Suppose we have a quantum system prepared in a state chosen at random from a known set. In the task of state discrimination we would attempt to determine which preparation took place. This can be achieved with certainty if and only if the potential states form an orthonormal set. A complementary problem is conclusive state exclusion, where we instead attempt to perform a measurement that allows us to exclude with certainty a single preparation from the given possibilities.
This problem can be formulated in the framework of Semidefinite Programming (SDPs). This enables us to derive necessary and sufficient conditions for a measurement to be optimal, and a necessary condition on the set of states for conclusive exclusion to be possible. The exclusion problem finds an application in a recent paper by Pusey, Barrett and Rudolph on the reality of the quantum state and we can use our SDP to show the necessity of a bound given therein.