## 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 $$r>1$$.

### MSC:

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