A (5,5) and (6,6) PPT edge state

Abstract

Entangled states with a positive partial transpose (PPTES) have interest both in quantum information and in the theory of positive maps. In $3\otimes 3$ there is a conjecture by Sanpera, Bruß and Lewenstein [PRA, 63, 050301] that all PPTES have Schmidt number two (or equivalently that every 2-positive map between $3\times 3$ matrices is decomposable). In order to prove or disprove the conjecture it is sufficient to look at edge PPTES. Here the rank m of the PPTES and the rank n of its partial transpose seem to play an important role. Until recently all known examples of edge PPTES had ranks (4,4) or (6,7). In a recent paper Ha and Kye [quant-ph/0509079] managed to find edge PPTES for all ranks except (5,5) and (6,6). Here we complement their work and present edge PPTES with those ranks.

5 pages, comments welcome

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• Searching for extremal PPT entangled states
• Classification of bi-qutrit positive partial transpose entangled edge states by their ranks
• Properties and construction of extreme bipartite states having positive partial transpose
• Existence of product vectors and their partial conjugates in a pair of spaces
• Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory
• Unextendible product bases, bound entangled states, and the range criterion
• Entangled symmetric states and copositive matrices
• Annihilating Entanglement Between Cones
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
• A family of generalized Horodecki-like entangled states
• Robust entanglement measure for mixed quantum states
• Entangled edge states of corank one with positive partial transposes
• Spectral characterizations of entanglement witnesses
• Bound entanglement in symmetric random induced states