Optimum unambiguous discrimination of two mixed states and application to geometrically uniform density operators

We study the measurement for the unambiguous discrimination of two mixed quantum states that are described by density operators of rank d, the supports of which jointly span a 2d-dimensional Hilbert space. Based on two conditions for the optimum measurement operators, minimizing the total probability of inconclusive results, and on a canonical representation for the density operators of the states, two equations are derived that allow the explicit construction of the optimum measurement, provided that the expression for the fidelity of the states has a specific simple form. The equations are applied to derive the complete solution for the optimum unambiguous discrimination of two geometrically uniform states. In particular, for the special case that these two states both occur with the same prior probability, we find that the optimum measurement always yields a probability of inconclusive results that is given by the fidelity