2008/06/30 by Paul B. Slater · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Applied mathematics #Classification of discontinuities #Combinatorics #Concurrence #Diagonal #Eigenvalues and eigenvectors #Matching (statistics) #Mathematical analysis #Mathematics #Multivariate statistics #Parameterized complexity #Piecewise #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Separable space #Statistics #Univariate #quant-ph
paper · pdf · doi:10.1088/1751-8113/41/50/505303
published as J. Phys. A: Math. Theor. 41 No 50 (19 December 2008) 505303 · 12 pages, 7 figures, new abstract, revised for J. Phys. A
arxiv created 2008/10/09 · openalex publication_date 2008/11/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The generic real (β = 1) and complex (β = 2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert–Schmidt and Bures metrics are obtainable as special cases of formulae of Życzkowski and Sommers. We claim that if one could determine certain metric- independent three-dimensional ' eigenvalue -parameterized separability functions' (EPSFs), S (1,β) 4 (λ 1 ...λ 4 ), then these formulae could be readily modified so as to yield the Hilbert–Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities ). Motivated by analogous earlier analyses of ' diagonal-entry -parameterized separability functions', we further explore the possibility that such three-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable—which we hypothesize to be the maximal concurrence (0 ⩽ C ⩽ 1) over spectral orbits. Extensive numerical results that we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at , as well as a number of additional matching discontinuities.