Suppose that a family of commutative measurement contexts is given, together with the corresponding classical probability laws. We study the problem under what conditions these contextwise descriptions can be combined into a single classical representation. We distinguish two ways in which this can be done. In {\em global integration}, the prescribed observable algebras are required to fit into a single commutative context and the corresponding probability laws to arise as marginals of one joint law. In {\em protocolwise integration}, by contrast, the context label is retained as part of the description, and the contextwise laws appear as conditional distributions within a single model of the measurement protocol. We show that global integration involves two distinct notions of compatibility: operator-algebraic compatibility of the underlying observables and probabilistic compatibility of their classicalised laws. The former implies the latter, but the converse fails; under the topological hypotheses stated below, probabilistic compatibility is characterised by a Kellerer-type marginal criterion. Protocolwise integration, on the other hand, always exists and is represented operator-algebraically by a direct-sum construction. When global integration exists, the two constructions arise as reductions of a common commutative refinement. We illustrate the distinction in the Bell--CHSH experiment, where Fine's theorem relates global probabilistic integration to Bell locality and the CHSH inequalities. This shows that the obstruction revealed by Bell--CHSH concerns not classical representation as such, but classical representation preserving specified cross-context identifications.
Suppose that a family of commutative measurement contexts is given, together with the corresponding classical probability laws. We study the problem under what conditions these contextwise descriptions can be combined into a single classical representation. We distinguish two ways in which this can be done. In {\em global integration}, the prescribed observable algebras are required to fit into a single commutative context and the corresponding probability laws to arise as marginals of one joint law. In {\em protocolwise integration}, by contrast, the context label is retained as part of the description, and the contextwise laws appear as conditional distributions within a single model of the measurement protocol. We show that global integration involves two distinct notions of compatibility: operator-algebraic compatibility of the underlying observables and probabilistic compatibility of their classicalised laws. The former implies the latter, but the converse fails; under the topological hypotheses stated below, probabilistic compatibility is characterised by a Kellerer-type marginal criterion. Protocolwise integration, on the other hand, always exists and is represented operator-algebraically by a direct-sum construction. When global integration exists, the two constructions arise as reductions of a common commutative refinement. We illustrate the distinction in the Bell--CHSH experiment, where Fine's theorem relates global probabilistic integration to Bell locality and the CHSH inequalities. This shows that the obstruction revealed by Bell--CHSH concerns not classical representation as such, but classical representation preserving specified cross-context identifications.
A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.
This preprint introduces the Trans–Integral (TI) framework, a reformulation of classical probability theory that removes the assumption of ownership—the requirement that all observables be defined on a single global probability space. Using a minimal finite example, the paper demonstrates that three locally classical and perfectly consistent probability models cannot be jointly embedded under ownership. The contradiction is expressed entirely at the level of expectations, revealing its structural equivalence to the Bell–CHSH inconsistency. To clarify the mechanism, the paper constructs context-dependent Markov semigroups whose invariant measures encode the relations A = B, B = C, and A ≠ C. The global obstruction arises only when these context-specific attractors are forcibly unified into a single stochastic world. The Trans–Integral framework resolves this obstruction by replacing each observable with a context-indexed family and defining expectations via a two-level integration over outcomes and contexts. The result preserves all local classical structures while eliminating the global contradiction. The work suggests a broader course correction for probability theory, with implications for stochastic processes, Markov semigroups, and the Kolmogorov extension theorem.
Locally classical behavior is often interpreted as evidence that a global classical description should exist. In particular, when individual processes admit classical (measure-and-prepare) realizations, it is natural to expect at least one compatible joint implementation that remains classical. We show that this intuition fails in a simple broadcast setting with one input system and two outputs. Specifically, we construct pairs of compatible channels that are entanglement-breaking individually and thus admit classical descriptions, yet every joint broadcast realization is necessarily Bell-nonlocal between the outputs, even for a maximally mixed input. This establishes a form of compatibility-induced activation of nonlocality: Bell nonlocality arises not from entanglement in the inputs or marginal channels, but solely from the requirement that these marginals admit a common global implementation. In this sense, compatibility is not merely a consistency condition on local descriptions, but a mechanism that can enforce nonclassical correlations at the global level.
This paper provides a systematic study of the operational idea that a quantum “state” is only defined up to what can be distinguished by a chosen family of observables. Concretely, any von Neumann algebra of observables M induces an equivalence relation on pure and mixed states by declaring two preparations indiscernible when they give identical statistics for every observable in M. The corresponding quotient, the Holevo space associated with M, is the effective (relational) state space of the experiment, explicitly dependent on the observer’s available measurements. We analyse the resulting geometry and topology of these quotients, and prove a context-complete classical representation theorem: for every von Neumann algebra M there is a canonical lift a↦â to bounded continuous functions on the Holevo space, reproducing expectation values pointwise. In the commutative case this reduces to ordinary probability theory on the joint spectrum. The framework is illustrated in explicit examples, including position measurements of a free particle and polarisation measurements in the qubit, Einstein–Podolsky–Rosen (EPR), and Bell settings. In particular, in the EPR scenario Charlie’s joint observable defines a simplex of joint outcome distributions, whereas the Alice/Bob marginal viewpoint collapses the effective description to a lower-dimensional space by “forgetting” the correlation parameter. We show that by varying the polariser settings, the indiscernibility classes become conjugated (and generically reshuffled), and different settings are typically incompatible at the level of observable algebras.
This paper argues that the quantum measurement problem is not a single explanatory gap but a collection of distinct tasks. We separate detector representability, probability structure, single-event actuality, and outcome-conditioned refinement, and distinguish all four from causal registration, the physical criterion by which an already-instantiated event becomes a completed measurement record. These are explanatory roles within one measurement phenomenon rather than successive temporal stages. The analysis begins from the contrast between a global quantum description containing several detector-correlated components and the single entry that appears in a realized measurement history. Rather than compressing this contrast into an unexplained transition often labeled “collapse,” we ask which specific explanatory constraints must be satisfied before the relation between the two descriptions can be diagnosed precisely. A finite-resolution alternating position–momentum history is used throughout to separate questions of record formation, probability, actuality, causal registration, and outcome-conditioned refinement. A continuum analysis shows that making the registered history arbitrarily dense does not by itself recover a classical trajectory. Several parts of the architecture are exact once their inputs are specified. A detector representation induces an exact quotient; a specified acquisition channel and record family induce system-side effects; and a specified quantum instrument determines a family of subnormalized conditional branches before any one outcome is instantiated. Given a realized outcome label, the same instrument determines the corresponding normalized outcome-conditioned state. A finite-dimensional acquisition model explicitly realizes the representability chain, while a record-algebra indistinguishability result shows that removing coherence between detector-record sectors can leave all expectation values in the lifted record algebra unchanged even when that coherence remains detectable in a richer context. These results sharpen, but do not dissolve, the measurement problem. Record structure does not derive Born weights; a probability law does not itself constitute the event realized in one run; a mathematically available conditional branch family does not select one branch condition as actual; and causal onset, stabilization, registration, and conditional refinement all presuppose or follow that event rather than produce it. The resulting constraint audit tests—without assuming—the hypothesis that global quantum multiplicity and detector-record definiteness may belong to different representational levels. The main conclusion is not a new collapse dynamics but a localization of the remaining explanatory burden. Once representability, probability, causal registration, and conditional refinement are assigned their proper roles, the unresolved actuality transition can no longer be hidden inside the single label “collapse.” In the typed measurement architecture considered here, no pre-event operation has the mathematical role of producing an individual event token without additional event-producing structure. This is version 1.0 of the manuscript.
In quantum physics, the term ‘contextual’ can be used in more than one way. One usage, here called ‘Bell contextual’ since the idea goes back to Bell, is that if A , B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C , whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the { A , B } context) or with C (the { A , C } context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C . A different definition, here called ‘globally (non)contextual’ refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.