2004/05/20 by Paul B. Slater, Slater, Paul B.
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Matrix Theory and Algorithms #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0405114
12 pages, 7 tables
arxiv created 2004/05/20 · openalex publication_date 2004/05/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N2-1)-dimensional volume and (N2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt2-1, appearing to play a fundamental role as regards the separable states.