A complete graphical calculus for Spekkens' toy bit theory

TL;DR

This paper introduces a complete graphical calculus for Spekkens' toy bit theory, enabling intuitive and rigorous analysis and comparison with quantum mechanics, particularly stabilizer quantum mechanics.

Contribution

It develops a novel, complete graphical language for Spekkens' toy theory, facilitating analysis and comparison with quantum theories using similar formalism.

Findings

The graphical language fully describes Spekkens' toy theory.

The language is complete: all derivable equalities can be shown graphically.

Enables direct comparison between Spekkens' toy theory and stabilizer quantum mechanics.

Abstract

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 {\psi}-ontic versus {\psi}-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 {\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 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.