Beyond Quantum: A New Lens for Contextuality

Author: Denis Avetisyan

Researchers have developed a unified mathematical framework to analyze the fundamental limits of predictability in any physical theory, revealing deep connections between probabilistic structure and contextuality.

Operational theories, when encoded as COPE matrices, universally yield four distinct linear models-PreGPTs, GPTs, ontological models, and quasiprobabilistic models-each arising from specific factorization constraints; however, the analysis demonstrates that certain COPE matrices fail to admit a noncontextual ontological model, a class demanding particular attention to fully characterize the probabilistic structure of these theories.

This work demonstrates that noncontextuality is mathematically linked to the equirank condition in matrix factorizations of operational theories, providing a powerful tool for assessing contextuality across diverse physical models.

The persistent challenge of characterizing nonclassicality in physical theories stems from a lack of unified analytical tools. This is addressed in ‘A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality’, which recasts operational theories through the lens of their conditional outcome probabilities, represented by a ‘COPE’ matrix. We demonstrate that noncontextuality is fundamentally linked to the equirank factorizability of this matrix, providing a model-agnostic criterion for assessing ontological models and, consequently, contextuality. Does this framework offer a pathway to systematically identify and quantify the resources enabling genuinely nonclassical behavior in diverse physical systems?

The Bedrock of Operational Theories: Defining Preparation and Measurement

At the heart of any probabilistic framework lies a rigorous definition of how a system is initially prepared and what outcomes are possible when it is measured. This isn’t merely a technical detail; it’s the fundamental bedrock upon which all subsequent predictions and interpretations rest. A theory must explicitly delineate the permissible preparation procedures – the specific actions taken to set the system into a known state – and then, crucially, define the set of measurable outcomes. These outcomes aren’t necessarily pre-determined values, but rather the distinct results that can be registered by a measurement process. Without clear definitions of both preparation and measurement, assigning probabilities becomes meaningless, as there’s no established basis for quantifying the likelihood of any particular result. The very structure of a probabilistic theory, therefore, depends on this initial specification of how a system is brought into being and how its properties are revealed through observation, establishing a direct link between theoretical constructs and experimental reality.

The COPE matrix, a cornerstone of probabilistic theory, provides a structured framework for representing the conditional probabilities that define any operational procedure. This matrix doesn’t simply list probabilities; it meticulously encodes the likelihood of obtaining specific measurement outcomes given a particular preparation procedure. Essentially, it’s a complete probabilistic recipe, detailing how a system is prepared and what results are anticipated. The power of the COPE matrix lies in its ability to fully capture the operational rules of a theory, allowing for precise calculations and predictions. By explicitly defining these conditional probabilities – represented mathematically as $P(outcome|preparation)$ – the matrix facilitates a rigorous analysis of the theory’s internal consistency and its capacity to describe observed phenomena. It is through this structured representation that the logical consequences of a theory can be explored and tested, ultimately revealing the assumptions embedded within its framework.

The way a theory encodes probabilities isn’t merely a mathematical detail; it fundamentally shapes the perceived reality that theory describes. Each probabilistic framework – be it quantum mechanics, Bayesian inference, or classical statistics – utilizes a specific method to represent conditional probabilities, and this representation dictates how information is structured and interpreted. A theory’s choice of representing $P(A|B)$ – the probability of event A given event B – influences not only calculations but also the very concepts of cause and effect, predictability, and even the existence of objective properties. Consequently, examining how a theory represents probabilities provides crucial insight into the ontological commitments embedded within its structure, revealing the nature of reality as conceived by that specific framework. The COPE matrix, for instance, offers a particular way to map these conditional probabilities, and its features illuminate the theoretical assumptions about how preparations lead to measurement outcomes, effectively defining the ‘rules of the game’ for that worldview.

The fragment COPE matrix represents a restricted subset of the full operational theory’s COPE matrix, and while the full matrix satisfies necessary assumptions, its fragment does not.

Beyond Axiomatic Constraints: Introducing PreGPTs

Standard probabilistic theories, such as classical probability and quantum mechanics, adhere to the principle that indistinguishable experimental procedures must be represented by identical mathematical objects. PreGPTs generalize this framework by removing this constraint; they allow for different representations of procedures that yield the same observed probabilities. This relaxation is achieved through a generalized representational structure where procedures are mapped to operators acting on a complex vector space, and the equivalence of indistinguishable procedures is no longer a foundational axiom. Consequently, PreGPTs can model scenarios where distinct physical implementations of the same probabilistic outcome are represented by non-identical mathematical structures, potentially offering a more flexible and expressive framework for representing ontological models.

The construction of PreGPTs necessitates a method for factorizing the COPE (Conditional Probability of Events) matrix, which mathematically represents the agent’s probabilistic knowledge. This factorization allows for the decomposition of complex probabilistic relationships into a more manageable set of underlying factors. A common technique employed for this purpose is Singular Value Decomposition (SVD), a matrix factorization method that expresses the COPE matrix as the product of three matrices: $U$, $S$, and $V^T$. The diagonal elements of the $S$ matrix, representing singular values, indicate the importance of each factor in representing the original probabilistic structure. Utilizing SVD or similar factorization techniques enables the generation of PreGPT representations that can model scenarios violating the standard axioms of probability theory.

Relaxing the constraint of identical representations for indistinguishable procedures, as enabled by PreGPTs, allows for the formulation of ontological models that are not expressible within standard probabilistic frameworks. Conventional frameworks, built on the assumption of equivalence for indistinguishable processes, inherently limit the scope of representable models to those conforming to this principle. By removing this limitation, PreGPTs facilitate the investigation of models where differing representations for equivalent procedures are permissible, potentially capturing nuances and complexities beyond the reach of traditional ontological representations. This expansion of representational capacity is crucial for exploring alternative models of reality and cognition that may exhibit behaviors not predicted by established theories, potentially leading to advancements in fields requiring nuanced modeling of complex systems.

Noncontextual ontological models are constructed by factorizing the COPE matrix into response functions and epistemic states, resulting in a polytope that, under the equirank condition, is coplanar with the response function polytope.

The Equirank Condition: A Defining Criterion for Ontological Models

An ontological model establishes a formal relationship between experimental preparations and observed outcomes with representations within an ontic variable space. This mapping isn’t merely descriptive; it aims to reveal the underlying reality by associating measurable quantities – such as probabilities of outcomes given specific preparations – with points or regions in a defined variable space. The ontic variable space represents the hypothesized properties of a system, independent of measurement. By constructing such a model, researchers attempt to move beyond predicting outcomes to understanding the system’s intrinsic properties and how these properties manifest in observable results. The model’s success is judged by its ability to consistently and accurately relate preparation settings to outcome distributions through this ontic representation, offering insights into the nature of physical reality being investigated.

The Equirank Condition, a fundamental requirement for noncontextual ontological models, stipulates that the COPE matrix, denoted as $C$, and its factorization matrices, $R$ and $P$, must all possess identical rank. Specifically, the condition is mathematically expressed as $rank(C) = rank(R) = rank(P)$. This equality ensures the uniqueness of the representation of preparations and outcomes within the ontic variable space. A shared rank guarantees that the information encoded in the COPE matrix is consistently and fully represented by its factored components, preventing ambiguity in the ontological model and allowing for a well-defined mapping between states and their underlying ontic descriptions.

Rank separation, occurring when $rank(C) \neq rank(R)$ or $rank(C) \neq rank(P)$ in the COPE matrix factorization (where C is the COPE matrix, R and P are its factorization matrices), indicates a failure of noncontextual ontological models to accurately represent the system. Noncontextual models assume that the ontic state is independent of the measurement context; rank separation demonstrates this assumption is invalid. Specifically, it suggests the existence of ontological contextuality, implying that the system’s underlying reality is influenced by the measurement choices made, and that a complete ontological description requires accounting for these contextual effects. This deviation from a unique, context-independent representation necessitates more complex ontological frameworks capable of capturing these contextual dependencies.

Illustrative Cases: Spekkens’ Theory and Discrete Qubits

Spekkens’ toy theory offers a compelling illustration of how operational theories can deviate from classical expectations regarding hidden variables. This specifically constructed model, designed to mimic certain quantum phenomena, demonstrably violates the Equirank Condition – a principle suggesting that all maximal measurement settings should be equivalent from an ontological perspective. By carefully defining the possible preparations and measurements within the toy theory, researchers have shown that assigning definite pre-existing values to system properties in a way consistent with all possible measurement outcomes is impossible. This violation isn’t a flaw in the mathematical framework, but rather a fundamental characteristic of the theory itself, indicating that the ontic space – the space of all possible system properties – must possess a certain minimal dimensionality. The theory, therefore, serves as a crucial stepping stone in understanding the limitations of classical intuitions when applied to the foundations of quantum mechanics and the nature of reality.

The Discrete Qubit Theory offers a compelling platform for investigating ontological contextuality, moving beyond abstract mathematical formulations to a concrete, operational model. This approach leverages the Hilbert-Schmidt product – a powerful tool for characterizing the relationships between quantum states – to define a specific ontic space where the properties of qubits are represented. By carefully constructing this space and analyzing the correlations between measurements, researchers can demonstrate how the very act of observation influences the realized properties of the system. This isn’t simply a quirk of quantum mechanics, but a fundamental feature embedded within the theory’s structure, revealing that certain properties only come into being when measured in relation to specific, contextual settings. Consequently, the Discrete Qubit Theory allows for a rigorous exploration of how contextuality arises from the underlying ontological assumptions of a theory, providing a valuable lens through which to understand the foundations of quantum mechanics and its departure from classical intuition.

The consistent violation of the Equirank Condition in theories like Spekkens’ toy model and discrete qubits isn’t merely a quirk of mathematical formalism, but a robust indicator of underlying ontological structure. This failure signifies that the properties of a system aren’t predetermined by a hidden variable residing in a low-dimensional space; instead, the ontic space – the space of all possible states of the system – must possess a minimum dimensionality dictated by the number of measurement settings. Specifically, research demonstrates that the dimensionality, denoted as $k$, must satisfy the inequality $k ≥ log₂(n)$, where $n$ represents the number of distinct measurement outcomes. This lower bound isn’t a theoretical limit, but a demonstrable requirement for any operational theory exhibiting such non-classical behavior, effectively establishing a fundamental connection between the informational content of a system and the complexity of its underlying reality.

Toward a Complete Framework: GPTs and Tomographic Completeness

Tomographic completeness represents a fundamental benchmark for any probabilistic theory, dictating its capacity to fully characterize physical systems through experimental observation. A theory exhibiting this property guarantees that the complete state of a system – defining its probabilities for all possible outcomes – can be unequivocally determined solely from a set of measurement results. Conversely, incomplete theories necessitate assumptions beyond the experimental data to reconstruct the system’s state, introducing potential ambiguities and limiting their predictive power. This characteristic is particularly relevant when comparing classical probability with quantum mechanics; while classical theories are inherently tomographically complete, quantum theory, despite its successes, requires careful consideration of measurement contexts to ensure uniquely defined states, a feature linked to the inherent ontological contextuality revealed by the structure of the COPE matrix and its limitations on factorization rank.

Generalized Probabilistic Theories, or GPTs, represent an evolution from earlier PreGPT frameworks, striving for a more comprehensive and subtle depiction of physical reality. While PreGPTs offered a valuable initial step in exploring alternatives to conventional probability, GPTs endeavor to overcome inherent limitations by incorporating a broader range of mathematical structures and operational constraints. This advancement allows for a more faithful representation of complex systems and phenomena, potentially capturing nuances lost in simpler models. The pursuit of GPTs isn’t merely mathematical exercise; it’s driven by the desire to create a theoretical landscape where quantum mechanics emerges as one possibility amongst many, enabling researchers to probe the fundamental principles governing information processing and the nature of physical laws themselves. This expanded framework facilitates the investigation of ontological contextuality and the limitations imposed by the structure of the COPE matrix, ultimately pushing the boundaries of what can be known about the universe.

A rigorous investigation into the interplay between tomographic completeness, PreGPTs, and GPTs offers a pathway towards resolving foundational questions in probability and quantum mechanics. This connection is particularly illuminated by analyzing the COPE (Contextuality Operational Probabilistic Entanglement) matrix; its inherent sparsity directly constrains the possible factorization rank of probabilistic models. This limitation isn’t merely a mathematical curiosity, but a signal of ontological contextuality – meaning the properties of a system aren’t predetermined but manifest only within specific measurement contexts. Consequently, exploring this relationship provides insights into how information is encoded in physical systems and challenges classical assumptions about objective reality, suggesting that the very act of observation fundamentally shapes the observed.

The pursuit of a unified framework, as detailed in this study concerning operational theories and the COPE matrix, echoes a fundamental principle of mathematical elegance. It seeks to distill complex physical models into a provable structure, independent of specific interpretations. As Albert Einstein once stated, “God does not play dice with the universe.” This sentiment aligns with the paper’s rigorous approach; it doesn’t merely demonstrate contextuality through observation, but establishes a link between noncontextuality and the equirank condition – a mathematically verifiable criterion. The emphasis on matrix factorization and probabilistic structure represents a commitment to establishing truth through logical deduction, rather than empirical approximation.

Beyond the Equirank: Charting Future Contexts

The presented framework, while establishing a precise link between operational theories, the COPE matrix, and matrix factorization – specifically the equirank condition – does not, of course, resolve the deeper philosophical questions. Establishing that noncontextuality mathematically manifests as a particular algebraic constraint is merely a first step. The enduring challenge remains: why should nature respect such constraints? The elegance of the formalism demands a corresponding elegance in the underlying physics, a connection currently lacking. Future work must move beyond simply detecting contextuality to elucidating its physical origin.

A critical limitation lies in the inherent difficulty of scaling these techniques to systems of substantial complexity. While the framework provides a rigorous method for analyzing small-dimensional systems, the computational cost associated with matrix factorization rapidly becomes prohibitive. Approximations will inevitably be necessary, and it is crucial to understand how these approximations affect the validity of the conclusions. A purely algebraic solution, devoid of physical insight, is ultimately unsatisfying.

Furthermore, the focus on probabilistic structures, while mathematically sound, may obscure potentially relevant ontological models. The framework, as presented, operates at a level of abstraction that risks losing sight of the physical degrees of freedom. The next logical progression requires a systematic exploration of how different ontological assumptions map onto the algebraic properties of the COPE matrix, providing a bridge between the formal and the physical. Only then can one hope to move beyond mere detection to genuine understanding.

Original article: https://arxiv.org/pdf/2512.10000.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/