Formulas for Generalized Two-Qubit Separability Probabilities. (24th May 2018) - Slater, Paul B.
Abstract : To begin, we find certain formulasQ ( k, α ) = G 1 k ( α ) G 2 k ( α ), fork = - 1, 0, 1, …, 9 . These yield that part of the total separability probability, P ( k, α ), for generalized (real, complex, q u a t e r n i o n i c, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequalityρ P T > ρ holds. Hereρ denotes a4 × 4 density matrix, obtained by tracing over the pure states in4 × ( 4 + k ) -dimensions, andρ P T denotes its partial transpose. Further, α is a Dyson-index-like parameter withα = 1 for the standard (15-dimensional) convex set of (complex) two-qubit states. Fork = 0, we obtain the previously reported Hilbert-Schmidt formulas, withQ ( 0, 1 / 2 ) = 29 / 128 (the real case), Q ( 0, 1 ) = 4 / 33 (the standard complex case), andQ ( 0, 2 ) = 13 / 323 (the quaternionic case), the three simply equallingP ( 0, α ) / 2 . The factorsG 2 k ( α ) are sums of polynomial-weighted generalized hypergeometric functions p F p - 1, p ≥ 7, all with argumentz = 27 / 64 = ( 3 / 4 ) 3 . We find number-theoretic-based formulas for the upper (u i k ) and lower (b i k ) parameter sets of these functions and, then, equivalently expressG 2 k ( α ) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms ofQ ( - 1, α ), Q ( 1, α ), andQ ( 3, α ), parallel to the one obtained previously (Slater 2013) forP ( 0, α ) = 2 Q ( 0, α ) . For nonnegative half-integer and integer values ofα, Q ( k, α ) (as well asP ( k, α ) ) has descending roots starting atk = - α - 1 . Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form forQ ( k, α ) itself. The possibility of an analogous "master" formula forP ( k, α ) is, then, investigated, and a number of interesting results are found.