# Spekkens' Toy Model, Finite Field Quantum Mechanics, and the Role of Linearity

## TL;DR

This paper maps Spekkens' toy model to finite field quantum mechanics, enabling linear combinations of states and revealing both agreements and differences in entanglement properties.

## Contribution

It introduces a finite field quantum framework for Spekkens' model, highlighting new linear structure and entanglement distinctions.

## Key findings

- Exact mapping for single elementary systems
- Linear combinations of states are well-defined in the mapped model
- Differences in entangled states between the models for two systems

## Abstract

We map Spekkens' toy model to a quantum mechanics defined over the finite field $\mathbb{F}_5$. This allows us to define arbitrary linear combinations of the epistemic states in the model. For Spekkens' elementary system with only $2^2=4$ ontic states, the mapping is exact and the two models agree completely. However, for a pair of elementary systems there exist interesting differences between the entangled states of the two models.