2004/10/31 by Paul B. Slater · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Applied mathematics #Combinatorics #Computer science #Discrete mathematics #Geometry #Haar measure #Mathematical analysis #Mathematics #Measure (data warehouse) #Metric (unit) #Monotone polygon #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Qubit #Qutrit #Rank (graph theory) #Separable space #quant-ph
paper · pdf · doi:10.1103/physreva.71.052319
published as Phys. Rev. A 71, 052319 (2005) · 36 pages, 15 figures, 11 tables, final PRA version, new last paragraph presenting qubit-qutrit probability ratios disaggregated by the two distinct forms of partial transposition
openalex publication_date 2005/05/16 · arxiv created 2005/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Paralleling our recent computationally intensive (quasi-Monte Carlo) work for the case N=4 (e-print quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and \ifmmode Z\else \.Z\fiyczkowski (e-print quant-ph/0304041) for the (N2\ensuremath-1)-dimensional volume and (N2\ensuremath-2)-dimensional hyperarea of the (separable and nonseparable) N\ifmmode×\else\texttimes\fiN density matrices, based on the Bures (minimal monotone) metric---and also their analogous formulas (e-print quant-ph/0302197) for the (nonmonotone) flat Hilbert-Schmidt metric. With the same seven 109 well-distributed (``low-discrepancy'') sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. 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 rank-6 (rank-5) density matrices. The (rank-6) separability probabilities obtained based on the 35-dimensional volumes appear to be---independently of the metric (each of the seven inducing Haar measure) employed---twice as large as those (rank-5 ones) based on the 34-dimensional hyperareas. (An additional estimate---33.9982---of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit simple exact formulas to our estimates of the Hilbert-Schmidt separable volumes and hyperareas in both the N=4 and N=6 cases.