A geometric characterization of invertible quantum measurement maps

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$H≤ ∞$, 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 $φ([ρ_1, ρ_2]) ⊆ [φ(ρ_1),φ(ρ_2)]$ for any $ρ_1, ρ_2 ∈ \mathcal S$ if and only if $φ$ has one of the following forms $$ρ \mapsto \fracMρ M^* tr(Mρ M^*)\quad \hboxor \quad ρ \mapsto \fracMρ^T M^* tr(Mρ^T M^*),$$ where $M$ is an invertible bounded linear operator and $ρ^T$ is the transpose of $ρ$ with respect to an arbitrarily fixed orthonormal basis.