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, to the classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an $\text{SU}\left(d\right)$ 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 insights seem to emerge from our study.

• Received 7 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.022308