Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices

TL;DR

This paper advances the understanding of quantum state separability by analyzing separability functions and probabilities across different metrics and parameterizations, highlighting Dyson index patterns and providing refined conjectures for two-qubit systems.

Contribution

It introduces new parameterizations and metrics for studying separability, confirms Dyson index patterns in separability probabilities, and refines conjectures for these probabilities in complex quantum systems.

Findings

Euler-angle separability function fits participation ratio to the sixth power

Dyson-index behavior observed in real and complex two-qubit scenarios

Proposed precise conjectures for HS and Bures separability probabilities

Abstract

We report substantial progress in the study of separability functions and their application to the computation of separability probabilities for the real, complex and quaternionic qubit-qubit and qubit-qutrit systems. We expand our recent work (arXiv:0704.3723), in which the Dyson indices of random matrix theory played an essential role, to include the use of not only the volume element of the Hilbert-Schmidt (HS) metric, but also that of the Bures (minimal monotone) metric as measures over these finite-dimensional quantum systems. Further, we now employ the Euler-angle parameterization of density matrices (rho), in addition to the Bloore parameterization. The Euler-angle separability function for the minimally degenerate complex two-qubit states is well-fitted by the sixth-power of the participation ratio, R(rho)=1/Tr(rho)^2. Additionally, replacing R(rho) by a simple linear transformation of the Verstraete-Audenaert-De Moor function (arXiv:quant-oh/0011111), we find close adherence to Dyson-index behavior for the real and complex (nondegenerate) two-qubit scenarios. Several of the analyses reported help to fortify our conjectures that the HS and Bures separability probabilities of the complex two-qubit states are 8/33 = 0.242424 and 1680 (sqrt{2}-1)/pi^8 = 0.733389, respectively. Employing certain regularized beta functions in the role of Euler-angle separability functions, we closely reproduce--consistently with the Dyson-index ansatz--several HS two-qubit separability probability conjectures.