We consider the problem of designing an optimal quantum detector that
distinguishes unambiguously between a collection of mixed quantum states. Using
arguments of duality in vector space optimization, we derive necessary and
sufficient conditions for an optimal measurement that maximizes the probability
of correct detection. We show that the previous optimal measurements that were
derived for certain special cases satisfy these optimality conditions. We then
consider state sets with strong symmetry properties, and show that the optimal
measurement operators for distinguishing between these states share the same
symmetries, and can be computed very efficiently by solving a reduced size
semidefinite program.Comment: Submitted to Phys. Rev.
