  In the task of quantum state exclusion we consider a quantum system, prepared
in a state chosen from a known set. The aim is to perform a measurement on the
system which can conclusively rule that a subset of the possible preparation
procedures can not have taken place. We ask what conditions the set of states
must obey in order for this to be possible and how well we can complete the
task when it is not. The task of quantum state discrimination forms a subclass
of this set of problems. Within this paper we formulate the general problem as
a Semidefinite Program (SDP), enabling us to derive sufficient and necessary
conditions for a measurement to be optimal. Furthermore, we obtain a necessary
condition on the set of states for exclusion to be achievable with certainty
and give a construction for a lower bound on the probability of error. This
task of conclusively excluding states has gained importance in the context of
the foundations of quantum mechanics due to a result of Pusey, Barrett and
Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well
a class of hidden variable models can perform at a particular task, proving an
analogue of Tsirelson's bound for the PBR experiment and the optimality of a
measurement given by PBR in the process. We also introduce variations of
conclusive exclusion, including unambiguous state exclusion, and state
exclusion with worst case error.
