Rains' bound is not additive

Rains’ bound is not additive Xin Wang1∗ and Runyao Duan1,2† 1

arXiv:1605.00348v1 [quant-ph] 2 May 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 Rains’ bound is arguably the best known upper bound of the distillable entanglement by operations completely preserving positivity of partial transpose (PPT) and was conjectured to be additive and coincide with the asymptotic relative entropy of entanglement. We disprove both conjectures by explicitly constructing a special class of mixed two-qubit states. We then introduce an additive semidefinite programming (SDP) lower bound EM for the asymptotic Rains’ bound, and it immediately becomes a computable lower bound for entanglement cost of bipartite states. Furthermore, EM is also proved to be the best known upper bound of the PPT-assisted deterministic distillable entanglement, and gives the asymptotic rates for all pure states and some class of genuinely mixed states. PACS numbers:

I.

INTRODUCTION

Entanglement plays a crucial role in quantum physics and is the key resource in many quantum information processing tasks. So it is quite natural and important to develop a theoretical framework to describe and quantify it. In spite of a series of remarkable recent progress in the theory of entanglement (for reviews see, e.g., Refs. [1–4]), many fundamental questions or challenges still remain open. One of the most significant questions is to determine the distillable entanglement ED , i.e. the highest rate at which one can obtain maximally entangled states from an entangled state by local operations and classical communication (LOCC) [5, 6]. This fundamental measure fully captures the ability of given state shared between distant parties (Alice and Bob) to generate strongly correlated qubits in order to allow reliable quantum teleportation or quantum cryptography. However, up to now, how to calculate ED for general quantum states still remains unknown. Also, in many practical applications, resources are finite and the number of entanglement are quite limited. It is also of imprtance to study the deterministic distillable entanglement of finite entanglement transformations. To evaluate the distillation rates efficiently, one possible way is to find computable upper bounds. A wellknown upper bound of the distillable entanglement is the relative entropy of entanglement (REE) [7–9], which expresses the minimal distinguishability between the given state and all possible separable states. An improved bound is the Rains’ bound [10], which is arguably the best known upper bound of distillable entanglement. The best known SDP upper bound is EW in Ref. [11] and it is an

∗ Electronic † Electronic

address: [email protected] address: [email protected]

improvement of the logarithmic negativity [12]. However, the relationship between Rains’ bound and EW remains unknown. Other known upper bounds of ED are studied in Refs. [13–16]. Unfortunately, most of these known upper bounds are difficult to compute [17] and usually easily computable only for states with high symmetries, such as Werner states, isotropic states, or the family of “iso-Werner” states [5, 14, 18, 19]. Entanglement cost EC [5, 20] is another fundamental measure in entanglement theory, which quantifies the rate for converting maximally entangled states to the given state by LOCC alone. However, computing EC is NP-hard [17] and the entanglement cost is knonwn only for a few of quantum states [21–23]. Even under the PPT operations, there are only bounds for the exact entanglement cost [24]. Since both distillable entanglement and the entanglement cost are important but difficult to compute, it is of great significance to find the best approach to efficiently evaluate them. As Rains’ bound is proved to be equal to the asymptotic relative entropy of entanglement for Werner states [25] and orthogonally invariant states [26], one open problem is to determine whether these two quantities always coincide [1]. Another significant open problem is whether Rains’ bound is additive, and it was conjectured in Ref. [26] that Rains’ bound might be additive for arbitrary quantum states . In this paper, we resolve the above two open problems about Rains’ bound by explicitly exhibiting a special class of two-qubit states whose Rains’ bound and relative entropy of entanglement are known. We show that the Rains’ bound is not additive and thereby it is reasonable to introduce the asymptotic Rains’ bound. Meanwhile, the asymptotic relative entropy of entanglement of these two-qubit states is strictly smaller than the Rains’ bound. Furthermore, a SDP lower bound EM for the asymptotic Rains’ bound is introduced and it is the first computable lower bound for entanglement cost of general bipartite

2 quantum states. Meanwhile, this bound is proved to be the best known upper bound of the deterministic distillable entanglement, which gives the PPT-assisted asymptotic rate for some states, including all the pure states and the mixed states ρ(α) (0 < α ≤ 0.2) in Ref. [11]. Before we present our main results, let us first 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 positive semidefinite operator EAB ∈ L(A ⊗ B) is said to be PPT TB if EAB ≥ 0, i.e., (|iA jB ihkA lB |)TB = |iA lB ihkA jB |. The concise definition of entanglement of distillation by LOCC is given in Ref. [1] as follows: rn ED (ρAB ) = sup{r : lim [inf kΛ(ρ⊗n AB ) − Φ(2 )k1 ] = 0}, n→∞ Λ

where Λ ranges over LOCC operations and Φ(d) = Pd 1/d i,j=1 |iiihjj| represents the standard d ⊗ d maximally entangled state. When Λ ranges over PPT operations, the PPT-assisted distillable entanglement is defined by EΓ . The Rains’ bound was introduced in Ref. [10] and refined in Ref. [26] as a convex optimization problem as follows: R(ρ) = min S(ρ||τ ) s.t. τ ≥ 0, Tr |τ TB | ≤ 1.

(2)

For a general bipartite state ρ, it holds that ER (ρ) ≥ R(ρ). However, ER (ρ) equals to R(ρ) for every two-qubit state ρ [27] or the bipartite state with one qubit subsystem [28]. In particular, a two-qubit full-rank state σ is the closest seperable state of any state ρ in the following form [27, 29]: ρ = σ − xG(σ),

(3)

Gi,j |vi ihvi |(|φihφ|)TB |vj ihvj |,

(4)

and G(σ) =

X

∞ ER (ρ) = inf

n≥1

1 ER (ρ⊗n ). n

(5)

The numerical estimation of relative entropy of entanglement with respect to the PPT states is introduced in Refs. [30, 31], i,e, can be estimated by a Matlab program. Suppose that the estimation of ER (ρ) by in Refs. [30, 31] + + is ER (ρ), and the inequality ER (ρ) = S(ρ||σ) ≥ ER (ρ) holds since the algorithm indeed provides a feasible PPT state σ which is almost optimal. This algorithm is implemented by CVX [33, 34] (a Matlab software for disciplined convex programming) and QETLAB [36]. In low dimensions, this algorithm provides an estimation + ER (ρ) with an absolute error smaller than 10−3 , i.e. + ER (ρ) + 10−3 ≥ ER (ρ) ≥ ER (ρ). The SDP upper bound on distillable entanglement EW (ρ) = log W (ρ) for a bipartite state ρ is introduced in Ref. [11], i.e., TB W (ρ) = max Tr ρRAB , |RAB | ≤ 1, RAB ≥ 0.

II. A.

(6)

MAIN RESULTS

Nonadditivity of Rains’ Bound

(1)

In this formula, S(ρ||σ) = Tr(ρ log ρ−ρ log σ) denotes the relative Von Neumann entropy. Rains’ bound is important in entanglement theory and the generalized Rains information of a quantum channel is recently proved to be a strong converse rate for quantum communication [32]. The relative entropy of entanglement (REE) [7–9] with respect to the PPT states is given by the following convex optimization problem: ER (ρ) = min S(ρ||σ) s.t. σ, σ TB ≥ 0, Tr(σ) = 1.

when λi 6= λj , where λi and |vi i are the eigenvalues and eigenvectors of σ, respectively. The asymptotic relative entropy of entanglement is given by

i,j

with span(|φi) is the kernel (or null space) of σ TB and Gi,j = λi when λi = λj and Gi,j = (λi −λj )/(ln λi −ln λj )

We first introduce a class of two-qubit states ρr whose closest separable states can be derived by the result in Ref. [27]. Thus, the Rains’ bound of ρr is exactly given. Then we apply the algorithm in Refs. [30, 31] to demonstrate the gap between R(ρ⊗2 ) and R(ρ). Theorem 1 There exists a two-qubit state ρ such that R(ρ⊗2 ) < 2R(ρ). Meanwhile, ∞ ER (ρ) < R(ρ).

Proof Firstly, we construct two-qubit states ρr and σr satisfying Eq. (3). Then we have R(ρr ) = S(ρr ||σr ). Suppose that 1 1 |00ih00| + |11ih11| + r|01ih01| 4 8 5 1 + ( − r)|10ih10| + √ (|01ih10| + |10ih01|). 8 4 2

σr =

√

√

The positivity of σr requires that 5−16 17 ≤ r ≤ 5+16 17 . Assume that r ≥ 5/8 − r and we can further choose 0.3125 ≤ r ≤ 0.57 for simplicity.

3 Meanwhile, let us choose 0.86

7 − 8x 1 |10ih10| ρr = |00ih00| + x|01ih01| + 8 8 2 32r − (6 + 32x)r + 10x + 1 √ + (|01ih10| + |10ih01|) 4 2

2R(ρ r)

0.85

2 E +R(ρ ⊗ ) r

0.84 0.83 0.82

with

0.81

32r2 − 10r + 1 (16r − 5)y −1 x = r+ + 2 256r − 160r + 33 32 ln (5/8 − y) − 32 ln (5/8 + y) and y = (4r2 −5r/2+33/64)1/2 . It is clear that Tr ρr = 1 and we set 0.3125 ≤ r ≤ 0.5480 to ensure the positivity of ρr . One can readily verify that ρr = σr −3G(σr )/2. Therefore, σr is the closest separable state (CSS) for ρr and we have that R(ρr ) = ER (ρr ) = S(ρr ||σr ).

(7)

In particular, let us first choose r0 = 0.547, the Rains’ bound of ρr0 is given by

0.8 0.79 0.78 0.77 0.46

For a bipartite quantum state ρ, we define

+ ⊗2 ⊗2 R(ρ⊗2 r0 ) ≤ ER (ρr0 ) ≤ ER (ρr0 ) < 2R(ρr0 ).

It is also easy to observe that 1 ER (ρ⊗2 r0 ) < R(ρr0 ). 2

(10)

where PAB is the projection onto the support of ρ. And the dual SDP is given by TB M (ρ) = min kRAB k∞ , s.t. RAB ≥ PAB .

(11)

The optimal values of the primal and the dual SDPs above coincide by strong duality. For any two bipartite states ρAB and σA0 B 0 , by utilizing semidefinite programming duality, it is not difficult to prove that EM (ρAB ⊗ σA0 B 0 ) = EM (ρAB ) + EM (σA0 B 0 ).

When 0.45 ≤ r ≤ 0.548, we show the gap between + ⊗2 2R(ρr ) and ER (ρr ) in FIG. 1. t u Since Rains’ bound is not additive, it is now reasonable to define the asymptotic Rains’ bound, i.e.,

B.

(9)

and

TB s.t. Tr |VAB | = 1, VAB ≥ 0,

The numerical value of relative entropy here is calculated based on the Matlab function “logm” [35] and the function “Entropy” in QETLAB [36]. In this case, the accuracy is guaranteed by the fact kelogm(σr0 ) −σr0 k1 ≤ 10−16 and kelogm(σ0 ) −σ0 k1 ≤ 10−14 . Noting that the difference + ⊗2 between 2R(ρr0 ) and ER (ρr0 ) is already 1.00691 × 10−2 , we have that

1 R(ρ⊗n ). n

0.54

M (ρ) = max Tr PAB VAB ,

+ ⊗2 ER (ρr0 ) = S(ρ⊗2 r0 ||σ0 ) ' 0.7683307.

n≥1

0.52

EM (ρ) = − log M (ρ),

Furthermore, applying the algorithm in Refs. [30, 31], we can find a PPT state σ0 such that

R∞ (ρ) = inf

0.5

FIG. 1: This plot demonstrates the difference between 2R(ρr ) + ⊗2 and ER (ρr ) for 0.45 ≤ r ≤ 0.548. The dashed line depicts + ⊗2 ER (ρr ) while the solid line depicts 2R(ρr ).

R(ρr0 ) = ER (ρr0 ) = S(ρr0 ||σr0 ) ' 0.3891999.

∞ ER (ρr0 ) ≤

0.48

r from 0.45 to 0.548

(8)

A SDP lower bound for entanglement cost

Since computing the entanglement cost of a bipartite state is very difficult, we introduce an efficiently computable lower bound to evaluate entanglement cost.

Furthermore, for any state bipartite ρ, EM (ρ) = 0 if and only if supp(ρ) contains the support of a PPT state σ, i.e. supp(σ) ⊆ supp(ρ). Too see this, if there exists PPT state σ such that supp(σ) ⊆ supp(ρ), then EM (ρ) = 0. On the other hand, if any state σ satisfies supp(σ) ⊆ supp(ρ) is NPPT. Let the optimal solution to SDP (10) be V , where V ≥ 0 and Tr |V TB | = 1. It is clear that Tr V ≤ 1. Thus, we have Tr V = 1 when EM (ρ) = 0. Hence, V is a PPT state and supp(V ) ⊆ supp(ρ). This leads to a contradiction. Theorem 2 For any bipartite state ρ, EM (ρ) ≤ R∞ (ρ) ≤ EC (ρ).

4 Proof Suppose that the optimal solution to Eq. (1) of R(ρ) is V , then V ≥ 0 and Tr |V TB | = t ≤ 1. Thus, V /t is a feasible solution to SDP (10) of M (ρ), this means that EM ≤ − log Tr P V /t = − log Tr P V + log t, where P is the projection onto supp(ρ). On the other hand, let N (σ) = P σP +(1−P )σ(1−P ), then by the monotonicity of quantum relative entropy, S(ρ||V ) ≥ S(N (ρ)||N (V )) = S(ρ||P V P ) PV P ) − log Tr P V = S(ρ|| Tr P V P ≥ − log Tr P V ≥ EM (ρ).

(12)

Noting that EM (·) is additive, we have that EM (ρ) ≤ inf

n≥1

1 R(ρ⊗n ) = R∞ (ρ). n

Finally, it is clear that ∞ EM (ρ) ≤ R∞ (ρ) ≤ ER (ρ) ≤ EC (ρ).

t u Remark As an application of this lower bound, one can also give an SDP lower bound for the entanglement cost of quantum channels [37], i.e. the rate of entanglement (ebits) needed to asymptotically simulate a quantum channel N with free classical communication. t u

C.

Hence, 1 log W0 (ρ⊗n ) n 1 ≤ lim − log M (ρ)n = EM (ρ). n→∞ n

E0,Γ (ρ) = lim − n→∞

Finally, to prove EM (ρ) ≤ EW (ρ), suppose that the optimal solution to SDP (11) is R, then we have R ≥ P ≥ 0. Let R0 = R/kRTB k∞ and it is easy to see the positivity of R0 and the fact that |R0TB | ≤ 1, which means that R0 is a feasible solution to SDP (6). Therefore, EW (ρ) ≥ log Tr ρR0 ≥ log Tr ρP/kRTB k∞ = − log kRTB k∞ = EM (ρ). t u Remark For any bipartite state ρ, if the support of ρ contains a PPT state σ, then EM (ρ) = 0 and we have that E0,Γ (ρ) = 0. Thus ρ is bound entanglement for exact distillation under both LOCC or PPT operations. t u We further show the estimation of Theorem 3 in Fig.3 by a class of 3 ⊗ 3 states in Ref. [11] defined by (α)

ρAB =

2 1 X † (X ⊗ X)m |ψ0 ihψ0 |(X ⊗ X † )m , 3 m=0

√ √ where |ψ0 i = α|00i + 1 − α|11i(0 < α ≤ 0.5) and P2 X = j=0 |j ⊕1ihj|. An interesting fact is that EM (ρ(α) ) is tight for E0,Γ (ρ(α) ) when 0 < α ≤ 1/5, which is proved in the following Proposition.

Deterministic distillable entanglement

In this section, we show that EM is the best upper bound on the deterministic distillable entanglement of bipartite states. The bipartite pure state case is completely solved in Refs. [38, 39]. For a general state, the PPT-assisted deterministic distillation rates depend only on the support of this state [11]. Note that the support supp(ρ) of a state ρ is defined to be the space spanned by the eigenvectors with with non-zero eigenvalues of ρ. The exact value of the one-copy PPT-assisted deterministic distillation rate of a given bipartite state ρ (1) is E0,Γ (ρ) = − log W0 (ρ) [11], where W0 (ρ) is given by W0 (ρ) = min kRTB k∞ , s.t. PAB ≤ R ≤ 1AB .

(13)

Here, PAB is the projection onto supp(ρ).

0.5 0.4 0.3

EW(ρ(α)) EM(ρ(α))

0.2

E(1) (ρ(α)) 0,Γ

0.1 0

0

0.1

0.2

0.3

0.4

0.5

α from 0 to 0.5

FIG. 2: This plot presents the estimation of EΓ (ρ(α) ) and E0,Γ (ρ(α) ) as well. The dot line depicts EW (ρ(α) ), the dash (1) line depicts E0,Γ (ρ(α) ) and the solid line depicts EM (ρ(α) ).

Theorem 3 For any bipartite state ρ, E0,Γ (ρ) ≤ EM (ρ) ≤ EW (ρ). Proof To prove E0,Γ (ρ) ≤ − log M (ρ), suppose that the optimal solution to SDP (13) of W0 (ρ) is R0 . It is clear that R0 is also a feasible solution to SDP (11) of M (ρ). Thus, W0 (ρ) = kR0 TB k∞ ≥ M (ρ). Furthermore, W0 (ρ⊗n ) ≥ M (ρ⊗n ) = M (ρ)n .

Proposition 4 For any bipartite state ρ with support projection P , suppose that the eigenvector |ψi of P TB with the eigenvalue kP TB k∞ is a product state, then E0,Γ (ρ) = EM (ρ) = − log kP TB k∞ ≤ EΓ (ρ). Proof In Ref. [11], it shows that E0,Γ (ρ) ≥ − log kP TB k∞ . If |ψihψ| is PPT, then we can choose

5 V = |ψihψ| and it is easy to see V is a feasible solution to SDP (10) of M (ρ). Thus, EM (ρ) ≤ − log Tr P TB |ψihψ| = − log kP TB k∞ . t u For any pure state |φihφ|, P suppose that |φi has the m Schmidt P decomposition |φi = i=1 λi |iiiPwith λ21 ≥ ... ≥ m m 2 2 λm and i=1 λi = 1. Then |φihφ|TB = i=1 λ2i |iiihii| + P TB 2 k∞ = λ1 and the correi6=j λi λj |jiihij|. Thus, kP sponding eigenvector is |11ih11|. Hence, by Proposition 4, E0,Γ (|φihφ|) = EM (|φihφ|) = − log k|φihφ|TB k∞ . This rate can be achieved by LOCC [39]. Example For the ρ(α) , when 0 < α ≤ 1/5, we have that E0,Γ (ρ(α) ) = EM (ρ(α) ) = − log(1 − α). Let U = X † ⊗ X, the projection onto supp(ρ(α) ) is Pα = P2 m † m m=0 U |ψ0 ihψ0 |(U ) . Therefore, p p PαTB =2 α(1 − α)|v1 ihv1 | − α(1 − α)(|v2 ihv2 | + |v3 ihv3 |) +

2 X

state. We further show that EM is the best upper bound for the deterministic distillable entanglement, which also gives the PPT-assisted deterministic distillation rate in some conditions, including all the pure states and some classes of the mixed states. It is of great interest to study the asymptotic Rains’ bound and compare it to the asymptotic relative entropy of entanglement. An ambitious goal is to determine whether the asymptotic Rains’ bound and the PPT distillable entanglement coincide. It would also be interesting to decide whether E0,Γ (ρ) = EM (ρ) and further compare Rains’ bound to EW . Relations between some fundamental entanglement measures and related quantities are showed in Fig. 3. We hope these new findings may be useful in studying other problems in quantum information theory.

U m [(1 − α)|11ih11| + α|00ih00|](U † )m ,

m=0

where |v1 i = √13 (|01i + |10i + |22i), |v2 i = √16 |01i + q 2 √1 |10i − √1 (|01i − |10i). When 0 < 3 |22i) and |v3 i = 6 2 p α ≤ 1/5, we always have 1−α ≥ 2 α(1 − α). Therefore, kPαTB k∞ = 1 − α and the corresponding eigenvector is |11ih11|. Applying Proposition 4, the proof is done. t u

III.

CONCLUSIONS AND DISCUSSIONS

We show that the Rains’ bound is neither additive nor equal to the asymptotic relative entropy of entanglement by explicitly constructing a special class of mixed twoqubit states. We also introduce the asymptotic Rains’ bound and give a SDP lower bound EM for it. These results solve two open problems in entanglement theory and provide an efficiently computable lower bound for the entanglement cost of general bipartite states for the first time. This bound also has desirable properties such as additivity under tensor product and vanishing if and only if the support of the given state contains some PPT

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FIG. 3: This plot presents relations between some fundamental entanglement measures and related quantities. EA → EB indicates that EA (ρ) ≥EB (ρ) for all ρ.

We were grateful to A. Winter and M. Tomamichel for helpful suggestions. This work was partly supported by the Australian Research Council (Grant Nos. DP120103776 and FT120100449).

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