可分和不可分量子位-量子系统空间的超区域和超区域：初步数值分析
Volumes and hyperareas of the spaces of separable and nonseparable qubit-qutrit systems: Initial numerical analyses
摘要：
在N=4的情况下，我们最近的计算密集型工作（quant-ph/0308037），我们承担了N=6的高数值精度计算任务，Sommers和Zyczkowski的公式（quant-ph/0304041）的（N^2-1）维体积和（N^2-2）维超面积，（可分离和不可分离）N × N密度矩阵，基于Bures（最小单调）度量。在同一时间，我们估计未知的体积和超面积的基础上，一些其他的单调度量的兴趣。此外，我们估计-但可能会减少准确性-所有这些体积和超面积的数量，当限制在“小”子集的6 × 6密度矩阵是可分离的（经典相关）的性质。然后，可分离体积与可分离体积加不可分离体积的比率产生相应的“可分离概率”的估计。我们特别感兴趣的是探索各种35维体积和34维超面积中的一些具有精确值的可能性-事实上，我们已经证明了量子比特-量子比特系统（N=4）的情况，“银平均”sqrt{2}-1似乎在可分离状态方面发挥了基本作用。
英文摘要：
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 (N^2-1)-dimensional volume and (N^2-2)-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'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
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