  Formalizing an old desire of Einstein, "psi-epistemic theories" try to
reproduce the predictions of quantum mechanics, while viewing quantum states as
ordinary probability distributions over underlying objects called "ontic
states." Regardless of one's philosophical views about such theories, the
question arises of whether one can cleanly rule them out, by proving no-go
theorems analogous to the Bell Inequality. In the 1960s, Kochen and Specker
(who first studied these theories) constructed an elegant psi-epistemic theory
for Hilbert space dimension d=2, but also showed that any deterministic
psi-epistemic theory must be "measurement contextual" in dimensions 3 and
higher. Last year, the topic attracted renewed attention, when Pusey, Barrett,
and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under
tensor product." In this paper, we prove that even without the Kochen-Specker
or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that
satisfy two reasonable conditions: (1) symmetry under unitary transformations,
and (2) "maximum nontriviality" (meaning that the probability distributions
corresponding to any two non-orthogonal states overlap). This no-go theorem
holds if the ontic space is either the set of quantum states or the set of
unitaries. The proof of this result, in the general case, uses some measure
theory and differential geometry. On the other hand, we also show the
surprising result that without the symmetry restriction, one can construct
maximally-nontrivial psi-epistemic theories in every finite dimension d.
