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arXiv:0910.0744

Non-existence of bipartite bound entanglement with negative partial transposition

Jan Sperling, W. Vogel

Published Oct 5, 2009• 0 citations
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Abstract

Bound entanglement with a nonpositive partial transposition (NPT) does not exist. For any NPT entangled state a distillation procedure can be based on a certain number of copies. This number is the minimal Schmidt rank of a pure state needed to witness the NPT entanglement under study.
DOI
10.48550/arxiv.0910.0744

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We prove the existence of bound entangled states with negative partial transpose (NPT) in any d × d(d ≥ 3) Hilbert space with a simple assumption on Schmidt rank two states. Obviously they belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. PACS number(s): 03.67.Hk, 03.65.Ud. The basic issue on the classification of mixed state entanglement at least on the level of bipartite systems solely depends upon whether there exist bound entangled states or not. The existence of PPT-bound (PPT means positive partial transpose) entangled states [1] and also the existence of NPT N−copy undistillable states [2, 3] for every positive integer N naturally indicates there may exist NPT-bound entangled states. In this work we are able to show the existence of NPT-bound entangled states with a simple assumption on Schmidt rank two states. We first briefly describe the issue and the importance of the problem. In recent years it is found that quantum entanglement is an useful resource in performing several tasks in quantum information theory and quantum communication [4]. Maximally entangled states shared between two parties are essential ingredients in this respect [5]. Now due to the interaction with environment, states are in practice found to be mixed. However there is a process called distillation by which we can distill sometimes maximally entangled
The discovery of entangled quantum states from which one cannot distill pure entanglement constitutes a fundamental recent advance in the field of quantum information. Such bipartite bound-entangled (BE) quantum states could fall into two distinct categories: (1) Inseparable states with positive partial transposition (PPT), and (2) states with negative partial transposition (NPT). While the existence of PPT BE states has been confirmed, only one class of conjectured NPT BE states has been discovered so far. We provide explicit constructions of a variety of multicopy undistillable NPT states, and conjecture that they constitute families of NPT BE states. For example, we show that for every pure state of Schmidt rank greater than or equal to 3, one can construct n-copy undistillable NPT states, for any n> 1.n>~1.n> 1. The abundance of such conjectured NPT BE states, we believe, considerably strengthens the notion that being NPT is only a necessary condition for a state to be distillable.
In this paper, existence of bound entangled states with nonpositive partial transpose (NPT) is considered. As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover, it would rule out a simple mathematical description of the set of distillable states. The particular state, known to be 1-copy nondistillable and supposed to be bound entangled, is considered. The problem of its two-copy distillability, which boils down to show that maximal overlap of some projectorQwith Schmidt rank two states does not exceed 1/2 (called thehalf-property), is studied. First, it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. Then, the problem in attacked twofold way: (a) the half-property is proved for some wide classes of Schmidt rank two states; (b) the overlap forallSchmidt rank two states is bounded from above bycA⊗I+I⊗BwithA,Btraceless 4 × 4 matrices, and TrAfA+ TrBfB=1/4.
Recently [quant-ph/0608250] again created a lot of interest to prove the existence of bound entangled states with negative partial transpose (NPT) in any d×d(d≥3)d \times d (d \geq 3)d×d(d≥3) Hilbert space. However the proof in quant-ph/0608250 is not complete but it shows some interesting properties of the Schmidt rank two states. In this work we are trying to probe the problem in a different angle considering the work by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. We have assumed that the Schmidt rank two states should satisfy some bounds. Under some assumptions with these bounds one could prove the existence of NPT bound entangled states. We particularly discuss the case of two copy undistillability of the conjectured family of NPT states. Obviously the class of NPT bound entangled states belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. However the problem of existence of NPT bound entangled states still remain open.
We exhibit a two-parameter family of bipartite mixed states ${\ensuremath{\rho}}_{\mathrm{bc}},$ in a $d\ensuremath{\bigotimes}d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\ensuremath{\bigotimes}2$ can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of ${\ensuremath{\rho}}_{\mathrm{bc}}$ using a projection on $2\ensuremath{\bigotimes}2.$ These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to a NPT state of the ${\ensuremath{\rho}}_{\mathrm{bc}}$ form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.
We analyze a class of entangled states for bipartite $d\ensuremath{\bigotimes}d$ systems, with ddd nonprime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such a simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition. Moreover, for a restricted set of such states, we prove that there exists an interval of the mixing parameter such that both partial transposition and realignment (i.e., all permutational criteria in the bipartite setting) fail to detect them as entangled. In the range in which the states are positive under partial transposition (PPT), they are not distillable; on the other hand, the states in the considered class are provably distillable as soon as they are nonpositive under partial transposition. The states are associated to any set of more than two pure states. The analysis is extended to the multipartite setting. By an opportune selection of the set of multipartite pure states, it is possible to construct mixed states which are PPT with respect to any choice of bipartite cuts and nevertheless exhibit genuine multipartite entanglement. Finally, we show that every kkk-positive but not completely positive map is associated to a family of nondecomposable maps.