A geometric characterization of invertible quantum measurement maps

Abstract

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 $[ρ_1, ρ_2]$ the line segment joining two elements $ρ_1, ρ_2$ in ${\mathcal S}(H)$. It is shown that a bijective map $ϕ:{\mathcal S}(H) \rightarrow {\mathcal S}(H)$ satisfies $ϕ([ρ_1, ρ_2]) \subseteq [ϕ(ρ_1),ϕ(ρ_2)]$ for any $ρ_1, ρ_2 \in {\mathcal S}$ if and only if $ϕ$ has one of the following forms $$ρ\mapsto \frac{MρM^*}{{\rm tr}(MρM^*)}\quad \hbox{or} \quad ρ\mapsto \frac{Mρ^T M^*}{{\rm tr}(Mρ^T M^*)},$$ where $M$ is an invertible bounded linear operator and $ρ^T$ is the transpose of $ρ$ with respect to an arbitrarily fixed orthonormal basis.