Optimal Unambiguous Discrimination of Pure Quantum States

A complete geometric view is presented for the optimal unambiguous discrimination among N > 2 pure states. A single intuitive picture contains all aspects of the problem: linear independence of the states, positivity of the detection operators, and a graphic method for finding and classifying the optimal solutions. The method is illustrated on the example of three states. We show that the problem depends on the phases of the complex inner products only through an invariant combination, the Berry phase phi, and present complete analytical results for phi = 0 and phi = pi. The optimal solution exhibits full permutational symmetry and is single valued for a large range of parameters. However, for phi = 0 it can be bivalued: beyond a critical value of the parameters a second, less symmetric solution becomes optimal. The bifurcation is analogous to a second-order symmetry-breaking phase transition. We conclude with a discussion of the unambiguous discrimination of N > 3 states.