Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States. (27th May 2015) - Slater, Paul B.; Dunkl, Charles F.
Abstract : Previously, a formula, incorporating a5 F 4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal momentsρ P T n ρ k / ρ k of4 × 4 density-matrices (ρ ) and their partial transposes (| ρ P T | ), was applied withk = 0 to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over "induced measures in the space of mixed quantum states." The associated induced-measure separability probabilities (k = 1, 2, … ) are found— via a high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (α = 1 / 2 ), (standard) two-qubit (α = 1 ), and two-quater[nionic]bit (α = 2 ) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed for general k . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.