Family of bound entangled states on the boundary of Peres set

TL;DR

This paper introduces a family of bound entangled states with positive partial transpose in odd-dimensional systems, constructed via unextendible product bases, revealing their geometric position on the boundary of the Peres set and implications for quantum state discrimination.

Contribution

It presents a new family of PPT-bound entangled states parameterized by $(d-3)/2$, constructed using unextendible product bases in odd dimensions, and analyzes their geometric and convex structure.

Findings

Family of PPT-BE states constructed in odd dimensions

States can be expressed as convex combinations of rank-4 PPT-BE states

The convex hull of these states forms a simplex on the boundary of the Peres set

Abstract

Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize them for an arbitrary composite quantum system. Here we present a $(d-3)/2$-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d$ odd and $d\ge 5$. The range of each such PPT-BE state is contained in a $2(d-1)$ dimensional entangled subspace whereas the associated UPB-subspace is of dimension $(d-1)^2+1$. We further show that each of these PPT-BE states can be written as a convex combination of $(d-1)/2$ number of rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set $\mathcal{P}$ of all PPT states in $\mathbb{C}^d\otimes\mathbb{C}^d$, namely the {\it Peres} set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the $(d-3)/2$-simplex -- is sitting on the boundary between the set $\mathcal{P}$ and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.