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arXiv:1007.1830

A problem of existence of bound entangled states with non-positive partial transpose and the Hilbert's 17th problem

Tohya Hiroshima

Published Jul 12, 2010• 0 citations
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Abstract

It is found that the problem of existence of bound entangled states with non-positive partial transpose (NPPT) has the intriguing relation to the Hilbert's 17th problem. More precisely, we compute the expectation value of the partially transposed Werner states by Schmidt rank-2 vectors for NPPT and 1-copy undistillable region. It is the positive polynomial but shown not to be expressed as the sum of squares of polynomials. A remedy for such pathological behavior as well as a similar but different approach to the problem is also mentioned.

DOI
10.48550/arxiv.1007.1830

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It is found that the problem of existence of bound entangled states with non-positive partial transpose (NPPT) has the intriguing relation to the Hilbert's 17th problem. More precisely, we compute the expectation value of the partially transposed Werner states by Schmidt rank-2 vectors for NPPT and 1-copy undistillable region. It is the positive polynomial but shown not to be expressed as the sum of squares of polynomials. A remedy for such pathological behavior as well as a similar but different approach to the problem is also mentioned.
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
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.
Equivalence between positive partial transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. Limited progress has been made so far, even on the seemingly simple case of the bound entanglement of Werner states. The primary challenge is to give a concise mathematical representation of undistillability. To this end, we propose a decomposition of the NNN-undistillability verification into log(N)\text{log}(N)log(N) repeated steps of 1-undistillability verification. For Werner states' NNN-undistillability verification, a parameter interval for NNN-undistillability is given, which is independent of the dimensionality of Werner states. Equivalent forms of inequalities for both rank 1 and 2 matrices are presented, before transforming the 2-undistillability case into a matrix analysis problem. A new perspective is also attempted by seeing it as a nonconvex multivariable function, proving its critical points and conjecturing Hessian positivity, which would make them local minimums.
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.
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.