A complete graphical calculus for Spekkens’ Toy Bit theory

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  • University of Oxford
  • King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia


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


Original languageEnglish
Pages (from-to)70-103
JournalFoundations of Physics
Issue number1
Early online date7 Oct 2015
Publication statusPublished - 1 Jan 2016


  • Quantum foundations, Spekkens’ toy theory, Stabilizer quantum mechanics, Graphical calculus, Categorical quantum mechanics, Graph states, ψ-epistemic theory