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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 Z. Páles [“Characterization of segment and convexity preserving maps”, arXiv:1212.1268] on convexity and segment preserving maps.

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L20 Operator ideals
81P15 Quantum measurement theory, state operations, state preparations
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