  A novel no-go theorem is presented which sets a bound upon the extent to
which '\Psi-epistemic' interpretations of quantum theory are able to explain
the overlap between non-orthogonal quantum states in terms of an experimenter's
ignorance of an underlying state of reality. The theorem applies to any Hilbert
space of dimension greater than two. In the limit of large Hilbert spaces, no
more than half of the overlap between quantum states can be accounted for.
Unlike other recent no-go theorems no additional assumptions, such as forms of
locality, invasiveness, or non-contextuality, are required.
