dorsal/arxiv
View SchemaVolumes and hyperareas of the spaces of separable and nonseparable qubit-qutrit systems: Initial numerical analyses
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0405114 |
| URL | https://arxiv.org/abs/quant-ph/0405114 |
Abstract
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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"abstract": "Paralleling our recent computationally-intensive work for the case N=4\n(quant-ph/0308037), we undertake the task for N=6 of computing to high\nnumerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041)\nfor the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the\n(separable and nonseparable) N x N density matrices, based on the Bures\n(minimal monotone) metric. At the same time, we estimate the unknown volumes\nand hyperareas based on a number of other monotone metrics of interest.\nAdditionally, we estimate -- but with perhaps unavoidably diminished accuracy\n-- all these volume and hyperarea quantities, when restricted to the ``small\u0027\u0027\nsubset of 6 x 6 density matrices that are separable (classically correlated) in\nnature. The ratios of separable to separable plus nonseparable volumes, then,\nyield corresponding estimates of the ``probabilities of separability\u0027\u0027. We are\nparticularly interested in exploring the possibility that a number of the\nvarious 35-dimensional volumes and 34-dimensional hyperareas, possess exact\nvalues -- which we had, in fact, conjectured to be the case for the qubit-qubit\nsystems (N=4), with the ``silver mean\u0027\u0027, sqrt{2}-1, appearing to play a\nfundamental role as regards the separable states.",
"arxiv_id": "quant-ph/0405114",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph"
],
"title": "Volumes and hyperareas of the spaces of separable and nonseparable qubit-qutrit systems: Initial numerical analyses",
"url": "https://arxiv.org/abs/quant-ph/0405114"
},
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