Deterministic quantum state transformations

We derive necessary conditions for the existence of a completely positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another. These conditions are also sufficie nt for linearly independent initial states. We also examine the issue of qu antum coherence, that is, when such operations maintain the purity of super positions. If, under any deterministic transformation from one linearly ind ependent set to another, even a single complete superposition maintains its purity, then the initial and final states an related by a unitary transfor mation. (C) 2000 Published by Elsevier Science B.V. All rights reserved.