# Ontological models for quantum theory as functors

## TL;DR

This paper models ontological theories of quantum mechanics as functors between categories, unifying previous results and exploring the limitations and possibilities of such models.

## Contribution

It formalizes ontological models as functors and analyzes constraints from prior no-go theorems, also demonstrating the feasibility of epistemic functors with signed kernels.

## Key findings

- Monoidal functors are ruled out by Pusey, Barrett, and Rudolph.
- Duality-preserving functors are ruled out by Leifer and Maroney.
- Functors satisfying Schrödinger dynamics are ruled out by Aaronson et al.

## Abstract

We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to the category of measurable spaces and Markov kernels. This uniformises several earlier results, that we analyse more closely: Pusey, Barrett, and Rudolph's result rules out monoidal functors; Leifer and Maroney's result rules out functors that preserve a duality between states and measurement; Aaronson et al's result rules out functors that adhere to the Schr\"odinger equation. We also prove that it is possible to have epistemic functors that take values in signed Markov kernels.