 − α)). Therefore, EN (ρ(α) ) < Eη (ρ(α) ). The irreversibility under PPT operations is presented in FIG. 2 for the state ρ(α) with 0.42 ≤ α ≤ 0.5. Exact values of EW (ρ(α) ) and Eη (ρ(α) ) can be calculated by CVX [35, 36] assisted with QETLAB [37]. From Theorem 5, we further consider the existence of entanglement irreversibility in a given subspace. To be specific, we derive a condition for the existence of irreversibility in a subspace S with support projection PAB by minimizing EW (ρ) over any ρ ∈ S: ˆW (PAB ) = min log kX TB k1 E s.t. X ≥ ρ ≥ 0, Tr ρ = Tr PAB ρ = 1.

E W(ρ

(α)

)

(α)

)

0.9 0.85 0.8 0.75 0.7 0.4

0.42

0.44

0.46

0.48

0.5

α from 0 to 0.5

FIG. 2: The solid line depicts EW (ρ(α) ) while the dash line depicts Eη (ρ(α) ). Note that EW is the upper bound for PPT distillable entanglement while Eη is the lower bound for PPT entanglement cost. When EW (ρ(α) ) < Eη (ρ(α) ), the asymptotic entanglement distillation under PPT operations is irreversible, i.e., ED,P P T (ρ(α) ) < EC,P P T (ρ(α) ).

distillable entanglement of any rank-two 3 ⊗ 3 antisymmetric state is strictly smaller than its PPT-assisted entanglement cost. A byproduct is that there is a gap between the regularized Rains’ bound and the asymptotic relative entropy of entanglement with respect to PPT states. Consequently, there is no unique entanglement measure in general for the asymptotic entanglement manipulation under PPT operations. Finally, we give an efficiently computable sufficient condition for the irreversibility under PPT operations. However, Eη is not tight for the 3 ⊗ 3 anti-symmetric state σa , i.e., Eη (σa ) = log 3/2 < ED,P P T (σa ) = EC,P P T (σa ) = log 5/3 [12]. How to improve the bound Eη remains an interesting open problem. RD would like to thank Andreas Winter for many inspirational discussions on the potential gap between the regularized Rains’ bound and the asymptotic relative entropy of entanglement. This work was partly supported by the Australian Research Council under Grant Nos. DP120103776 and FT120100449.

(13) ∗

Corollary 6 For a bipartite subspace S with support proˆW (PAB ) < Eη (PAB ), then there exists jection PAB , if E a state ρ ∈ S such that ED,P P T (ρ) < EC,P P T (ρ). Discussions We prove that distillable entanglement can be strictly smaller than the entanglement cost under PPT operations, which implies the irreversibility in asymptotic manipulation of entanglement under PPT operations. In particular, we prove that the PPT-assisted

†

[1] [2] [3] [4] [5] [6]

Electronic address: [email protected] Electronic address: [email protected] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). G. Vidal and J. I. Cirac, Phys. Rev. Lett. 86, 5803 (2001). G. Vidal and J. I. Cirac, Phys. Rev. A 65, 12323 (2001). G. Vidal, W. Dr, and J. I. Cirac, Phys. Rev. Lett. 89, 27901 (2002). K. G. H. Vollbrecht, R. F. Werner, and M. M. Wolf, Phys.

6 Rev. A 69, 62304 (2004). [7] M. F. Cornelio, M. C. de Oliveira, and F. F. Fanchini, Phys. Rev. Lett. 107, 20502 (2011). [8] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998). [9] D. Yang, M. Horodecki, R. Horodecki, and B. SynakRadtke, Phys. Rev. Lett. 95, 190501 (2005). [10] E. M. Rains, IEEE Trans. Inf. Theory 47, 2921 (2001). [11] T. Eggeling, K. G. H. Vollbrecht, R. F. Werner, and M. M. Wolf, Phys. Rev. Lett. 87, 257902 (2001). [12] K. Audenaert, M. B. Plenio, and J. Eisert, Phys. Rev. Lett. 90, 27901 (2003). [13] F.G.S.L. Brand˜ ao and M. B. Plenio, Nat. Phys. 4, 873 (2008). [14] F.G.S.L. Brand˜ ao and M. B. Plenio, Commun. Math. Phys. 295, 829 (2010). [15] F.G.S.L. Brand˜ ao and G. Gour, Phys. Rev. Lett. 115, 70503 (2015). [16] O. Krueger and R. F. Werner, arXiv Prepr. quantph/0504166 (2005). See also URL: http://qig.itp. uni-hannover.de/qiproblems/20. [17] M. B. Plenio and S. Virmani, Quantum Inf. Comput. 7, 1 (2007). [18] L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996). [19] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 59, 1070 (1999). [20] P. M. Hayden, M. Horodecki, and B. M. Terhal, J. Phys. A. Math. Gen. 34, 6891 (2001).

[21] K. Audenaert, B. De Moor, K. G. H. Vollbrecht, and R. F. Werner, Phys. Rev. A 66, 32310 (2002). [22] X. Wang and R. Duan, arXiv:1605.00348 (2016). [23] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997). [24] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [25] V. Vedral, M. B. Plenio, K. Jacobs, and P. L. Knight, Phys. Rev. A 56, 4452 (1997). [26] G. Vidal and R. F. Werner, Phys. Rev. A 65, 32314 (2002). [27] M. B. Plenio, Phys. Rev. Lett. 95, 90503 (2005). [28] X. Wang and R. Duan, arXiv:1601.07940 (2016). [29] M. Hayashi, Quantum Information (Springer, 2006). [30] E. Chitambar and R. Duan, Phys. Rev. Lett. 103, 110502 (2009). [31] J. Watrous, Theory of Quantum Information, University of Waterloo (2011). [32] G. Lindblad, Commun. Math. Phys. 40, 147 (1975). [33] A. Uhlmann, Commun. Math. Phys. 54, 21 (1977). [34] F. Yura, J. Phys. A Math. Gen. 36, 15 (2003). [35] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, http://cvxr.com (2014). [36] M. C. Grant and S. P. Boyd, in Recent Adv. Learn. Control (Springer, 2008), pp. 95110. [37] Nathaniel Johnston, QETLAB: A MATLAB toolbox for quantum entanglement, http://qetlab.com (2015).

Supplemental Material The additivity of Eη (ρ) under tensor product

To see the additivity of Eη (ρ), we reformulate it as Eη (ρ) = − log η(ρ), where η(ρ) = min t TB s.t. − YAB ≤ PAB ≤ YAB ,

− t1 ≤

TB YAB

(14)

≤ t1,

where PAB is the projection onto supp(ρ). The dual SDP of η(ρ) is given by η(ρ) = max Tr PAB (VAB − FAB )TB , s.t. VAB + FAB ≤ (WAB − XAB )TB , Tr(WAB + XAB ) ≤ 1,

(15)

VAB , FAB , WAB , XAB ≥ 0. The optimal values of the primal and the dual SDPs above coincide by strong duality. Proposition 7 For any two bipartite states ρ1 and ρ2 , we have that Eη (ρ1 ⊗ ρ2 ) = Eη (ρ1 ) + Eη (ρ2 ). Proof

On one hand, suppose that the optimal solution to SDP (14) of η(ρ1 ) and η(ρ2 ) are {t1 , Y1 } and {t2 , Y2 },

7 respectively. It is easy to see that T

1 T T [(Y1 + P1TB ) ⊗ (Y2 + P2 B0 ) + (Y1 − P1TB ) ⊗ (Y2 − P2 B0 )] ≥ 0, 2 1 T T = [(Y1 + P1TB ) ⊗ (Y2 − P2 B0 ) + (Y1 − P1TB ) ⊗ (Y2 + P2 B0 )] ≥ 0. 2

Y1 ⊗ Y2 + P1TB ⊗ P2 B0 = T

Y1 ⊗ Y2 − P1TB ⊗ P2 B0

T

Then, we have that −Y1 ⊗ Y2 ≤ P1TB ⊗ P2 B0 ≤ Y1 ⊗ Y2 . Moreover, T

T

kY1TB ⊗ Y2 B0 k∞ ≤ kY1TB k∞ kY2 B0 k∞ ≤ t1 t2 , T

which means that −t1 t2 1 ≤ Y1TB ⊗ Y2 B0 ≤ t1 t2 1. Therefore, {t1 t2 , Y1 ⊗ Y2 } is a feasible solution to the SDP (14) of η(ρ1 ⊗ ρ2 ), which means that η(ρ1 ⊗ ρ2 ) ≤ t1 t2 = η(ρ1 )η(ρ2 ).

(16)

On the other hand, suppose that the optimal solutions to SDP (15) of η(ρ1 ) and η(ρ2 ) are {V1 , F1 , W1 , X1 } and {V2 , F2 , W2 , X2 }, respectively. Assume that V = V1 ⊗ V2 + F1 ⊗ F2 , F = V1 ⊗ F2 + F1 ⊗ V2 , W = W1 ⊗ W2 + X1 ⊗ X2 , X = W1 ⊗ X2 + X1 ⊗ W2 . It is easy to see that V + F = (V1 + F1 ) ⊗ (V2 + F2 ) ≤ (W1 − X1 )TB ⊗ (W2 − X2 )TB0 = (W − X)TBB0 and Tr(W + X) = Tr(W1 + X1 ) ⊗ (W2 + X2 ) ≤ 1. Thus, {V, F, W, X} is a feasible solution to the SDP (15) of η(ρ1 ⊗ ρ2 ). This means that η(ρ1 ⊗ ρ2 ) ≥ Tr(P1 ⊗ P2 )(V − F )TBB0 = Tr(P1 ⊗ P2 )((V1 − F1 )TB ⊗ (V2 − F2 )TB0 ) = η(ρ1 )η(ρ2 ).

(17)

Hence, combining Eq. (16) and Eq. (17), it is clear that η(ρ1 ⊗ ρ2 ) = η(ρ1 )η(ρ2 ), which means that Eη (ρ1 ⊗ ρ2 ) = Eη (ρ1 ) + Eη (ρ2 ). t u Irreversibility for any rank-two 3 ⊗ 3 antisymmetric state

Proposition 8 For any ρ with spectral decomposition ρ = p|u1 ihu1 | + (1 − p)|u2 ihu2 | (0 < p < 1), where √ √ |u1 i = (|01i − |10i)/ 2, |u2 i = (|abi − |bai)/ 2, it holds that ED,P P T (ρ) < 1 = EC,P P T (ρ). Proof Suppose that |ai = a0 |0i + a1 |1i + a2 |2i and |bi = b0 |0i + b1 |1i + b2 |2i. Noting that hu1 |u2 i = 0, we have a0 b1 − a1 b0 = 0. Thus, with simple calculation, it is easy to see that √ |u2 i = [(a0 |0i + a1 |1i) ⊗ b2 |2i + a2 |2i ⊗ (b0 |0i + b1 |1i) − (b0 |0i + b1 |1i) ⊗ a2 |2i − b2 |2i ⊗ (a0 |0i + a1 |1i)]/ 2. Then, one can simplify |u2 i to √ |u2 i = [(cos θ|0i + sin θ|1i) ⊗ |2i − |2i ⊗ (cos θ|0i + sin θ|1i)]/ 2 (0 ≤ θ ≤ π/2),

8 where θ is determined by |ai and |bi. We assume that PAB = |u1 ihu1 | + |u2 ihu2 |. It is can be calculated that kρTB k1 < 2 for any 0 < p < 1 and 0 ≤ θ ≤ π/2, which means that ED,P P T (ρ) < 1. Moreover, let us choose 1 TB Y = PAB + (|00i + |11i + |22i)(h00| + h11| + h22|). 2 TB TB It is clear that Y ≥ PAB and it can be easily checked that −Y ≤ PAB . Thus, Y is a feasible solution to the SDP (14) of Eη (ρ), which means that

Eη (ρ) ≥ − log kY TB k∞ = 1. ∞ Finally, noting that EF (ρ) = 1 [5], we have that EC,P P T (ρ) = ER,P P T (ρ) = Eη (ρ) = 1.

t u