Recent theories developing broad notions of context and its effects on inference are becoming increasingly important in fields as diverse as cognitive psychology, information science and quantum information theory and computing. Here we introduce a novel and general approach to the characterisation of contextuality using the techniques of Chu spaces and Channel Theory viewed as general theories of information flow. This involves introducing three essential components into the formulism: events, conditions and measurement systems. Incorporating these factors in relationship to conditional probabilities leads to information flows both in the setting of Chu spaces and Channel Theory. The latter provides a representation of semantic content using local logics from which conditionals can be derived. We employ these features to construct cone-cocone diagrams, commutativity of which enforces inferential coherence. With these we build a scale-free architecture incorporating a Bayesian-like hierarchical structure, in which there is an interpretation of active inference and Markov blankets. We compare this architecture with other theories of contextuality which we briefly review. We also show that this development of ideas conveniently accommodates negative probabilities, leading to the notion of signed information flow, and address how quantum contextuality can be interpreted within this model. Finally, we relate contextuality to the Frame Problem, another way of characterising a fundamental limitation on the observational and inferential capabilities of finite agents.
Recent theories developing broad notions of context and its effects on inference are becoming increasingly important in fields as diverse as cognitive psychology, information science and quantum information theory and computing. Here we introduce a novel and general approach to the characterisation of contextuality using the techniques of Chu spaces and Channel Theory viewed as general theories of information flow. This involves introducing three essential components into the formulism: events, conditions and measurement systems. Incorporating these factors in relationship to conditional probabilities leads to information flows both in the setting of Chu spaces and Channel Theory. The latter provides a representation of semantic content using local logics from which conditionals can be derived. We employ these features to construct cone-cocone diagrams, commutativity of which enforces inferential coherence. With these we build a scale-free architecture incorporating a Bayesian-like hierarchical structure, in which there is an interpretation of active inference and Markov blankets. We compare this architecture with other theories of contextuality which we briefly review. We also show that this development of ideas conveniently accommodates negative probabilities, leading to the notion of signed information flow, and address how quantum contextuality can be interpreted within this model. Finally, we relate contextuality to the Frame Problem, another way of characterising a fundamental limitation on the observational and inferential capabilities of finite agents.
The context-sensitivity of cognition has been demonstrated across a wide range of cognitive functions such as perception, memory, judgement and decision making. A related term, 'contextuality', has appeared from the field of quantum cognition, with mounting empirical evidence demonstrating that cognitive phenomena are sometimes contextual. Contextuality is a subtle notion that influences how we must view the properties of the cognitive phenomenon being studied. This article addresses the questions: What does it mean for a cognitive phenomenon to be contextual? What are the implications of contextuality for probabilistic models of cognition? How does contextuality differ from context-sensitivity? Starting from George Boole's "conditions of possible experience", we argue that a probabilistic model of a cognitive phenomenon is necessarily subject to an assumption of realism. By this we mean that the phenomenon being studied is assumed to have cognitive properties with a definite value independent of observation. In contrast, quantum cognition holds that a cognitive property maybe indeterminate, i.e., its properties do not have well established values prior to observation. We argue that indeterminacy is sufficient for incompatibility between cognitive properties. In turn, incompatibility is necessary for their contextuality. The significance of this argument for cognitive psychology is the following:if a cognitive phenomenon is found to be contextual, then there is reason to believe it may be indeterminate. We illustrate by means of two crowdsourced experiments how context-sensitivity and contextuality of cognitive properties in the form of facial trait judgements can be characterized from empirical data. Finally, we conceptually and formally contrast contextuality with context-sensitivity. We propose that both involve a form of context dependence, with causality being the differentiating factor: the context dependence in context-sensitivity has a causal basis, whereas the context dependence in contextuality is acausal. The resulting implications for probabilistic models of cognition are discussed.
Context-sensitive behavior can be modeled by enriching an internal state, by allowing a response rule to access context directly, or by preserving a shared state while introducing an auxiliary criterion or control variable. This paper isolates an information-theoretic constraint on the third architecture. Let C denote context, S a candidate internal or latent state, O an observable response, and M an auxiliary variable such that O⊥C∣(S,M). Then I(C;O∣S)≤I(C;M∣S)≤H(M∣S). Once a shared state has been specified, residual context dependence in behavior therefore lower-bounds both the context information and the conditional entropy that an auxiliary context-mediating mechanism must carry. The bound is representation-relative rather than a measure of state-space size or a universal contextuality measure. A worked recognition-memory example shows how the quantity can be computed for payoff-induced criterion shifts and compared across alternative representational allocations. Ontological contextuality and quantum probability are treated as a separate specialization rather than as prerequisites for the cognitive interpretation. More broadly, the framework provides a basis for analyzing context-memory-control trade-offs in cognitive models and artificial agents that must maintain coherent behavior across changing contexts under bounded internal representations.
The main subjects of this thesis are non-locality and contextuality, two fundamental features of quantum mechanics that constitute valuable resources for quantum computation. Our analysis is based on Abramsky & Brandenburger's sheaf theoretic framework, which captures both these phenomena in a unified treatment and in a very general setting. This high-level description transcends quantum physics and allows to precisely characterise the notion of contextuality as the apparent paradox realised by data being locally consistent but globally inconsistent. More specifically, we aim to develop a deeper understanding of so-called logical forms of contextuality, i.e. situations where the phenomenon can be witnessed using purely logical arguments, disregarding probabilities. The sheaf theoretic description of logical contextuality has recently inspired the development of a topological treatment of the phenomenon based on sheaf cohomology. In this thesis, we embark on a detailed analysis of the cohomology of contextuality, exposing key shortcomings in the current methods, and introducing an (almost) complete cohomological characterisation of logical forms of contextuality. More specifically, we show that, in its current formulation, sheaf cohomology does not constitute a complete invariant for contextuality, not even in its strongest forms, and that higher cohomology groups cannot be used to study the phenomenon. Then, we solve these issues by introducing a novel construction, which derives refined versions of the presheaves describing empirical models to expose their deeper extendability properties, resulting in a sheaf cohomological invariant which is applicable to the vast majority of empirical models, and conjectured to work universally. We propose a general theory of contextual semantics using the language of valuation algebras. In particular, we give a general definition of contextual behaviour as a fundamental gap between local agreement and global disagreement of information sources. Not only does this formalism aptly capture and generalise the known instances of contextuality beyond quantum theory, but it also provides inspiration for further applications of the phenomenon, and paves the way for the transfer of results and techniques between different fields. We give a prime example of this potential by developing faster algorithms to detect contextuality based on mainstream methods of generic inference. Finally, we turn our attention back to instances of contextuality in quantum physics, and study strong contextuality in multi-qubit states. We give a complete combinatorial characterisation of All-vs-Nothing proofs of strong contextuality in stabiliser quantum mechanics. This allows to produce the complete list of all stabiliser states exhibiting this kind of contextuality, which consitututes an important resource in certain models of quantum computation. Then, we extend our search for strongly contextual behaviour beyond stabiliser states, and identify the minimum quantum resources needed to realise strong non-locality. Additional results include a partial classification of strongly non-local models comprised of three-qubit states and local projective measurements, and the introduction of a new infinite family of strongly non-local three-qubit states.
We generalize the Abramsky--Brandenburger sheaf-theoretic treatment of contextuality by replacing probability distributions with distributions valued in a convex effect algebra A. This yields a notion of A-valued measurement model encompassing probabilistic, deterministic, and quantum measurement structures within a single framework.
States σ:A→[0,1] induce ordinary empirical models, allowing observable behaviour to be viewed as arising from effect-valued measurement data. This leads to a distinction between internal contextuality of A-models and observable contextuality after state evaluation. We analyze the relationship between these notions and identify coherence conditions under which observable classical explanations assemble into internal ones.
Using the ordered-vector-space representation of effect modules, we show that non-contextuality is characterized by feasibility of an associated cone program, generalizing the linear-programming formulation of contextuality in the probabilistic case. We also introduce an effect-valued contextual fraction and study its relation to observable contextuality witnesses. We show how cone duality leads to a notion of Bell witnesses for contextuality as a direct generalization of Bell inequalities.
Finally, we analyze sharp realizability and uniform dilation for measurement models, clarifying the relationship between general POVMs and projective measurements, and show how resource-indexed effect structures induce graded monads generalizing the quantum monad.
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.