Qubit-Qutrit Separability-Probability Ratios

Abstract

Paralleling our recent computationally-intensive (quasi-Monte Carlo) 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 -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion 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-six (rank-five) density matrices. The (rank-six) 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-five ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable volumes and 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 exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.

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

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Cited by in corpus (24)

• Quantum entanglement
• Manipulating biphotonic qutrits
• Universal observable detecting all two-qubit entanglement and determinant based separability tests
• Relating the Bures measure to the Cauchy two-matrix model
• On the structure of the body of states with positive partial transpose
• Bell inequality violation versus entanglement in presence of local decoherence
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Geometric quantum computation using fictitious spin- 1/2 subspaces of strongly dipolar coupled nuclear spins
• Eigenvalues, Separability and Absolute Separability of Two-Qubit States
• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• On the Bures Volume of Separable Quantum States
• Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors
• Two-Qubit Separability Probabilities and Beta Functions
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities
• An Explicit Computation of the Bures Metric Over the Space of $N$-Dimensional Density Matrices
• Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities
• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Polytope structures for Greenberger-Horne-Zeilinger diagonal states
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
• Hilbert-Schmidt Geometry of n-Level Jakobczyk-Siennicki Two-Dimensional Quantum Systems