Three-by-three bound entanglement with general unextendible product bases

TL;DR

This paper characterizes the structure of certain low-rank entangled states in 3x3 systems using unextendible product bases, confirming numerical findings with algebraic geometry tools.

Contribution

It proves that rank-4 non-separable PPT states are locally equivalent to projections orthogonal to general unextendible product bases, extending previous orthogonal cases.

Findings

Rank-4 non-separable PPT states are locally equivalent to orthogonal projections.

Product vectors in the kernels can be transformed to orthogonal form.

The results confirm recent numerical observations by Leinaas et al.

Abstract

We discuss the subject of Unextendible Product Bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e. states of rank 4 in 3 x 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal Unextendible Product Basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general Unextendible Product Bases, nevertheless they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's Theorem from algebraic geometry.