[Paper Review] Representing probabilistic data via ontological models

Nicholas Harrigan, Terry Rudolph+1 more|ArXiv.org|Sep 7, 2007

Quantum Mechanics and Applications16 references22 citations

TL;DR

This paper proposes a framework for representing probabilistic quantum data using ontological models based on positive probability distributions and indicator functions, enabling a realist interpretation of quantum mechanics. It demonstrates how any finite set of quantum preparation and measurement statistics can be modeled using three distinct factorization methods, shows how indeterministic models can be made deterministic, and explores minimal ontic state representations relevant to classical simulation of quantum systems.

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.

Motivation & Objective

• To develop a formalism for representing finite sets of quantum preparation and measurement statistics using ontological models grounded in positive probability distributions.
• To investigate how contextuality and deficiency—key features of ontological models—emerge in discrete, finite scenarios.
• To demonstrate that any indeterministic ontological model can be converted into a deterministic one, preserving empirical predictions.
• To explore the minimal number of ontic states required to represent quantum data, with implications for classical simulation of quantum systems.
• To connect the ontological model framework to matrix factorization techniques such as completely positive and non-negative matrix factorization (NMF), enabling compression and structural insight.

Proposed method

• Represent empirical data from finite preparation and measurement procedures as a matrix D, where D_ij = P(m_j | p_i) gives the probability of outcome j given preparation i.
• Introduce an ontological factorization (OF) of D as D = M P, where M is a matrix of preparation-to-ontic-state probabilities and P is a matrix of measurement-outcome indicator functions.
• Propose three distinct factorizations (including one based on singular value decomposition and another using non-negative matrix factorization) that can represent any empirical data matrix D.
• Apply a transformation technique inspired by Bell to convert an indeterministic ontological model into a deterministic one by introducing additional ontic states.
• Use the concept of deficiency to characterize contextuality, showing that the set of ontic states supporting a preparation is strictly smaller than the set that would guarantee a certain measurement outcome.
• Leverage connections to completely positive matrix factorization and NMF to analyze the minimal number of ontic states (Ω) required, linking this to the cp-rank and potential compression of quantum data representations.

Experimental results

Research questions

• RQ1 Can any finite set of quantum preparation and measurement statistics be represented by an ontological model using only positive probability distributions and indicator functions?
• RQ2 How can contextuality and deficiency be explicitly demonstrated in a finite, discrete ontological model framework?
• RQ3 Is it possible to transform an indeterministic ontological model into a deterministic one while preserving empirical predictions?
• RQ4 What is the minimal number of ontic states required to represent a given data table, and how does this relate to matrix factorization ranks?
• RQ5 Can techniques from non-negative matrix factorization (NMF) and completely positive factorization be used to compress ontological models and inform classical simulations of quantum systems?

Key findings

• Any finite data table of preparation and measurement probabilities can be represented by an ontological model using a matrix factorization D = M P with non-negative M and P.
• Three distinct factorization methods—based on SVD, CP decomposition, and NMF—can be used to construct such models, with the NMF approach preserving column stochasticity when required.
• An indeterministic ontological model can be converted into a deterministic one by introducing additional ontic states, following a technique analogous to Bell’s construction.
• Contextuality is manifested through the property of deficiency, where the set of ontic states supporting a preparation is strictly smaller than the set that would yield a certain measurement outcome.
• The number of ontic states Ω corresponds to the cp-rank of the data matrix D, and upper bounds on cp-rank suggest that some families of quantum data may admit polynomial-sized ontological models in the Hilbert space dimension.
• The connection to NMF suggests that ontological models can be interpreted as decomposing measurement data into fundamental, interpretable components, analogous to identifying features in image analysis.