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arXiv:1702.00231 · Physical Review A

Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario

Yinan Li, Xin Wang, Runyao Duan

Published May 30, 2017• 20 citations
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Abstract

A bipartite subspace $S$ is called strongly positive-partial-transpose (PPT) unextendible if for every positive integer $k$, there is no PPT operator supporting on the orthogonal complement of ${S}^{\ensuremath{\bigotimes}k}$. 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). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an $m\ensuremath{\bigotimes}n$ system is $m+n\ensuremath{-}1$, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].

DOI
10.1103/physreva.95.052346

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A bipartite subspace SSS is called strongly positive-partial-transpose (PPT) unextendible if for every positive integer kkk, there is no PPT operator supporting on the orthogonal complement of S\ensuremath⨂k{S}^{\ensuremath{\bigotimes}k}S\ensuremath⨂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). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an m\ensuremath⨂nm\ensuremath{\bigotimes}nm\ensuremath⨂n system is m+n\ensuremath−1m+n\ensuremath{-}1m+n\ensuremath−1, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].
A bipartite subspace SSS is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer kkk, there is no PPT state supporting on the orthogonal complement of S⊗kS^{\otimes k}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⊗nm\otimes nm⊗n system is m+n−1m+n-1m+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)].
In this work, we explore the notions unextendible product basis and uncompletability for operators which remain positive under partial transpose. Then, we analyze their connections to the ensembles which are many-copy indistinguishable under local operations and classical communication (LOCC). We show that the orthogonal complement of any bipartite pure entangled state is spanned by product states which form a nonorthogonal unextendible product basis (nUPB) of maximum cardinality. This subspace has one to one correspondence with the maximum dimensional subspace where there is no orthonormal product basis. Due to these, the proof of indistinguishability of a class of ensembles under LOCC in many-copy scenario becomes simpler. Furthermore, it is now clear that there are several many-copy indistinguishable ensembles which are different construction-wise. But if we consider the technique of proving their indistinguishability property under LOCC, then, for many of them it can be done using the general notion of unextendible product basis. Explicit construction of the product states, forming nUPBs is shown. Thereafter, we introduce the notion of positive partial transpose uncompletability to unify different many-copy indistinguishable ensembles. We also report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing number of mixed states.
We study the distinguishability of bipartite quantum states by positive operator-valued measures with positive partial transpose (PPT POVMs). The contributions of this paper include: 1) we give a negative answer to an open problem of showing a limitation of a previous known method for detecting nondistinguishability; 2) we show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, cannot be distinguished by the PPT POVMs, even unambiguously. This result is much stronger than the previous known ones; and 3) we study the entanglement cost of distinguishing quantum states. It is proved that √{2/3}|00〉+√{1/3}|11〉 is sufficient and necessary for distinguishing three Bell states by the PPT POVMs. An upper bound of entanglement cost of distinguishing a d ⊗ d pure state and its orthogonal complement is obtained for separable operations. Based on this bound, we are able to construct two orthogonal quantum states, which cannot be distinguished unambiguously by separable POVMs, but finite copies would make them perfectly distinguishable by local operations and classical communication. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a d ⊗ d pure state and its orthogonal complement by the PPT POVMs, no matter the value of d. In sharp contrast, an entangled state with Schmidt number at least d is always needed for distinguishing such two states by separable POVMs. As an application, we show that the entanglement cost of distinguishing a d ⊗ d maximally entangled state and its orthogonal complement must be a maximally entangled state for d=2, which implies that teleportation is optimal, and in general, it could be chosen as O(logd/d).
It is known how to construct, in a bipartite quantum system, a unique low-rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by nonsingular unitary or nonunitary product transformations. The characteristic property of a state $\ensuremath{\rho}$ in such a family is that its kernel $\text{Ker} \ensuremath{\rho}$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\mathrm{Im} \ensuremath{\rho}$, the orthogonal complement of $\text{Ker} \ensuremath{\rho}$. The generalized UPB in $\text{Ker} \ensuremath{\rho}$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\ifmmode\times\else\texttimes\fi{}3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, nonorthogonal UPBs in similar ways.
In bipartite quantum systems of dimension $3\ifmmode\times\else\texttimes\fi{}3$, entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPBs). As discussed in a previous publication, all the lowest rank entangled PPT states of this system seem to be equivalent, under $\mathrm{SL}\ensuremath{\bigotimes}\mathrm{SL}$ transformations, to states that are constructed in this way. Here we consider a possible generalization of the UPB constuction to low-rank entangled PPT states in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the subspace defined by the UPB. We examine first this generalization for the $3\ifmmode\times\else\texttimes\fi{}3$ system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of similar form. Based on these results we suggest that the UPB construction of the lowest rank entangled states in the $3\ifmmode\times\else\texttimes\fi{}3$ system can be generalized to higher dimensions, with the use of nonorthogonal UPBs.
It is known that, in an $(m\ensuremath{\bigotimes}n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is $(m\ensuremath{-}1)(n\ensuremath{-}1)$. We show that the exact same bound is tight if we require the stronger condition that every state with range in the subspace has non-positive partial transpose. As an immediate corollary of our result, we solve an open question that asks for the maximum number of negative eigenvalues of the partial transpose of a quantum state. In particular, we give an explicit method of construction of a bipartite state whose partial transpose has $(m\ensuremath{-}1)(n\ensuremath{-}1)$ negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large mmm and nnn.