On the geometric probability of entangled mixed states

The state space of a composite quantum system, the set of density matrices $\mathfrak P_+$, is decomposable into the space of separable states $\mathfrak S_+$ and its complement, the space of entangled states. An explicit construction of such a decomposition constitutes the so-called separability problem. If the space $\mathfrak P_+$ is endowed with a certain Riemannian metric, then the separability problem admits a measurement-theoretical formulation. In particular, one can define the “geometric probability of separability” as the relative volume of the space of separable states $\mathfrak S_+$ with respect to the volume of all states. In the present note, based on the Peres–Horodecki positive partial transposition criterion, the measurement theoretical aspects of the separability problem are discussed for bipartite systems composed either of two qubits or of qubit-qutrit pairs. The necessary and sufficient conditions for the $2$-qubit state separability are formulated in terms of local $\mathrm{SU(2)\otimes SU(2)}$ invariant polynomials, the determinant of the correlation matrix, and the determinant of the Schlienz–Mahler matrix. Using the projective method of generation of random density matrices distributed according to the Hilbert–Schmidt or Bures measure, the separability (including the absolute separability) probabilities of $2$-qubit and qubit-qutrit pairs have been calculated.