Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose

Abstract

We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decomposition, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three qubit case, generic five dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five dimensional simplex with six vertices. As a byproduct, we construct three qubit entangled PPT edge states of rank four with explicit formulae. This covers those entanglement which cannot be constructed from unextendible product basis.

19 pages, 1 figure

References in corpus (3)

• Generic local distinguishability and completely entangled subspaces
• The Structure of Qubit Unextendible Product Bases
• Extremal states of positive partial transpose in a system of three qubits

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• New techniques for bounding stabilizer rank
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• Dimension formula for induced maximal faces of separable states and genuine entanglement