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While quantum theory cannot be described by a local hidden variable model, itis nevertheless possible to construct such models that exhibit featurescommonly associated with quantum mechanics. These models are also used toexplore the question of {\psi}-ontic versus {\psi}-epistemic theories forquantum mechanics. Spekkens' toy theory is one such model. It arises fromclassical probabilistic mechanics via a limit on the knowledge an observer mayhave about the state of a system. The toy theory for the simplest possibleunderlying system closely resembles stabilizer quantum mechanics, a fragment ofquantum theory which is efficiently classically simulable but also non-local.Further analysis of the similarities and differences between those two theoriescan thus yield new insights into what distinguishes quantum theory fromclassical theories, and {\psi}-ontic from {\psi}-epistemic theories.  In this paper, we develop a graphical language for Spekkens' toy theory.Graphical languages offer intuitive and rigorous formalisms for the analysis ofquantum mechanics and similar theories. To compare quantum mechanics and a toymodel, it is useful to have similar formalisms for both. We show that ourlanguage fully describes Spekkens' toy theory and in particular, that it iscomplete: meaning any equality that can be derived using other formalisms canalso be derived entirely graphically. Our language is inspired by a similargraphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'toy bit theory and stabilizer quantum mechanics can be analysed and comparedusing analogous graphical formalisms.

A complete graphical calculus for Spekkens' toy bit theory