# Spekkens' toy model in all dimensions and its relationship with stabilizer quantum mechanics

## TL;DR

This paper extends Spekkens' toy model to all dimensions, derives measurement update rules, and demonstrates its operational equivalence to stabilizer quantum mechanics across all odd dimensions, enhancing understanding of quantum foundations.

## Contribution

It develops measurement update rules for Spekkens' model and proves its equivalence to stabilizer quantum mechanics in all odd dimensions, including non-prime and composite dimensions.

## Key findings

- Spekkens' model extended to all dimensions, prime and non-prime.
- Measurement update rules derived for the model.
- Operational equivalence with stabilizer quantum mechanics established in all odd dimensions.

## Abstract

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.