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

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.

36 pages

References in corpus (9)

• Understanding entanglement as resource: locally distinguishing unextendible product bases
• Generic local distinguishability and completely entangled subspaces
• Bell inequalities with no quantum violation and unextendible product bases
• A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces
• Distillability and PPT entanglement of low-rank quantum states
• Unextendible product bases and extremal density matrices with positive partial transpose
• Numerical studies of entangled PPT states in composite quantum systems
• Description of rank four PPT entangled states of two qutrits
• Low rank positive partial transpose states and their relation to product vectors

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