## 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
Full Text: