Relative volume of separable bipartite states

TL;DR

This paper investigates the proportion of separable states within bipartite quantum systems by analyzing the structure of quantum state spaces as SU(d) orbits, providing detailed results for two-qubit systems and Monte Carlo estimates for higher dimensions.

Contribution

It introduces a novel geometric approach to quantify the relative volume of separable states in bipartite systems, extending analysis from two-qubit to higher dimensions.

Findings

Two-qubit separable state volume analyzed in detail.

Monte Carlo methods used for higher-dimensional systems.

New insights into the structure of quantum state spaces.

Abstract

Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several new insights seem to emerge from our study.