Volumes of conditioned bipartite state spaces

TL;DR

This paper investigates the geometric properties of conditioned bipartite quantum state spaces, revealing that their volume and separability probability are largely independent of the conditioned state, with analytical proofs for X-states.

Contribution

It provides analytical and numerical analysis of the volume and separability probability of conditioned bipartite state spaces, highlighting their independence from the conditioned state for certain cases.

Findings

Volume of conditioned state spaces is a polynomial of the radius of the conditioned state.

Separability probability is independent of the conditioned state (except pure states).

Analytical proofs are provided for X-states, supported by numerical investigations.

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.