A concise formula for generalized two-qubit Hilbert–Schmidt separability probabilities
Paul B. Slater · Journal of Physics A Mathematical and Theoretical · 2013
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, , (15-dimensional) two-qubit systems, and (27-dimensional) two-quater(nionic)bit systems, . 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 7 F 6 hypergeometric functions (all with argument ). 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 . From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) 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.