  In this article, by treating minimum error state discrimination as a
complementarity problem, we obtain the geometric optimality conditions. These
can be used as the necessary and sufficient conditions to determine whether
every optimal measurement operator can be nonzero. Using these conditions and
an inductive approach, we demonstrate a geometric method and the intrinsic
polytope for N-qubit mixed-state discrimination. When the intrinsic polytope
becomes a point, a line segment, or a triangle, the guessing probability, the
necessary and sufficient condition for the exact solution, and the optimal
measurement are analytically obtained. We apply this result to the problem of
discrimination to arbitrary three-qubit mixed states with given a priori
probabilities and obtain the complete analytic solution to the guessing
probability and optimal measurement.
