Separable states with unique decompositions

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.

26 pages, 1 figure. A part on SPA conjecture is withdrawn

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Cited by in corpus (24)

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• Exposedness of Choi type entanglement witnesses and applications to lengths of separable states
• Checking the optimality of entanglement witnesses: an application to structural physical approximations
• The role of phases in detecting three-qubit entanglement
• Boundary of the set of separable states
• Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
• Faces for two qubit separable states and the convex hulls of trigonometric moment curves
• Separability criterion for three-qubit states with a four dimensional norm
• Conditions for uniqueness of product representations for separable quantum channels and separable quantum states
• Physical realization of realignment criteria using structural physical approximation
• Dimension formula for induced maximal faces of separable states and genuine entanglement
• Length filtration of the separable states
• Construction of exposed indecomposable positive linear maps between matrix algebras
• Three qubit separable states of length ten with unique decompositions
• Separability of multi-qubit states in terms of diagonal and anti-diagonal entries
• Structural physical approximation of partial transposition makes possible to distinguish SLOCC inequivalent classes of three-qubit system
• Bayesian model selection in the $\mathcal{M}$-open setting -- Approximate posterior inference and probability-proportional-to-size subsampling for efficient large-scale leave-one-out cross-validation
• The number of product vectors and their partial conjugates in a pair of spaces
• Indecomposable exposed positive bi-linear maps between two by two matrices
• Global geometric difference between separable and Positive partial transpose states