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A geometric characterization of invertible quantum measurement maps. (English) Zbl 1283.47024
Each bijective map on the state space $S(H)$ (nuclear positive operators with unitary trace) of a Hilbert space $H$, preserving segments in the sense that $\phi[\rho_1,\rho_2]\subseteq [\phi(\rho_1),\phi(\rho_2)]$ for any $\rho_i\in S(H)$, is characterized as follows: $$ \rho\mapsto \frac{M\rho M^*}{\mathrm{tr}(M\rho M^*)}\qquad\text{or}\qquad \rho\mapsto \frac{M\rho^T M^*}{\mathrm{tr}(M\rho^T M^*)}, $$ where $\rho^T$ is the transpose of $\rho$ and $M=M(\phi)$ is an invertible operator in $B(H)$. The proof depends on a result by {\it Z. Páles} [“Characterization of segment and convexity preserving maps”, \url{arXiv:1212.1268}] on convexity and segment preserving maps.

47B10Operators belonging to operator ideals
47L20Operator ideals
81P15Quantum measurement theory
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