  We investigate the strengths and limitations of the Spekkens toy model, which
is a local hidden variable model that replicates many important properties of
quantum dynamics. First, we present a set of five axioms that fully encapsulate
Spekkens' toy model. We then test whether these axioms can be extended to
capture more quantum phenomena, by allowing operations on epistemic as well as
ontic states. We discover that the resulting group of operations is isomorphic
to the projective extended Clifford Group for two qubits. This larger group of
operations results in a physically unreasonable model; consequently, we claim
that a relaxed definition of valid operations in Spekkens' toy model cannot
produce an equivalence with the Clifford Group for two qubits. However, the new
operations do serve as tests for correlation in a two toy bit model, analogous
to the well known Horodecki criterion for the separability of quantum states.
