2006/02/14 by Paul B. Slater, Slater, Paul B.
Computer Science · Physics and Astronomy · #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0602109
16 pages, modest revisions, retitled, some new references incorporated, including to quant-ph/0609006, "Two-Qubit Separability Probabilities and Beta Functions"
openalex publication_date 2006/02/14 · arxiv created 2006/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure μHaar (common to each metric) and 3-dimensional measures μmetric (specific to each metric) over the 3d-simplex formed by the four eigenvalues of ρ. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] μmetric's and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of μHaar over the twelve Euler angles. We do this by fitting W so that the conjectured HS separable volumes and hyperareas (quant-ph/0410238; cf. quant-ph/0609006) are reproduced. We further evaluate a number of possible choices of W by seeing how well they also yield the conjectured separable volumes for the Bures, Kubo-Mori, Wigner-Yanase and (arithmetic) average monotone metrics and the conjectured separable Bures hyperarea (quant-ph/0308037,Table VI). We, in fact, find two such exact (rather similar) choices that give these five conjectured (non-HS) values all within 5%. In addition to the above-mentioned Euler angle parameterization of ρ, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.