Separable states with unique decompositions

TL;DR

This paper investigates the structure of separable quantum states with unique decompositions, identifying specific faces of the convex set of separable states that are affinely isomorphic to simplices, and explores their properties in various quantum systems.

Contribution

It introduces new classes of faces of the separable state set that are isomorphic to simplices, providing explicit constructions and analyzing their properties in two-qutrit and qubit-qudit systems.

Findings

Identified faces isomorphic to 5- and 9-simplices in two-qutrit states.

Showed that every rank-4 edge state arises from these faces.

Constructed PPT states of type (9,5) and generalized to higher dimensions.

Abstract

We search for faces of the convex set consisting of all separable states, which are affinely isomorphic to simplices, to get separable states with unique decompositions. In the two-qutrit case, we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5-dimensional simplex with six vertices, under suitable linear independence assumption. If the partial conjugates of six product vectors also span a 5-dimensional space, then this face is inscribed in the face for PPT states whose boundary shares the fifteen 3-simplices on the boundary of the 5-simplex. The remaining boundary points consist of PPT entangled edge states of rank four. We also show that every edge state of rank four arises in this way. If the partial conjugates of the above six product vectors span a 6-dimensional space then we have a face isomorphic to 5-simplex, whose interior consists of separable states with unique decompositions, but with non-symmetric ranks. We also construct a face isomorphic to the 9-simplex. As applications, we give answers to questions in the literature \cite{chen_dj_semialg,chen_dj_ext_PPT}, and construct $3\ot 3$ PPT states of type (9,5). For the qubit-qudit cases with $d\ge 3$, we also show that $(d+1)$-dimensional subspaces give rise to faces isomorphic to the $d$-simplices, in most cases.