# A Functorial Construction of Quantum Subtheories

## TL;DR

This paper develops a functorial approach to constructing quantum subtheories using geometric quantization, connecting toy models with actual quantum mechanics through algebraic structures.

## Contribution

It introduces a novel functorial framework linking epistemically restricted toy theories to quantum subtheories via symplectic groupoids and Frobenius algebras.

## Key findings

- Constructs algebraic structures of quadrature subtheories in continuous systems.
- Establishes a functor from toy theories to stabilizer quantum mechanics in finite systems.
- Bridges toy models and quantum theory through geometric and algebraic methods.

## Abstract

We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra in \textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.