  In this paper, a characterization of maps between quantum states that
preserve pure states and strict convex combinations is obtained. Based on this
characterization, a structural theorem for maps between multipartite quantum
states that preserve separable pure states and strict convex combinations is
established. Then these results are applied to characterize injective (local)
quantum measurements and answer some conjectures proposed in
[J.Phys.A:Math.Theor. 45 (2012) 205305].
