Representing probabilistic data via ontological models

TL;DR

This paper explores ontological models for probabilistic data, presenting three universal models, discussing their determinism and contextuality, and investigating minimal ontic state representations.

Contribution

It introduces three models capable of describing any empirical data from finite preparations and measurements, and discusses how to minimize ontic states and address contextuality.

Findings

Three models can describe any empirical data set.

Deterministic versions of models can be constructed.

Explicit example demonstrates contextuality in ontological models.

Abstract

Ontological models are attempts to quantitatively describe the results of a probabilistic theory, such as Quantum Mechanics, in a framework exhibiting an explicit realism-based underpinning. Unlike either the well known quasi-probability representations, or the "r-p" vector formalism, these models are contextual and by definition only involve positive probability distributions (and indicator functions). In this article we study how the ontological model formalism can be used to describe arbitrary statistics of a system subjected to a finite set of preparations and measurements. We present three models which can describe any such empirical data and then discuss how to turn an indeterministic model into a deterministic one. This raises the issue of how such models manifest contextuality, and we provide an explicit example to demonstrate this. In the second half of the paper we consider the issue of finding ontological models with as few ontic states as possible.