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.