Relative volume of separable bipartite states

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 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 insights seem to emerge from our study.