On independent permutation separability criteria

Recently P. Wocjan and M. Horodecki [quant-ph/0503129] gave a characterization of combinatorially independent permutation separability criteria. Combinatorial independence is a necessary condition for permutations to yield truly independent criteria meaning that that no criterion is strictly stronger that any other. In this paper we observe that some of these criteria are still dependent and analyze why these dependencies occur. To remove them we introduce an improved necessary condition and give a complete classification of the remaining permutations. We conjecture that the remaining class of criteria only contains truly independent permutation separability criteria. Our conjecture is based on the proof that for two, three and four parties all these criteria are truly independent and on numerical verification of their independence for up to 8 parties. It was commonly believed that for three parties there were 9 independent criteria, here we prove that there are exactly 6 independent criteria for three parties and 22 for four parties.Comment: Revtex4, 7 pages, minor correction

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

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, Bru{\ss} 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.Comment: 5 pages, comments welcom

Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory

PhD thesis (University of York). The thesis covers in a unified way the material presented in quant-ph/0403073, quant-ph/0502040, quant-ph/0504160, quant-ph/0510035, quant-ph/0512012 and quant-ph/0603283. It includes two large review chapters on entanglement and distillation.Comment: 192 page