Volumes of conditioned bipartite state spaces

Abstract

We analyse the metric properties of $\text{conditioned}$ quantum state spaces ${\mathcal{M}}_{\eta }^{(n\times m)}$. 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 ${\mathcal{M}}_{\eta }^{(2\times m)}$ 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 ${\mathcal{M}}_{\eta }^{(2\times m)}$ 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.

23 pages, 7 figures

References in corpus (2)

• Classical and quantum correlations under decoherence
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities

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