Quasirandom estimations of two-qubit operator-monotone-based separability probabilities

TL;DR

This paper estimates two-qubit state separability probabilities using quasirandom methods across various measures, supporting some previous conjectures and highlighting discrepancies possibly due to infinite volume issues.

Contribution

It provides new quasirandom estimations of two-qubit separability probabilities across multiple operator monotone measures, extending prior results and addressing inconsistencies.

Findings

Supports known Hilbert-Schmidt and Bures separability probabilities

Suggests Wigner-Yanase probability is approximately 1/20

Identifies potential issues with infinite volume for certain measures

Abstract

We conduct a pair of quasirandom estimations of the separability probabilities with respect to ten measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parameterization. The measures include the (non-monotone) Hilbert-Schmidt one, plus nine others based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert-Schmidt and Bures (minimal monotone) separability probabilities are $\frac{8}{33} \approx 0.242424$ and $\frac{25}{341} \approx 0.0733138$, respectively, as well as suggestive of the Wigner-Yanase counterpart being $\frac{1}{20}$. However, one result appears inconsistent (much too small) with an earlier claim of ours that the separability probability associated with the operator monotone (geometric-mean) function $\sqrt{x}$ is $1-\frac{256}{27 \pi ^2} \approx 0.0393251$. But a seeming explanation for this disparity is that the volume of states for the $\sqrt{x}$-based measure is infinite. So, the validity of the earlier conjecture--as well as an alternative one, $\frac{1}{9} \left(593-60 \pi ^2\right) \approx 0.0915262$, we now introduce--can not be examined through the numerical approach adopted, at least perhaps not without some truncation procedure for extreme values.