Low rank positive partial transpose states and their relation to product vectors

Abstract

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.

29 pages, 4 figures

References in corpus (8)

• Experimental multiparticle entanglement dynamics induced by decoherence
• All bipartite entangled states display some hidden nonlocality
• Experimental bound entanglement in a four-photon state
• Three-by-three bound entanglement with general unextendible product bases
• Extreme points of the set of density matrices with positive partial transpose
• 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

Cited by in corpus (11)

• Three-by-three bound entanglement with general unextendible product bases
• Family of bound entangled states on the boundary of Peres set
• Facial structures for various notions of positivity and applications to the theory of entanglement
• Separable states with unique decompositions
• Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
• Existence of product vectors and their partial conjugates in a pair of spaces
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
• Extremal states of positive partial transpose in a system of three qubits
• Necessary conditions for optimality of decomposable entanglement witnesses
• Extremal entanglement witnesses
• Nongeneric positive partial transpose states of rank five in $3\times 3$ dimensions