A geometric characterization of invertible quantum measurement maps

A geometric characterization is given for invertible quantum measurement maps. Denote by S(H)S(H) the convex set of all states (i.e., trace 1 positive operators) on Hilbert space H   with dimH⩽∞, and [ρ1,ρ2][ρ1,ρ2] the line segment joining two elements ρ1,ρ2ρ1,ρ2 in S(H)S(H). It is shown that a bijective map ϕ:S(H)→S(H)ϕ:S(H)→S(H) satisfies ϕ([ρ1,ρ2])⊆[ϕ(ρ1),ϕ(ρ2)]ϕ([ρ1,ρ2])⊆[ϕ(ρ1),ϕ(ρ2)] for any ρ1,ρ2∈Sρ1,ρ2∈S if and only if ϕ has one of the following formsρ↦MρM⁎tr(MρM⁎)orρ↦MρTM⁎tr(MρTM⁎), where M   is an invertible bounded linear operator and ρTρT is the transpose of ρ with respect to an arbitrarily fixed orthonormal basis.