2 00 4 A three state invariant

For triples of probability measures, pure quantum states and mixed quantum states we obtain the exact constraints on the fidelities of pairs in the sequence. We show that it is impossible to decide between a quantum model, either pure or mixed, and a classical model on the basis of the fidelities alone. Next, we introduce a quantum three state invariant called phase and show that any sequence of pure quantum states is uniquely reconstructible given the fidelities and phases. The transition probability or overlap between states is one of the fundamental ingredients of quantum mechanics. It lies at the basis of its probabilistic interpretation. Two states with an overlap close to one are almost equal. This motivates the term fidelity which will be used as a synonym of transition probability.