A Geometric Characterization of Invertible Quantum Measurement Maps

A geometric characterization is given for invertible quantum measurement maps. Denote by S(H) the convex set of all states (i.e., trace-1 positive operators) on Hilbert space H with dimH ≤ ∞, and [ρ1, ρ2] the line segment joining two elements ρ1, ρ2 in S(H). It is shown that a bijective map φ : S(H) → S(H) satisfies φ([ρ1, ρ2]) ⊆ [φ(ρ1), φ(ρ2)] for any ρ1, ρ2 ∈ S if and only if φ has one of the following forms ρ 7→ MρM ∗ tr(MρM∗) or ρ 7→ Mρ TM∗ tr(MρTM∗) , where M is an invertible bounded linear operator and ρ is the transpose of ρ with respect to an arbitrarily fixed orthonormal basis.