A Qualitative Approach to Quantum Probability

A likelihood order is defined over linear subspaces of a finite dimensional Hilbert space. The question arises as to when such an order can be represented by a quantum probability. We introduce a few behaviorally plausible axioms that provide the answer in two cases: pure state and uniform measure. The general problem is answered by using duality-like conditions. The general problem of characterizing the partial orders that admit a quantum representation by behaviorally justified axioms remains open.