Volumes of conditioned bipartite state spaces

Abstract

We analyze the metric properties of conditioned quantum state spaces These spaces are the convex sets of density matrices that, when partially traced over m degrees of freedom, respectively yield the given n × n density matrix η. For the case n = 2, the volume of equipped with the Hilbert–Schmidt measure can be conjectured to be a simple polynomial of the radius of η in the Bloch-ball. Remarkably, for we find numerically that the probability to find a separable state in is independent of η (except for η pure). For , the same holds for , the probability to find a state with a positive partial transpose in . These results are proven analytically for the case of the family of 4 × 4 X-states, and thoroughly numerically investigated for the general case. The important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed.