A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality

TL;DR

The paper introduces the COPE matrix as a unifying, statistics-first representation of operational theories, showing that five model classes correspond to distinct factorizations of COPE. A central result is that noncontextual ontological models exist if and only if COPE admits an equirank nonnegative factorization (ENMF); failure of equirank (rank separation) implies contextuality. The authors demonstrate rank separation via two methods: a geometric, nested-polytopes approach with boxworld, and a dimensionality argument for discrete-qubit theory, linking contextuality to ontic excess baggage. They further show that convexity-preserving linear maps can relate GPTs and ontological models, and that at least one of the maps from GPTs to ontological models can be set-valued, ensuring universality of ontological representations for GPTs. Overall, the framework provides a GPT-agnostic, matrix-analytic route to quantify and study contextuality and nonclassical resources, with clear directions for extensions to continuous variables and quantitative measures of 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.

Figures (3)

• Figure 1: Illustration of a fragment COPE matrix $C_{\rm F}$. In considering fragments, the faithfulness assumption implies that the COPE matrix is a submatrix of the full operational theory's COPE matrix $D$ restricted to preparations ${\cal P}_{\rm F}\subset{\cal P}$ and measurements ${\cal M}_{\rm F}\subset{\cal M}$. Although the full COPE matrix $D$ satisfies Assumption \ref{['ass:COPE_comp']}, its fragment $C_{\rm F}$ fails to do so.
• Figure 2: Models of COPE matrices of operational theories. Given the probabilistic structure of an operational theory encoded in its COPE matrix, one can universally obtain four types of linear models. Each model arises from a particular factorization subject to specific constraints: PreGPTs (no constraint), GPTs (quotiented), ontological models (nonnegative with $\mathbf{1}^\mathsf{T}$ as unit effect), and quasiprobabilistic models (quotiented with $\mathbf{1}^\mathsf{T}$ as unit effect). However, a COPE matrix may fail to admit a noncontextual ontological model (quotiented and nonnegative with $\mathbf{1}^\mathsf{T}$ as unit effect). This contrasting class is the focus of our analysis in Secs. \ref{['sec:OM']} and \ref{['sec:Rank_sep']}.
• Figure 3: Illustration of the geometry of noncontextual ontological models. Factorizing the COPE matrix $C$ into nonnegative matrices of response functions, $R$, and epistemic states, $P$, gives an ontological model. Given the number of outcomes $Jn$, there is a simplex $\Delta\in{\mathbb{R}}^{Jn}$ which contains the polytope $\mathbf{R}$ constructed from the columns of $R$. The polytope $\mathbf{C}$ obtained from columns of the COPE matrix is obtained by convexly mixing the vertices of $\mathbf{R}$. The mixing weights are given by the epistemic states in $P$. The equirank condition of Eq. \ref{['eq:OM_equirank']} implies that in a noncontextual ontological model $\mathbf{C}$ and $\mathbf{R}$ are coplanar.

Theorems & Definitions (53)

• Example 1
• Corollary 1
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Corollary 2
• Definition 5
• Lemma 1
• proof