# Techniques: Ontological models for quantum theory as functors

## What the paper contributes (one paragraph)
This is a quantum foundations paper that recasts ontological (hidden variable) models of finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to a category of measurable spaces and Markov kernels. The categorical language uniformises several existing no-go results: the Pusey-Barrett-Rudolph theorem becomes a statement that no epistemic functor preserves tensor products (monoidality), Leifer and Maroney's result becomes the non-existence of a functor preserving state-measurement duality, and Aaronson et al's result rules out a maximally nontrivial equivariant functor. The one constructive result shows that if the target category is enlarged from Markov kernels to signed Markov kernels (quasiprobabilities), an epistemic functor does exist, built explicitly from discrete Wigner functions and phase-space point operators. The work is conceptual and foundational rather than a protocol or algorithm aimed at hardware or applications.

## Techniques offered
- **Categorical (functorial) reformulation of ontological models** - reframes the question of ontic versus epistemic interpretations as the existence of structure-preserving functors between an operational source category and a probability-theoretic target category, into which existing no-go theorems are translated as constraints on the functor.
  - guarantee: structural / no-go results (PBR rules out monoidal functors, Leifer-Maroney rules out duality-preserving functors, Aaronson et al rules out maximally nontrivial equivariant functors); these are negative theorems, not a positive capability supplied to an external result.
  - quote: "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."
- **Discrete Wigner-function (quasiprobability) representation** - represents mixed states and completely positive maps by signed quasiprobability vectors and transfer matrices over phase-space point operators, giving a composition-preserving classical stochastic image of finite-dimensional quantum channels.
  - guarantee: the construction is a well-defined functor that is maximally epistemic up to a multiplicative constant, so quasiprobability variation distance matches quantum trace distance up to a factor.
  - quote: "We interpret this functor as assigning Wigner functions to quantum states."

## Where it could apply
- Target primitive(s): none (no VQE / QAOA / QPE / sampling / Trotterised target); the discrete Wigner construction is adjacent to classical simulation of finite-dimensional channels.
- Target application group(s): platforms_tooling (foundations and representation theory), at most adjacent to classical-simulation tooling.

## Caveats
This is a quantum foundations and category theory paper. Its principal results are no-go theorems (PBR, Leifer-Maroney, Aaronson et al) recast functorially, which are guarantees and non-supplies rather than techniques usable on an external vendor or application result. The only constructive content is a discrete Wigner / signed-Markov-kernel representation; it is a known quasiprobability tool restated categorically and is not packaged as a runnable simulation, mitigation, or verification method. There is no decoder, error mitigation, accreditation, compilation, or benchmarking capability here, and nothing is supplied directly to a quantum computing or quantum machine learning result.