Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments

TL;DR

This paper investigates the determinantal moments of two-rebit and two-qubit quantum systems under Bures and Hilbert-Schmidt measures, aiming to improve understanding and calculation of separability probabilities using numerical and symbolic methods.

Contribution

It provides new insights and conjectures on the exact values of Bures determinantal moments, revealing underlying polynomial ratio patterns and enhancing probabilistic analysis of quantum states.

Findings

Exact first moment of |rho^{PT}| for two-qubit systems is -1/256.

Approximate first moment for two-rebit systems is -0.00309594.

In Bures case, moments are ratios of 5th-degree polynomials, unlike the Hilbert-Schmidt case.

Abstract

We seek to gain insight into the nature of the determinantal moments of generic (9-dimensional) two-rebit and (15-dimensional) two-qubit systems (rho). Such information-as it has proved to be in the Hilbert-Schmidt counterpart--should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining improved, or possibly even exact Bures 2 x 2 separability probabilities for such systems. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert-Schmidt moments (J. Phys. A, 45, 095305 [2012]). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures moments. In particular, the first moment (average) of |rho^{PT}| (where |rho^{PT}| in [-1/16, 1/256]), PT denoting partial transpose, for the two-qubit systems is, remarkably, -1 / 256. The analogous value for the two-rebit systems is -2663/(2^(13) x 3 x 5 x 7) approx -0.00309594. While, an important function in the Hilbert-Schmidt case is the ratio of 3n-degree polynomials, it appears to be the ratio of 5n-degree polynomials in the Bures case.