A map ψ 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 ψ(tx + (1 t)y) = sψ(x) + (1 s) ψ (y). Every quantum measurement described by a measurement operator M introduces a map (i.e.σ →M _σM*/Tr(M _σM*) if M _σM* ≠ 0) 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 H _1?H _2 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 σ → (S ? T)σ (S ? T)*/Tr(S ? T)σ (S ? T)* with S, T invertible) and some of the following maps: the transpose, the partial transpose and the swap.
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