Dyson-Index-Like Behavior of Bures Separability Functions

TL;DR

This paper investigates the behavior of Bures separability functions in quantum states, revealing near-proportionalities related to Dyson indices, which are exact in the Hilbert-Schmidt case but approximate in the Bures case.

Contribution

It extends the analysis of separability functions from the Hilbert-Schmidt to the Bures metric, showing approximate Dyson index proportionalities in simple quantum scenarios.

Findings

Dyson index proportionalities are nearly preserved in Bures separability functions.

Exact proportionalities hold in Hilbert-Schmidt case, approximate in Bures case.

Results aid in calculating quantum state separability probabilities.

Abstract

We conduct a study based on the Bures (minimal monotone) metric, analogous to that recently reported for the Hilbert-Schmidt (flat or Euclidean) metric (arXiv:0704.3723v2). Among the interesting results obtained there had been proportionalities--in exact correspondence to the Dyson indices \beta = 1, 2, 4 of random matrix theory--between the fourth, second and first powers of the separability functions S_{type}(\mu) for real, complex and quaternionic qubit-qubit scenarios, Here \mu=\sqrt{\frac{\rho_{11} \rho_{44}}{\rho_{22} \rho_{33}}}, with \rho being a 4 x 4 density matrix. Separability functions have proved useful--in the framework of the Bloore (correlation coefficient/off-diagonal scaling) parameterization of density matrices--for the calculation of separability probabilities. We find--for certain, basic simple scenarios (in which the diagonal entries of \rho are unrestricted, and one or two off-diagonal [real, complex or quaternionic] pairs of entries are nonzero) --that these proportionalities no longer strictly hold in the Bures case, but do come remarkably close to holding.