Volumes of conditioned bipartite state spaces

Abstract: We analyse the metric properties of $\textit{conditioned}$ quantum state spaces $\mathcal{M}^{(n\times m)}{\eta}$. These spaces are the convex sets of $nm \times nm$ density matrices that, when partially traced over $m$ degrees of freedom, respectively yield the given $n\times n$ density matrix $\eta$. For the case $n=2$, the volume of $\mathcal{M}^{(2\times m)}{\eta}$ equipped with the Hilbert-Schmidt measure is a simple polynomial of the radius of $\eta$ in the Bloch-Ball. Remarkably, the probability $p_{\mathrm{sep}}^{(2\times m)}(\eta)$ to find a separable state in $\mathcal{M}^{(2\times m)}_{\eta}$ is independent of $\eta$ (except for $\eta$ pure). Both these results are proven analytically for the case of the family of $4\times 4$ $X$-states, and thoroughly numerically investigated for the general case. The important implications of these results for the clarification of open problems in quantum theory are pointed out and discussed.