On the Geometric Probability of Entangled Mixed States

Abstract

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 measuretheoretic 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, using the Peres–Horodecki positive partial transposition criterion, we discuss the measure-theoretic aspects of the separability problem for bipartite systems composed either of two qubits or of a qubit-qutrit pair. Necessary and sufficient conditions for the separability of a two-qubit state are formulated in terms of local SU(2) ⊗ SU(2) invariant polynomials, the determinant of the correlation matrix, and the determinant of the Schlienz–Mahler matrix. Using the projective method of generating random density matrices distributed according to the Hilbert–Schmidt or Bures measure, we calculate the probability of separability (including that of absolute separability) of a two-qubit and qubit-qutrit pair. Bibliograhpy: 47 titles.