Quantum Steering Ellipsoid Volume as a Measure on the Two-Qubit States and Associated Separability and Absolute Separability Ratios

TL;DR

This paper uses quantum steering ellipsoid volume to measure two-qubit states, estimating the ratio of separable states and exploring related separability properties and measures.

Contribution

It introduces the quantum steering ellipsoid volume as a new measure for two-qubit states and estimates separability ratios under this measure.

Findings

Separable states ratio under QSE measure is approximately 0.0288.

Compared to Hilbert-Schmidt and Bures measures, QSE measure yields a smaller separability ratio.

Separable probability increases near the pure state boundary in the QSE setting.

Abstract

Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.