  The Pusey-Barrett-Rudolph theorem has recently provoked a lot of discussion
regarding the reality of the quantum state. In this article we focus on a
property called antidistinguishability, which is a main component in
constructing the proof for the PBR theorem. In particular we study algebraic
conditions for a set of pure quantum states to be antidistinguishable, and a
novel sufficient condition is presented. We also discuss a more general
criterion which can be used to show that the sufficient condition is not
necessary. Lastly, we consider how many quantum states needs to be added into a
set of pure quantum states in order to make the set antidistinguishable. It is
shown that in the case of qubit states the answer is one, while in the general
but finite dimensional case the answer is at most $n$, where $n$ is the size of
the original set.
