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A concise formula for generalized two-qubit Hilbert–Schmidt separability probabilities

Paul B. Slater
Organization: Universty of California, Santa Barbara, California
Journal / Anthology

Journal of Physics A:Mathematical and Theoretical
Year: 2013
Volume: 46
Issue: 44

We report major advances in the research program initiated in ‘Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2×2 separability probabilities’ (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). A highly succinct separability probability function P(α) is put forth, yielding for generic (nine-dimensional) two-rebit systems, P(12 ) = 29 64 , (15-dimensional) two-qubit systems, P(1) = 8 33 and (27-dimensional) two-quater(nionic)bit systems, P(2) = 26 323 . This particular form of P(α) was obtained by Qing- Hu Hou by applying Zeilberger’s algorithm (‘creative telescoping’) to a fully equivalent—but considerably more complicated—expression containing six 7F6 hypergeometric functions (all with argument 27 64 = ( 3 4 )3). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7501 determinantal moments of partially transposed 4 × 4 density matrices, parameterized by α = 12 , 1, 32 , 2, . . . , 32. From these computations, exact rational-valued separability probabilities were discernible. The (integral/halfintegral) sequences of 32 rational values then served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P(α) emerged.

*Science > Physics

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