Quantum measurement and maps preserving convex combinations of separable states

A map ϕ between convex sets is said to be convex combination preserving if, for any elements x, y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ(tx + (1 − t)y) = sϕ(x) + (1 − s)ϕ(y). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H1⊗H2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S, T invertible) and some of the following maps: the transpose, the partial transpose and the swap.