A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality

TL;DR

This paper introduces a linear algebraic framework using matrix factorizations to unify and analyze various models of operational theories and contextuality, providing new criteria and methods for detecting nonclassicality.

Contribution

It develops a model-agnostic, matrix-based criterion for noncontextuality using equirank nonnegative matrix factorizations, unifies different models under a common framework, and operationalizes the analysis of contextuality.

Findings

Established that the boxworld theory is ontologically contextual.

Derived a lower bound on ontological dimensionality for sparse COPE matrices.

Provided a new proof that a discrete qubit theory exhibits contextuality.

Abstract

We develop a bottom-up, statistics-first framework in which the full probabilistic content of an operational theory is encoded in its matrix of conditional outcome probabilities of events (COPE). Within this setting, five model classes (preGPTs, GPTs, quasiprobabilistic, ontological, and noncontextual ontological) are unified as constrained factorizations of the COPE matrix. We identify equirank factorizations as the structural core of GPTs and noncontextual ontological models and establish their relation to tomographic completeness. This yields a simple, model-agnostic criterion for noncontextuality: an operational theory admits a noncontextual ontological model if and only if its COPE matrix admits an equirank nonnegative matrix factorization (ENMF). Failure of the equirank condition in all ontological models therefore establishes contextuality. We operationalize rank separation via two complementary methods provided by the linear-algebraic framework. First, we use ENMF to interpret noncontextual ontological models as nested polytopes. This allows us to establish that the boxworld operational theory is ontologically contextual. Second, we apply techniques from discrete mathematics to derive a lower bound on the ontological dimensionality of COPE matrices exhibiting sparsity patterns, and use this bound to establish a new proof that a discrete version of qubit theory exhibits ontological contextuality. By reframing contextuality as a problem in matrix analysis, our work provides a unified structure for its systematic study and opens new avenues for exploring nonclassical resources.