What connects minimal decision-making to contextual probability? We address this question through the Tug-of-War (TOW) principle, a resource-coupled mechanism for multi-armed bandit decision-making. Starting from classical probability, we show that when reward-learning functionality and specified information requirements are fixed, minimizing the necessary relaxation of common event semantics can lead to nonclassical probability. We derive the exact minimum semantic repair and identify the conditions under which information efficiency, calibration, and compositional consistency select contextual statistics. We then construct a single quantum Tug-of-War (QTOW) instrument in which the same decision--reward process both improves stationary decision performance and drives the unconditional learned state beyond the Klyachko--Can--Binicioğlu--Shumovsky (KCBS) noncontextuality bound. A finite diagnostic based on the instrument's recovery operations also yields a corrected contextuality witness. These results connect minimal decision-making, operational economy, and the emergence of contextual probability.
Comments
Substantially revised. Derived the minimum semantic repair for contextual probability and demonstrated that QTOW reward learning reaches KCBS-violating states. Added numerical validation and supplementary analysis
What connects minimal decision-making to contextual probability? We address this question through the Tug-of-War (TOW) principle, a resource-coupled mechanism for multi-armed bandit decision-making. Starting from classical probability, we show that when reward-learning functionality and specified information requirements are fixed, minimizing the necessary relaxation of common event semantics can lead to nonclassical probability. We derive the exact minimum semantic repair and identify the conditions under which information efficiency, calibration, and compositional consistency select contextual statistics. We then construct a single quantum Tug-of-War (QTOW) instrument in which the same decision--reward process both improves stationary decision performance and drives the unconditional learned state beyond the Klyachko--Can--Binicioğlu--Shumovsky (KCBS) noncontextuality bound. A finite diagnostic based on the instrument's recovery operations also yields a corrected contextuality witness. These results connect minimal decision-making, operational economy, and the emergence of contextual probability.
Quantum cognition often explains order effects, contextuality, and violations of the law of total probability by replacing classical probability with quantum probability on a fixed event structure. This paper proposes a different interpretation: quantum probability is the fixed-spacetime projection of contextual spacetime formation under finite-state requirements.
The framework begins not with time, space, objects, or probabilities, but with requirements such as finite representational capacity, single-state semantic stability, context-sensitive intervention, avoidance of explicit context labels, coherent world-formation, and intersubjective transformability. When these requirements cannot be realized within a single global Boolean event structure, the mismatch appears, under fixed-spacetime projection, as noncommutativity, interference, and quantum-like probability.
Building on prior single-state approaches to contextuality, we reinterpret classical contextual bookkeeping cost as the fixed-spacetime shadow of contextual spacetime formation. Auxiliary memory or context labels in a classical representation correspond, in this account, to holonomy-like mismatch among locally Boolean logic-worlds. The interference term is the cross term generated when locally classical realization contributions are nontrivially glued and projected back into a fixed classical spacetime form.
The result is a transcendental-operational realist account: objecthood, eventhood, probability, and spacetime are treated as forms of realization under requirements, while objectivity is defined by invariants preserved across observer- and history-dependent spacetime formations.
We propose an information -- theoretic framework for quantifying Kochen-Specker contextuality. Two complementary measures are introduced: the mutual information energy, a state-independent quantity inspired by Onicescu's information energy that captures the geometric overlap between joint eigenspaces within a context; and an operational measure based on commutator expectation values that reflects contextual behavior at the level of measurement outcomes. We establish a hierarchy of bounds connecting these measures to the Robertson uncertainty relation, including spectral, purity-corrected, and operator norm estimates. The framework is applied to the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario for spin-1 systems, where all quantities admit closed-form expressions. The Majorana-stellar representation furnishes a common geometric platform on which both the operational measure and the uncertainty products can be analyzed. For spin-1, this representation yields a three-dimensional Euclidean-like visualization of the Hilbert space in which, states lying on a plane exhibit maximum uncertainty for the observable along the perpendicular direction; simultaneous optimization across all KCBS contexts singles out a unique state on the symmetry axis. Notably, states achieving the optimal sum of uncertainty products exhibit vanishing operational contextuality, while states with substantial operational contextuality satisfy a nontrivial Robertson bound -- the two extremes are achieved by distinct quantum states.
Contextuality is a central feature distinguishing quantum from classical probability theories, but its operational meaning is often stated only qualitatively. In this Letter, we study a simple information-theoretic question: how much additional contextual information must a classical simulation introduce when it tries to keep a shared internal description fixed across contexts? To make this question precise, we analyze a minimal external-label simulation model in which the remaining context dependence is carried only by an auxiliary label. For this model, we define an obstruction cost as the minimum mutual information between the context and the auxiliary label required to reproduce the observed statistics. We then prove a conservative quantitative lower bound: any linear witness that separates the observed statistics from the zero-obstruction set yields a positive lower bound on this cost. We do not claim that this bound is tight, and we do not claim that the simulation model covers every possible classical architecture. Its role is narrower and more explicit: under fixed shared-state semantics, contextuality can be read as a certificate of irreducible external bookkeeping cost in a simple and well-defined simulation model.
Contextuality is a defining feature that separates the quantum from the classical descriptions of physical systems. Within the marginal-scenario framework, noncontextual models are characterized by the existence of a single joint probability distribution consistent with all measurable contexts, while contextual models violate this condition. Building on this approach, we introduce a general method to analyze contextuality in terms of stochastic linear maps that effectively model invasive measurements on an otherwise classical statistics. These maps transform probabilities within the noncontextuality polytope, which includes all classical probabilities, into probabilities that may lie outside the polytope, while preserving the compatibility structure of the scenario at hand. We derive general consistency conditions that such maps must satisfy to represent admissible invasive measurements, and we fully identify them for a paradigmatic example of contextuality for a single three-level quantum system. Furthermore, we introduce a quantifier of contextuality based on the minimal invasiveness required to reproduce a given probability distribution, which offers a distinct approach on how to evaluate the degree of contextuality in a general scenario.
Generalisation in machine learning often relies on the ability to encode structures present in data into an inductive bias of the model class. To understand the power of quantum machine learning, it is therefore crucial to identify the types of data structures that lend themselves naturally to quantum models. In this work we look to quantum contextuality -- a form of nonclassicality with links to computational advantage -- for answers to this question. We introduce a framework for studying contextuality in machine learning, which leads us to a definition of what it means for a learning model to be contextual. From this, we connect a central concept of contextuality, called operational equivalence, to the ability of a model to encode a linearly conserved quantity in its label space. A consequence of this connection is that contextuality is tied to expressivity: contextual model classes that encode the inductive bias are generally more expressive than their noncontextual counterparts. To demonstrate this, we construct an explicit toy learning problem -- based on learning the payoff behaviour of a zero-sum game -- for which this is the case. By leveraging tools from geometric quantum machine learning, we then describe how to construct quantum learning models with the associated inductive bias, and show through our toy problem that they outperform their corresponding classical surrogate models. This suggests that understanding learning problems of this form may lead to useful insights about the power of quantum machine learning.
Literary interpretation changes as readers encounter new passages, answer interpretive questions, and revisit earlier material in light of later context. Quantum cognition provides a probability calculus for judgments whose probabilities depend on context and probe order, but the use of Hilbert-space notation alone does not establish nonclassical behavior. This article develops Quantum Semantic Dynamics (QSD) as a constrained quantum-probabilistic model and distinguishes three claim levels: quantum-inspired representation, quantum-probabilistic structure, and intrinsic nonclassicality. The present paper advances only the second. The confirmatory implementation restricts the local semantic state to a two-dimensional space, fixes probe geometry in an independent pilot sample, reconstructs a population-level 2 × 2 density state from three linearly independent binary probe bases, and gives the off-diagonal state components an operational meaning through cross-basis judgments. Contextual evolution is restricted to a one-parameter completely positive trace-preserving contraction toward a preregistered context-target state rather than an arbitrary latent transition. Two probes form the randomized A → B/B → A sequence used for the quantum question-order (QQ) diagnostic; the third is measured in an independent arm for state identification. The paper derives the QQ equality, states its assumptions, and adds an uncertainty-style consistency constraint for the frozen noncommuting probe geometry. QSD is compared with explicitly defined classical comparator classes ranging from a static commutative model to matched sequential latent-state models. Compatibility with QQ or the Hilbert representation is not treated as proof of intrinsic nonclassicality; such a claim would require a separate contextuality analysis against an explicitly defined classical representability criterion. The resulting architecture makes the mathematical status, empirical burden, and rejection conditions of QSD explicit before data collection.