  Spekkens' toy model is a non-contextual hidden variable model with an
epistemic restriction, a constraint on what an observer can know about reality.
The aim of the model, developed for continuous and discrete prime degrees of
freedom, is to advocate the epistemic view of quantum theory, where quantum
states are states of incomplete knowledge about a deeper underlying reality.
Many aspects of quantum mechanics and protocols from quantum information can be
reproduced in the model. In spite of its significance, a number of aspects of
Spekkens' model remained incomplete. Formal rules for the update of states
after measurement had not been written down, and the theory had only been
constructed for prime-dimensional, and infinite dimensional systems. In this
work, we remedy this, by deriving measurement update rules, and extending the
framework to derive models in all dimensions, both prime and non-prime.
Stabilizer quantum mechanics is a sub-theory of quantum mechanics with
restricted states, transformations and measurements. First derived for the
purpose of constructing error correcting codes, it now plays a role in many
areas of quantum information theory. Previously, it had been shown that
Spekkens' model was operationally equivalent in the case of infinite and odd
prime dimensions. Here, exploiting known results on Wigner functions, we extend
this to show that Spekkens' model is equivalent to stabilizer quantum mechanics
in all odd dimensions, prime and non-prime. This equivalence provides new
technical tools for the study of technically difficult compound-dimensional
stabilizer quantum mechanics.
