Asymptotic Indistinguishability of Bipartite Quantum States by Operations Completely Preserving Positivity of Partial Transpose Yinan Li1 ,∗ Xin Wang1 ,† and Runyao Duan1,2‡ 1
arXiv:1702.00231v1 [quant-ph] 1 Feb 2017
Centre for Quantum Software and Information (CQSI), 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 A bipartite subspace S is called strongly positive-partial-transpose-unextendible (PPTunextendible) if for every positive integer k, there is no PPT state supporting on the orthogonal complement of S ⊗k . We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a m ⊗ n system is m + n − 1, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].
I.
INTRODUCTION
One fascinating phenomenon of quantum mechanics is the quantum nonlocality, that is, there exist some global quantum operations on a composite system that cannot be implemented by the owners of the subsystems using local operations and classical communication (LOCC) only. A general strategy to study studying quantum nonlocality is to consider what kind of information processing tasks can be achieved by LOCC. Roughly speaking, if a certain task is accomplished with different optimal global and local efficiencies, then we can construct a class of quantum operations that cannot be realized by LOCC. There is no doubt that the discrimination of orthogonal quantum states is an effective one and received considerable attention in the past decades. See Refs. [1, 4–21] for a partial list. The set up of local distinguishability of quantum states is simple: two or more spatially separated observers share a composite quantum system prepared in one of many known mutually orthogonal quantum states. Their goal is to identify the unknown states by LOCC. A set of orthogonal quantum states is said to be perfectly (or unambiguously) distinguishable if every state can be identified with certainty (resp. some non-zero probability). It is well-known that if globe operations are permitted, the perfect distinguishability of a set of quantum states is completely determined by their orthogonality. Mean-
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address:
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while, the local discrimination of two orthogonal multipartite pure states can always be achieved perfectly [4]. Nevertheless, if we have more than two states to distinguish, they cannot be perfectly distinguished by LOCC if one or more states are entangled [6]. This phenomenon is percipient since entanglement has been shown to ensure difficulty in state discrimination [12], while it is not always necessary. It has been shown that there exist sets of orthogonal product states that cannot be discriminated perfectly by LOCC [1–3]. However, the local indistinguishability might be overcome, when multiple copies of quantum states are provided. For example, four 2⊗2 Bell states can be perfectly distinguished within two copies [8], while it is indistinguishable for a single copy [7, 8, 10]. In fact, a general bound can be obtained. It is shown that N orthogonal pure states can always be perfectly distinguished by LOCC when N − 1 copies of unknown states are provided [18]. In contrast, it turns out that there exists a set of orthogonal quantum states, not all of which are pure, cannot be perfectly (or even unambiguously) distinguished by LOCC no matter how many copies are supplied. Surprisingly, such a set can be minimal, consisting of only two orthogonal quantum states, one of which is necessarily mixed [18]. Consequently, the local indistinguishability of orthogonal quantum states is more robust in mixed states for it persists even in the domain of multiple copies, whereas in the case of pure states it does not. To obtain a deeper understanding of quantum nonlocality, it is effective to study the asymptotic local indistinguishability of quantum states. That is, sets of quantum states which cannot be distinguished perfectly by LOCC, even arbitrary many numbers of copies are pro-
2 vided. In general, proving local indistinguishability is hard since the knowledge about LOCC is limited. To circumvent the difficulty, one approach is to show indistinguishability by operations completely preserving positivity of partial transpose (denote it shortly by PPT operations), and local indistinguishability automatically follows since the set of all LOCC operations will also preserving positivity of partial transpose. The advantage of this approach is that the set of PPT operations enjoys a tractable mathematical structure. It has been shown that the bipartite maximally entangled state and the normalized projection onto its orthogonal complement cannot be distinguished by PPT operations asymptotically [22], which is the first known example of asymptotically locally indistinguishable quantum states which is constructed without invoking unextendible product bases (UPBs). Meanwhile, the notion of PPT plays a significant role in quantum information theory. It has been used to provide some convenient criterion for the separability of quantum states [31–33] and study the problem of entanglement distillation, pure state transformation and communication over quantum channels (e.g. [34–40]). In this paper, we systematically study the asymptotic indistinguishability of bipartite quantum states by PPT operations. In particular, we provide a sufficient criterion for verifying the asymptotic PPT indistinguishability. This criterion is based on the observation that if a set of orthogonal quantum states contains a state whose support is strongly PPT-unextendible, then this set of states cannot be (unambiguously) distinguished by PPT operations asymptotically. Strongly PPT-unextendible subspace is a subspace S of some bipartite Hilbert space such that for any positive integer k, there is no PPT state supporting orthogonal complement of S ⊗k . It is worth noting that this concept is a natural generalization of the concept of strongly unextendible subspace, introduced in Ref. [23]. Given a subspace, the property of strongly PPT-unextendible can be efficiently verified by checking whether there is a PPT-definite operator, which is a positive semidefinite operator whose partial transpose is positive definite, supporting on it. This can be efficiently checked by semidefinite programming (SDP) [24]. As an application, we show that any entangled pure states and the normalized projector onto their orthogonal complement cannot be (unambiguously) distinguished by PPT operations asymptotically, which is a far-reaching extension of one of the main results in Ref. [22]. Meanwhile, we show that the minimum dimension of strongly PPTunextendible subspaces in a m ⊗ n system is m + n − 1, which extends the result of the minimum dimension of the PPT-unextendible subspace in Ref. [26].
II.
PRELIMINARIES
We review some notations and definitions. In the following, we will consider bipartite case only, and will frequently use symbols such as A (or A′ ) and B (or B ′ )
to denote (finite-dimensional) Hilbert spaces associated with Alice and Bob, respectively. The dimension of A and B are denoted by dA and dB . We say two subspaces S1 and S2 of some Hilbert space are orthogonal, denoted by S1 ⊥ S2 , if for any ∣ψ1 ⟩ ∈ S1 and ∣ψ2 ⟩ ∈ S2 , ⟨ψ1 ∣ψ2 ⟩ = 0. The orthogonal complement of a subspace S is denoted by S ⊥ = {∣ψ⟩ ∶ ⟨ψ∣φ⟩ = 0 ∀∣φ⟩ ∈ S}. The space of all linear operators over A is denoted by L(A). A quantum state is characterized by its density operator ρ ∈ L(A), which is a positive semidefinite operator with trace unity. The support of ρ, which is denoted by supp(ρ), is defined to be the space which is spanned by the eigenvectors of ρ with positive eigenvalues. We say a positive semidefinite operator X is supporting on some subspace S of A, if supp(X) ⊂ S. For convenience, we use λmax (X) and λmin (X) to denote the maximum eigenvalue and the minimum eigenvalue of some operator X ∈ L(A). A bipartite positive semidefinite operator EAB ∈ L(A⊗ B) is said to be Positive-Partial-Transpose (PPT) if TB EAB is positive semidefinite, where the action of partial transpose (with respect to B) is defined as (∣iA ⟩⟨kA ∣ ⊗ ∣jB ⟩⟨lB ∣)TB = ∣iA ⟩⟨kA ∣ ⊗ ∣lB ⟩⟨jB ∣. Moreover, a PPT operTB ator EAB ∈ L(A ⊗ B) is said to be PPT-definite, if EAB is positive definite. In this paper, the PPT operations used for distinguishing a set of n orthogonal quantum states S = {ρ1 , . . . , ρn } can be defined as a n-tuple of matrices, (Mk )k=1,...,n , where Mk ∈ L(A ⊗ B) is PPT for k = 1, . . . , n and n ∑k=1 Mk = 1A⊗B . Then S is said to be (i) perfectly distinguishable by PPT operations, if there exist (Mk )k=1,...,n , where Mk ∈ L(A ⊗ B) is PPT for k = 1, . . . , n and ∑nk=1 Mk = 1A⊗B , such that Tr(Mi ρj ) = δij , for any 1 ≤ i, j ≤ n; (ii) unambiguously distinguishable by PPT operations, if there exist (Mk )k=1,...,n , where Mk ∈ L(A ⊗ B) is PPT for k = 1, . . . , n and ∑nk=1 Mk = 1A⊗B , such that Tr(Mi ρj ) = pi δij , pi > 0 for any 1 ≤ i, j ≤ n; (iii) indistinguishable by PPT operations, if it is not unambiguously distinguishable by PPT (LOCC) operation. These definitions can be naturally generalized to the multiple copy cases. In addition, we say a set of orthogonal quantum states {ρ1 , ..., ρn } is indistinguishable by PPT operations asymptotically, if for any positive integer ⊗k k, {ρ⊗k 1 , ..., ρn } is indistinguishable by PPT operations. In the end, we introduce the concepts of PPTextendible subspace, PPT-unextendible subspace and the strongly PPT-unextendible subspace.a bipartite subspace S of A ⊗ B is said to be PPT-extendible, if there exists a PPT operator σ ∈ L(A ⊗ B), such that σ is in the orthogonal complement of S, i.e., supp(σ) ⊥ S. S is PPT-unextendible if it is not PPT-extendible, and is strongly PPT-unextendible if for any positive integer k, S ⊗k is not PPT-extendible.
3 III. A.
Proof Assume that there exists k such that supp(ρ⊗k ) is PPT-extendible and ρ⊥ ∈ L(A⊗k ⊗ B ⊗k ) is the PPT operator satisfying Tr(ρ⊗k ρ⊥ ) = 0. Moreover, we have supp(σ ⊗k ) ⊂ supp(ρ⊗k ) and
MAIN RESULTS
Indistinguishability by PPT operations
The main result of this paper is a sufficient criterion for verifying the PPT indistinguishability of orthogonal quantum states: Theorem 1 For a set of orthogonal bipartite quantum states S = {ρ1 , ..., ρn }, if there is a PPT-definite operator supporting on the support of some ρk ∈ S, then S is asymptotically indistinguishable by PPT operations. The property of PPT-definite can be verified easily by semidefinite programming (SDP) [24], which is a powerful tool in quantum information theory with many applications (e.g., [41–48]). There are known polynomial-time algorithms for semidefinite programming [49]. To be specific, for any positive semidefinite operator σ ∈ L(A ⊗ B), suppose that P is the projection onto supp(σ), then σ is PPT-definite if and only if T (σ) > 0, where T (σ) = max t, s.t. 0 ≤ R ≤ P, RTB ≥ t1.
(1)
To prove this theorem, we first show the following lemma. Lemma 2 For a set of orthogonal quantum states S = {ρ1 , ..., ρn }, if there exists k ∈ {1, . . . , n} such that supp(ρk ) is strongly PPT-unextendible, then S is asymptotically indistinguishable by PPT operations. Proof We only need to show that if there exists k ∈ {1, . . . , n} such that supp(ρ⊗m k ) is PPT-unextendible. ⊗m Then {ρ⊗m , ..., ρ } cannot be unambiguously distinn 1 guished by PPT operations. Suppose that Pk is the projection onto supp(ρ⊗m k ) for any 1 ≤ k ≤ n. by the definition of unambiguously distinguishablility by PPT operations, there exists a set of PPT operators {Ek ∈ L(A⊗m ⊗ B ⊗m ) ∶ k = 1, . . . , n} for each 1 ≤ k ≤ n such that Tr(Ek Pj⊗m )
= pk δkj , pk > 0, j = 1, . . . , m.
⊥ It is clear that Ek ∈ supp(ρ⊗m j ) , then Ek / Tr(Ek ) is a ⊗m ⊥ PPT operator in supp(ρj ) for all j ≠ k. Hence, for ⊥ each 1 ≤ k ≤ n, supp(ρ⊗m k ) contains at least one PPT operator, which leads to a contradiction. ⊔ ⊓ Now, it is sufficient to show that the property of strongly-unextendible can be observed by PPT-definite operator:
Lemma 3 For any quantum state ρ ∈ L(A ⊗ B), if there is a PPT-definite operator σ ∈ L(A ⊗ B) such that supp(σ) ⊂ supp(ρ), then supp(ρ) is strongly PPTunextendible.
Tr(σ ⊗k ρ⊥ ) = Tr((σ ⊗k )TB (ρ⊥ )TB ) > 0,
(2)
which will be a contradiction if the last inequality holds. Now we will prove (σ ⊗k )TB is still positive definite. In fact, we are going to show that the function T (⋅) has great properties such as the super-multiplicativity, i.e. T (σ1 ⊗ σ2 ) ≥ T (σ1 )T (σ2 ). To prove this, we can assume that the optimal solutions to SDP (1) of T (σ1 ) and T (σ2 ) are {R1 , t1 } and {R2 , t2 }, respectively. It is clear that 0 ≤ R1 ⊗ R2 ≤ P1 ⊗ P2 T T and R1 B1 ⊗ R2 B2 ≥ t1 t2 1. Then {R1 ⊗ R2 , t1 t2 } is a feasible solution to SDP (1) of T (σ1 ⊗ σ2 ), which means that T (σ1 ⊗ σ2 ) ≥ t1 t2 > 0. It follows immediately that (σ ⊗n )TB1 ...Bn is also positive definite, since T (σ ⊗n ) ≥ T (σ)n . ⊔ ⊓ B.
Examples of sets of quantum states which are asymptotically PPT indistinguishable
It has been shown that the bipartite maximally entangled state and the normalized projection onto its orthogonal complement cannot be distinguished by PPT operations asymptotically, which is one of the main results in Ref. [22]. Nevertheless, it is easy to verify that the projection onto the orthogonal complement of a bipartite maximally entangled state is PPT-definite. In fact, we have the following more general result. Proposition 4 Let ∣φ⟩⟨φ∣ ∈ L(A⊗B) be the density operator of any entangled state ∣φ⟩ (need not to be maximally entangled). The projection onto the orthogonal complement of span{∣φ⟩}, after normalization, is a density operator φ⊥ = d21−1 (1A⊗B − ∣φ⟩⟨φ∣). Then {∣φ⟩⟨φ∣, φ⊥ } cannot be distinguished by PPT operations asymptotically. Proof We assume dA = dB = m. It is easy to see that λmin ((1 − ∣φ⟩⟨φ∣)TB ) > 0 is equivalent to λmax (∣φ⟩⟨φ∣TB ) < 1. Suppose that ∣φ⟩ has the Schmidt decomposition ∣φ⟩ = m m ∑i=1 λi ∣ii⟩ with λ21 ≥ ... ≥ λ2m and ∑i=1 λ2i = 1. Then m
∣φ⟩⟨φ∣TB = ∑ λ2i ∣ii⟩⟨ii∣ + ∑ λi λj ∣ji⟩⟨ij∣. i=1
i≠j
Noting that ∣φ⟩ is an entangled pure state, then λmax (∣φ⟩⟨φ∣TB ) = λ21 < 1. Therefore, 1 − ∣φ⟩⟨φ∣ is PPT-definite and the result fol⊔ ⊓ lows directly from Theorem 1. Moreover, Theorem 1 can be used to construct more general sets of quantum states which are asymptotically PPT indistinguishable without invoking unextendible product bases (UPBs). The following is another example:
4 Proposition 5 Let dA = dB = d. Choose a set of orthogonal basis {∣φi ⟩ ∈ A ⊗ B ∶ i = 1, . . . , d2 } of A ⊗ B such that ∣φi ⟩ is maximally entangled for any i = 1, . . . , d2 . For any positive integer m > d2 − d + 1, the projection P onto the subspace span{∣φi ⟩ ∶ i = 1, . . . , m} is PPTdefinite. Moreover, for any k = 2, . . . , m, we can construct {ρi ∶ i = 1, . . . , k} which is a set of asymptotically PPT indistinguishable orthogonal states. Proof
2
Notice that P = Id2 − ∑di=m+1 ∣φi ⟩⟨φi ∣, then λmin (P
TB
Surprisingly, using Lemma 3, we are able to show that the subspace S ⊥ is strongly PPT-unextendible, which illustrated that the minimum dimension of strongly PPTunextendible subspaces in A ⊗ B is also dA + dB − 1.
Theorem 6 Suppose that dA = m, dB = n (2 ≤ m ≤ n) and S = span{∣j⟩∣k + 1⟩ − ∣j + 1⟩∣k⟩ ∶ 0 ≤ j ≤ m − 2, 0 ≤ k ≤ n − 2}. S ⊥ is strongly PPT-unextendible.
d2
) = 1 − λmax ( ∑ ∣ψi ⟩⟨ψi ∣TB ) i=m+1
d2
≥ 1 − ∑ λmax (∣ψi ⟩⟨ψi ∣TB )
(3)
First, we denote S ⊥ by Smn with respect to the dimension of A and B. Smn can be obtained by solving a system of linear equations:
i=m+1 2
≥1−
Proof We will first show an explicit form of S ⊥ and then use induction method to prove that S ⊥ contains a PPT-definite operator by an explicit construction.
d −m 1 ≥ > 0. d d
Then the maximal indistinguishable set of k states {ρ1 , . . . , ρk } can be chosen as:
m−1−s
Smn =span{∣ψs ⟩ = ∑ ∣j⟩∣m − 1 − s − j⟩ ∶ s = 0, . . . , m − 1; j=0
ρ1 =
1 d2 − d + 1
d2 −d+1
min{n−1,t}
∑ ∣φi ⟩⟨φi ∣,
∣φt ⟩ =
i=1
ρ2 =∣φd2 −d+2 ⟩⟨φd2 −d+2 ∣, ... ρk =∣φd2 −d+k ⟩⟨φd2 −d+k ∣,
∑
j=t−m+1
∣t − j⟩∣j⟩ ∶ t = m, . . . , m + n − 2}.
We claim that there exists positive real numbers x0 , x1 , . . . , xm−1 , ym , . . . , ym+n−2 such that ⊔ ⊓ m−1
m+n−2
s=0
t=m
ρmn = ∑ xm−1−s ∣ψs ⟩⟨ψs ∣ + ∑ yt ∣φt ⟩⟨φt ∣ C.
Minimum dimension of strongly PPT-unextendible subspace
(4)
is PPT-definite. So far we have shown that the notion of strongly PPT-unextendible subspace plays an important role in constructing and verifying asymptotic PPT indistinguishability of bipartite quantum states. Thus, it is of great interest to study the properties of strongly PPTunextendible subspaces. Since any set of N orthogonal bipartite pure states can always be perfectly distinguished by LOCC if N −1 copies [18], it indicates that the dimension of strongly PPT-unextendible subspace cannot be too small. In fact, if we consider the dimension of bipartite subspaces in A ⊗ B of which there is no product state in the orthogonal complement, it has been shown in Ref. [25] that the minimum dimension of such a subspace is dA + dB − 1. Since there is no product state in the orthogonal complement of PPT-unextendible subspaces, it is easy to obtain the minimum dimension of the strongly PPT-unextendible subspace is at least dA + dB − 1. Moreover, a PPT-unextendible subspace with dimension exact equals to dA + dB − 1 can be explicitly constructed [26]. The be specific, for dA = m and dB = n satisfying 2 ≤ m ≤ n, the subspace is the orthogonal complement of S = span{∣j⟩∣k+1⟩−∣j+1⟩∣k⟩ ∶ 0 ≤ j ≤ m−2, 0 ≤ k ≤ n−2}.
B We rewrite the eigenvectors of ρTmn under the computational basis {∣jk⟩ ∶ j = 0, . . . , m − 1, k = 0, . . . , n − 1}. Notice that
n−1−s
∣ψs ⟩⟨ψs ∣TB = ∑ ∣j1 ⟩⟨j2 ∣ ⊗ ∣m − 1 − s − j2 ⟩⟨m − 1 − s − j1 ∣, j1 ,j2 =0
min{n−1,t}
∣φt ⟩⟨φt ∣TB =
∑
j1 ,j2 =t−m+1
∣t − j1 ⟩⟨t − j2 ∣ ⊗ ∣j2 ⟩⟨j1 ∣. (5)
B Now we consider the matrix form of ρTmn under the computational basis. Divide {∣jk⟩ ∶ j = 0, . . . , m − 1, k = 0, . . . , n − 1} into the two following families
Pa = {∣m − 1 − a + t⟩∣t⟩ ∶ 0 ≤ t ≤ a}; Qb = {∣r⟩∣r + b⟩ ∶ 0 ≤ r ≤ min{n − 1 − b, m − 1}},
(6)
where a = 0, . . . , m − 1 and b = 1, . . . , n − 1. The submatrices spanned by Pa and Qb are denoted by Pa and Qb .
5 More precisely, Pa and Qb have the following form: ⎛ xa xa−1 ⋱ ⎜x Pa = ⎜ a−1 ⎜ ⋱ ⋱ ⎝ x0 ym
⋯ x0 ⎞ ⋱ ym ⎟ ⎟ ,0 ≤ a ≤ m − 1 ⋱ ⋮ ⎟ ⋯ ym+a−1 ⎠(a+1)×(a+1)
⎛xm−1−b ⎜ ⋮ ⎜ x ⎜ 0 Qb = ⎜ ⎜ ym ⎜ ⎜ ⋮ ⎝ ym+b−1
⋯ ⋱ ⋱ ⋱ ⋱ ⋯
x0 ⋱ ⋱ ⋱ ⋱ ⋯
ym ⋱ ⋱ ⋱ ⋱ ⋯
⋯ ym+b−1 ⎞ ⋱ ⋮ ⎟ ⎟ ⋱ ⋮ ⎟ ⎟ ,1 ≤ b ≤ n − m ⎟ ⋱ ⋮ ⎟ ⎟ ⋱ ⋮ ⋯ y2m+b−2 ⎠(m)×(m)
⎛xm−1−b ⎜ ⋮ ⎜ x ⎜ 0 Qb = ⎜ ⎜ ym ⎜ ⎜ ⋮ ⎝ yn−1
⋯ ⋱ ⋱ ⋱ ⋱ ⋯
x0 ⋱ ⋱ ⋱ ⋱ ⋯
ym ⋱ ⋱ ⋱ ⋱ ⋯
⋯ yn−1 ⎞ ⋱ ⋮ ⎟ ⋱ ⋮ ⎟ ⎟ ⎟ ,b > n − m ⋱ ⋮ ⎟ ⎟ ⋱ ⋮ ⎟ ⋯ y2n−b−2 ⎠(n−b)×(n−b) (7)
Moreover, we have n−1 B ρTmn = (⊕m−1 a=0 Pa ) ⊕ (⊕b=1 Qb ). B To construct ρTmn which is positive definite, it is equivalent to construct Pa and Qb which are positive definite for a = 0, . . . , m − 1 and b = 1 . . . , n − 1. The case m = n = 2 is easy to verify. Notice that S22 = span{∣00⟩, ∣01⟩ + ∣10⟩, ∣11⟩} and we can easily choose a operator ρ = 2(∣00⟩⟨00∣+∣11⟩⟨11∣)+(∣01⟩+∣10⟩)(⟨01∣+⟨10∣) which is PPT-definite. The case when m = n can be firstly proved by matheB matical induction. We are going to prove if ρTmm is posTB itive definite, than there exists ρm+1m+1 which is also positive definite. Notice that when m = n, we can assume xa = ym−1+a , then we have Pa = Qm−1−a for B a = 0, . . . , m − 2 and ρTmm = Pm−1 ⊕ [⊕m−2 k=0 (Pk ⊕ Qm−1−k )]. B Then assume ρTmm is positive definite, there exists positive x0 , . . . , xm−1 such that
⎛ xk xk−1 ⋱ ⎜x Pk = ⎜ k−1 ⎜ ⋮ ⋱ ⎝ x0 x1
⋯ x0 ⎞ ⋱ x1 ⎟ ⎟ ⋱ ⋮ ⎟ ⋯ xk ⎠(k+1)×(k+1)
is positive definite for k = 0, . . . , m − 1. Consider B ρT(m+1)(m+1) , we want to find x′0 , . . . , x′m such that
′ B ρTm+1m+1 = x′m ∣ψm ⟩⟨ψm ∣ + ∑m−1 k=0 xk (∣ψk ⟩⟨ψk ∣ + ∣φk ⟩⟨φk ∣) is positive definite, i.e. we need ′ ′ ⎛ xk xk−1 ′ ⋱ ⎜x Pk′ = ⎜ k−1 ⎜ ⋮ ⋱ ⎝ x′0 x′1
⋯ x′0 ⎞ ⋱ x′1 ⎟ ⎟ >0 ⋱ ⋮ ⎟ ′⎠ ⋯ xk (k+1)×(k+1)
for k = 0, . . . , m. Let x′k = xk for k = 0, . . . , m − 1, then Pk′ > 0 since Pk′ = Pk for k = 0, . . . , m − 1. We only need
to find a positive x′m such that ′ ⎛ xm xm−1 ⋱ ⎜x ′ Pm = ⎜ m−1 ⎜ ⋮ ⋱ ⎝ x0 x1
⋯ x0 ⎞ ⋱ x1 ⎟ ⎟ ⋱ ⋮ ⎟ ⋯ x′m ⎠(k+1)×(m+1)
is positive definite. This can be done by considering the ′ leading principal minors of Pm are all positive. Thus we have m − 1 linear constraints and a quadratic constraint. Notice that for all linear constraints, the coefficient of xm should be positive, which can be easily derived since Pk′ are positive definite for k = 1, . . . , m − 1. Moreover the coefficient of x2 in the quadratic constraint is also positive since the following matrix ⎛xm−2 ⎜ ⋮ ⎜ ⎜ x1 ⎝ x0
⋯ ⋱ ⋱ x1
x1 x0 ⎞ x0 x1 ⎟ ⎟ ⋱ ⋮ ⎟ ⋯ xm−2 ⎠
is also positive definite. It is not difficult to obtain that the set of positive real numbers which satisfy the above constraints are always non-empty. Thus we can choose B one to make sure ρT(m+1)(m+1) is positive definite. Finally, the case m ≠ n can be proved by similar technique. Assume we have already known x′0 , . . . , x′m−1 such that ρmm is PPT-definite. For ρmn where n > m, we are going to find x0 , . . . , xm−1 and ym , . . . , ym+n−2 such that ρmn is PPT-definite too. Now let xi = x′i for i = 0, . . . , m − 1 and ym+j = xj+1 for j = 0, . . . , m − 2. This will guarantee Pa for a = 0, . . . , m − 1 in Eq. 7 are positive definite. Then we consider Qb where b = 1 . . . , n − m. When b = 1, we only need to determine y2m−1 such that y2m−1 > 0 and Q1 is positive definite. This can be done by only guarantee the determinant of Qb is always positive, which is a linear constraint for y2m−1 . This is true since the first m − 1 leading principle minors of Q1 are leading principle minors of Pm−2 , which is automatically positive. Then we can determine y2m−2 , . . . , ym+n−2 with the same technique. We now need to prove that Qb is positive definite too for b = n − m + 1, . . . , n − 1. In fact, this is the m − 1th leading principle minors of some Qn−m−2 , which is positive definite. Then we can prove Qb is positive definite for any b = n − m + 1, . . . , n − 1, which finish our proof. In conclusion, for any 2 ≤ m ≤ n, we can construct a PPT-definite operator whose support is containing in Smn . Then Smn turns out to be a strongly PPTunextendible subspace with dimension m + n − 1. ⊔ ⊓ IV.
CONCLUSIONS AND DISCUSSIONS
In summary, we study the asymptotic indistinguishability of bipartite quantum states by PPT operations. By introducing the concept of strongly PPT-unextendible subspace, we first show that such subspace can be served as a witness of asymptotic PPT indistinguishability.
6 Driven by that, we introduce the PPT-definite operator for deciding whether a subspace is strongly PPTunextendible. Remarkably, verifying the property of PPT-definite can be formalized as a semidefinite program, which can be solved efficiently. Utilizing these results, we show that any entangled pure state and the normalized projector onto its orthogonal complement cannot be distinguished by PPT operations asymptotically. This result actually reveals the nature of why a maximally entangled state and its orthogonal complement are asymptotically PPT indistin-
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The authors would like to thank Yuan Feng, Andreas Winter and Dong Yang for helpful discussions. This work was partly supported by the Australian Research Council under Grant Nos. DP120103776 and FT120100449.
