New family of bound entangled states residing on the boundary of Peres set. (arXiv:1801.00405v1 [quant-ph])

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 all these
states 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 $d\ge 5$. We also provide the `tile structure' corresponding to these
UPBs. 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.

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