Volumes of conditioned bipartite state spaces. (English) Zbl 1315.81018

Let \(\mathcal{M}^{(n,m)}\) be the space of states (density matrices) of a bipartite quantum system whose parts \(S\) and \(R\) have finite dimensions \(n\) and \(m\) respectively. Given a (reduced) state \(\eta\) of \(S\), let \(\mathcal{M}^{(n,m)}_{\eta}\) be the subspace conditioned to consist of all \(\rho \in \mathcal{M}^{(n,m)}\) with partial trace \(\mathrm{Tr}_R\rho = \eta\). The authors study the volumes of these subspaces w.r.t. the Hilbert-Schmidt measure for \(n=2\). They test numerically the conjectures that such a volume equals a polynomial of the radius \(r\) of the Bloch sphere of \(\eta\) and that the relative volume of separable states within \(\mathcal{M}^{(n,m)}_{\eta}\) is independent of \(r\) for \(r>1\).


81P16 Quantum state spaces, operational and probabilistic concepts
81P40 Quantum coherence, entanglement, quantum correlations
65C05 Monte Carlo methods
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