Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
Kil-Chan Ha, Seung-Hyeok Kye · Journal of Physics A Mathematical and Theoretical · 2014
We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decompositions, 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. We construct, as a byproduct, three-qubit entangled positive partial transpose edge states of rank 4 with explicit formulas. This covers those entanglements which cannot be constructed from an unextendible product basis.