A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality

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.

💡 Research Summary

The paper introduces a “statistics‑first” framework for operational theories in which the complete probabilistic content is captured by a single object: the matrix of conditional outcome probabilities of events (COPE). By focusing on the simplest prepare‑measure causal structure, the authors treat preparations as columns and measurement outcomes as rows of the COPE matrix C, which may be infinite‑dimensional but is assumed to contain all observable statistics.

Within this setting five families of physical models are unified as constrained factorizations of C: (i) pre‑GPTs (unconstrained factorisations), (ii) GPTs (factorisations with equal rank to C, i.e., equirank), (iii) quasiprobabilistic models (allowing signed entries), (iv) ontological models (non‑negative factorisations), and (v) non‑contextual ontological models (non‑negative equirank factorisations, abbreviated ENMF). Each class corresponds to a familiar notion in the foundations literature, but the unified linear‑algebraic description places them on equal footing and makes the relationships between them explicit.

The central technical result is that the existence of a non‑contextual ontological model for an operational theory is equivalent to the existence of an ENMF of its COPE matrix. In other words, the “rank‑separation” condition—failure of the equirank property in any non‑negative factorisation—exactly characterises contextuality. The authors prove that the equirank condition is mathematically identical to tomographic completeness: a theory is tomographically complete precisely when its COPE matrix admits an equirank factorisation. Consequently, rank‑separation provides a model‑agnostic, operational criterion for contextuality that does not rely on pre‑specified operational equivalences or an underlying GPT.

Two complementary methods for demonstrating rank‑separation are developed. The first interprets an ENMF geometrically as a pair of nested polytopes (the state and effect polytopes) and shows that, for the “boxworld” theory, no such pair can be constructed; thus boxworld is ontologically contextual. The second method exploits sparsity patterns in COPE matrices to derive lower bounds on the ontological dimension (the number of ontic states required). By applying discrete‑mathematics techniques, the authors prove that a discrete version of qubit theory violates the equirank condition, yielding a new proof of its ontological contextuality.

Beyond these examples, the paper analyses the most general transformations between model classes, showing that linear set‑valued functions constitute the full class of convexity‑preserving maps. This insight allows the authors to construct explicit embeddings of GPTs into contextual ontological models even when a non‑contextual ontological model does not exist.

The computational complexity of finding ENMFs is discussed: the problem is NP‑hard in general, but the authors point out that existing results from graph theory and computational geometry provide efficient algorithms for special cases (e.g., highly sparse COPE matrices). They also outline a practical workflow for applying the framework to experimental data, enabling researchers to determine directly from observed statistics which model classes are admissible.

In summary, by recasting contextuality as a problem of matrix factorisation, the work offers a unified, mathematically precise, and operationally transparent toolkit for detecting non‑classicality. It bridges foundational questions with concrete linear‑algebraic methods, opening new avenues for the systematic study of contextuality and the exploitation of non‑classical resources in quantum information processing.