A geometric characterization is given for invertible quantum measurement maps. Denote by ${ \mathcal S}(H)$ the convex set of all states (i.e., trace-1 positive operators) on Hilbert space $H$ with dim$H\leq \infty$, and $[\rho_1, \rho_2]$ the line segment joining two elements $\rho_1, \rho_2$ in ${ \mathcal S}(H)$. It is shown that a bijective map $\phi:{ \mathcal S}(H) \rightarrow { \mathcal S}(H)$ satisfies $\phi([\rho_1, \rho_2]) \subseteq [\phi(\rho_1),\phi(\rho_2)]$ for any $\rho_1, \rho_2 \in { \mathcal S}$ if and only if $\phi$ has one of the following forms $$\rho \mapsto \frac{ M\rho M^*}{ { \rm tr}(M\rho M^*)}\quad \hbox{ or} \quad \rho \mapsto \frac{ M\rho^T M^*}{ { \rm tr}(M\rho^T M^*)},$$ where $M$ is an invertible bounded linear operator and $\rho^T$ is the transpose of $\rho$ with respect to an arbitrarily fixed orthonormal basis.
Categories: quant-ph math.OA