On the Existence of Physical Transformations Between Sets of Quantum States

Let A={ρ1,…,ρn} be a given set of quantum states. We consider the problem of finding necessary and sufficient conditions on another set B={σ1,…,σn} that guarantee the existence of a physical transformation taking ρi to σi for all i. Uhlmann has given an elegant such condition when both sets comprise pure states. We give a simple proof of this condition and develop some consequences. Then we consider multi-probabilistic transformations between sets of pure states which leads to conditions for the problem of transformability between A and B when one set is pure and the other is arbitrary.