  The problem addressed is to design a detector which is maximally sensitive to
specific quantum states. Here we concentrate on quantum state detection using
the worst-case a posteriori probability of detection as the design criterion.
This objective is equivalent to asking the question: if the detector declares
that a specific state is present, what is the probability of that state
actually being present? We show that maximizing this worst-case probability
(maximizing the smallest possible value of this probability) is a quasiconvex
optimization over the matrices of the POVM (positive operator valued measure)
which characterize the measurement apparatus. We also show that with a given
POVM, the optimization is quasiconvex in the matrix which characterizes the
Kraus operator sum representation (OSR) in a fixed basis. We use Lagrange
Duality Theory to establish the optimality conditions for both deterministic
and randomized detection. We also examine the special case of detecting a
single pure state. Numerical aspects of using convex optimization for quantum
state detection are also discussed.
