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QubitQutrit SeparabilityProbability Ratios  Paul B. Slater
;  Date: 
28 Oct 2004  Journal:  Phys. Rev. A 71, 052319 (2005)  Subject:  quantph  Abstract:  Paralleling our recent computationallyintensive (quasiMonte Carlo) work for the case N=4 (quantph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quantph/0304041) for the (N^21)dimensional volume and (N^22)dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric  and also their analogous formulas (quantph/0302197) for the (nonmonotone) HilbertSchmidt metric. With the same seven billion welldistributed (``lowdiscrepancy’’) sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the KuboMori and WignerYanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically ranksix (rankfive) density matrices. The (ranksix) separability probabilities obtained based on the 35dimensional volumes appear to be  independently of the metric (each of the seven inducing Haar measure) employed  twice as large as those (rankfive ones) based on the 34dimensional hyperareas. Accepting such a relationship, we fit exact formulas to the estimates of the Bures and HilbertSchmidt separable volumes and hyperareas.(An additional estimate  33.9982  of the ratio of the rank6 HilbertSchmidt separability probability to the rank4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the HilbertSchmidt metric, but not the others. We fit exact formulas for the HilbertSchmidt separable volumes and hyperareas.  Source:  arXiv, quantph/0410238  Services:  Forum  Review  PDF  Favorites 


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