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 sufficient for linearly independent initial states. We also examine the issue of quantum coherence, that is, when such operations maintain the purity of superpositions. If, under any deterministic transformation from one linearly independent set to another, even a single complete superposition maintains its purity, then the initial and final states an related by a unitary transformation. (C) 2000 Published by Elsevier Science B.V. All rights reserved.