On independent permutation separability criteria

Authors:
Lieven Clarisse;Paweł Wocjan
Affiliations:
Department of Mathematics, The University of York, Heslington, York, U.K.;Computer Science Department & Institute for Quantum Information, California Institute of Technology, Pasadena, CA
Venue:
Quantum Information & Computation
Year:
2006

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Abstract

Recently, P. Wocjan and M. Horodecki [Open Syst. Inf. Dyn. 12, 331 (2005)] gave a characterizationof combinatorially independent permutation separability criteria. Combinatorialindependence is a necessary condition for permutations to yield truly independentcriteria meaning that no criterion is strictly stronger that any other. In this paper weobserve that some of these criteria are still dependent and analyze why these dependenciesoccur. To remove them we introduce an improved necessary condition and give acomplete classification of the remaining permutations. We conjecture that the remainingclass of criteria only contains truly independent permutation separability criteria. Ourconjecture is based on the proof that for two, three and four parties all these criteria aretruly 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, herewe prove that there are exactly 6 independent criteria for three parties and 22 for fourparties.