Abstract Relative-entropy measures of contextuality quantify the statistical distance between an empirical model and the noncontextual set. We study how this established quantity changes when an experiment is observed through a characterized loss mechanism. Each attempted trial is retained in the analysis, with missed outcomes represented by context-labeled no-click symbols, and the same stochastic observation channel is applied to the target model and every noncontextual comparator. Under this common-channel transportability assumption, the data-processing inequality gives an observed-data contextual divergence that cannot exceed its complete-data value. We define the absolute contextual-information loss and its normalized ratio, the fraction of missing contextual information (FMCI). We prove general reference-set contraction, exact scaling under missing completely at random loss, a cellwise erasure sandwich, a no-exact-hiding result under strictly positive retention, and additive and multiplicative decompositions for sequential design and detector channels. We also separate response dependence from disturbance of shared marginals and clarify why outcomes of unchosen incompatible contexts are not model-neutral missing entries. The theory is evaluated on the perfect-anticorrelation three-cycle, the Klyachko–Can–Binicioğlu–Shumovsky pentagon, and the Peres–Mermin magic square. The hidden-variable and inequality-constrained optimizations agree to within 10 − 12 nats. Exhaustive enumeration of all 104 context subcovers shows that, for these three canonical benchmark points, every proper subcover is noncontextual. A 51 × 51 scan of a tunable outcome-dependent loss family confirms that exact population hiding occurs only on the zero-retention boundary, while finite-sample detectability can deteriorate well before that boundary. For a known finite-alphabet erasure channel with cellwise retention probabilities we give an exact identification criterion, regular asymptotic and Fieller inference for FMCI, and diagnostics that withhold scalar ratio inference near the noncontextual boundary. Unknown outcome-dependent loss is handled by a calibration-constrained sensitivity workflow and union confidence sets. FMCI is therefore a conditional diagnostic of information attenuation under a declared observation channel, not a replacement for loophole-free contextuality certification.
Abstract Relative-entropy measures of contextuality quantify the statistical distance between an empirical model and the noncontextual set. We study how this established quantity changes when an experiment is observed through a characterized loss mechanism. Each attempted trial is retained in the analysis, with missed outcomes represented by context-labeled no-click symbols, and the same stochastic observation channel is applied to the target model and every noncontextual comparator. Under this common-channel transportability assumption, the data-processing inequality gives an observed-data contextual divergence that cannot exceed its complete-data value. We define the absolute contextual-information loss and its normalized ratio, the fraction of missing contextual information (FMCI). We prove general reference-set contraction, exact scaling under missing completely at random loss, a cellwise erasure sandwich, a no-exact-hiding result under strictly positive retention, and additive and multiplicative decompositions for sequential design and detector channels. We also separate response dependence from disturbance of shared marginals and clarify why outcomes of unchosen incompatible contexts are not model-neutral missing entries. The theory is evaluated on the perfect-anticorrelation three-cycle, the Klyachko–Can–Binicioğlu–Shumovsky pentagon, and the Peres–Mermin magic square. The hidden-variable and inequality-constrained optimizations agree to within 10 − 12 nats. Exhaustive enumeration of all 104 context subcovers shows that, for these three canonical benchmark points, every proper subcover is noncontextual. A 51 × 51 scan of a tunable outcome-dependent loss family confirms that exact population hiding occurs only on the zero-retention boundary, while finite-sample detectability can deteriorate well before that boundary. For a known finite-alphabet erasure channel with cellwise retention probabilities we give an exact identification criterion, regular asymptotic and Fieller inference for FMCI, and diagnostics that withhold scalar ratio inference near the noncontextual boundary. Unknown outcome-dependent loss is handled by a calibration-constrained sensitivity workflow and union confidence sets. FMCI is therefore a conditional diagnostic of information attenuation under a declared observation channel, not a replacement for loophole-free contextuality certification.
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicioglu-Shumovsky, Einstein-Podolsky-RosenBell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected (i.e., it allows for all possible forms of "disturbance" or "signaling" in them). Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L-1-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals. We establish that, in relation to the maximally tight Bell-type CbD inequality for (generally, inconsistently connected) cyclic systems, the measure of contextuality is proportional to the absolute value of the difference between its two sides. For noncontextual cyclic systems, the measure of noncontextuality is shown to be proportional to the smaller of the same difference and the L-1-distance to the surface of the box circumscribing the noncontextuality polytope. These simple relations, however, do not generally hold beyond the class of cyclic systems, and noncontextuality of a system does not follow from noncontextuality of its cyclic subsystems.
In this paper we analyze the notion of measurement noncontextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any noncontextual ontological model fails to explain the statistics of outcomes of a single carefully constructed positive-operator-valued measure (POVM) executed sequentially on a quantum system. The context of measurement arises from the different configurations in which the device can be used. We develop an inequality from the noncontextual (NC) ontic model, and construct a quantum situation involving measurements from the Klyachko-Can-Binicio\ifmmode \check{g}\else \v{g}\fi{}lu-Shumovsky (KCBS) inequality. We show that the resultant statistics arising from this device violate our NC inequality. This device can be generalized by incorporating measurements from arbitrary n-cycle contextuality inequalities of which n=5 corresponds to the KCBS inequality. We show that the NC and quantum bounds for various scenarios can be derived more easily using only the functional relationships between the outcomes for larger values n. This makes it one of the simpler contextual inequalities to analyze.
Abstract We develop a convex-analytic framework that turns algebraic obstructions to global sections in sheaf models of quantum contextuality into certifiable signatures on quantum state space. For a finite family of projective measurement contexts we consider, for each context, the entropy increase incurred by dephasing in that context and the lower envelope of these associated classicalisation costs. Mathematically, we show that this envelope is convex exactly when a single classical frame can explain all statistics; genuine contextuality generically produces kinks and strictly positive Jensen gaps along chords, an epigraph-union manifestation of the failure of a global section. We obtain dimension- and overlap-dependent lower and upper bounds on these gaps using relative-entropy inequalities, and show that the resulting convexity defects are monotone under dephasing-covariant quantum channels. Operationally, the framework requires only context-wise frequency tables, yields finite-sample-robust witnesses and visual diagnostics, and is directly implementable on cloud quantum devices. A contextual Landauer principle links algebraic incompatibility of contexts to a nonzero minimal erasure work in branched-then-erase protocols. Taken together, these results provide algebraic and geometric invariants of measurement covers that translate foundational features of contextuality into experiment-ready benchmarks for contemporary quantum hardware.
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected (i.e., it allows for all possible forms of ``disturbance'' or ``signaling'' in them). Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L1-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals. We establish that, in relation to the maximally tight Bell-type CbD inequality for (generally, inconsistently connected) cyclic systems, the measure of contextuality is proportional to the absolute value of the difference between its two sides. For noncontextual cyclic systems, the measure of noncontextuality is shown to be proportional to the smaller of the same difference and the L1-distance to the surface of the box circumscribing the noncontextuality polytope. These simple relations, however, do not generally hold beyond the class of cyclic systems, and noncontextuality of a system does not follow from noncontextuality of its cyclic subsystems.
We establish a formally verified three-layer separation theorem demonstrating that contextual information residue—the non-trivial Čech cohomology class of a Boolean 1cochain on a finite Z/2 cycle graph—is structurally invisible to both Shannon mutual information and bounded-degree polynomial computation. Specifically: (1) Shannon mutual information of a binary symmetric channel achieves AUC = 0.500 for contextual/noncontextual classification, exhibiting perfect orthogonality; (2) multilinear polynomials of degree d < n on n-bit cochains produce FN = 4088/4092 on the exhaustive corpus, a consequence of the Minsky–Papert degree barrier; and (3) exact rational symbolic evaluation Q via the parity formula parity(b) = (1 − i (1 − 2bi ))/2 ∈ Q achieves perfect classification with zero error. The algebraic core—including H 1 (Cn ; Z/2) ∼ = Z/2 and the non-coboundary witness for the triangle and square parity cochains—is verified in Lean 4 with Mathlib. The corpus comprises all 2n Boolean 1-cochains on cycles Cn for n ∈ {3, . . . , 12}, totaling 8,184 cochains exhaustively enumerated. All proofs, experiments, and data are publicly available. Published by The Institute for Applied Ontological Mathematics.
Contextuality is a feature of quantum correlations. It is crucial from a foundational perspective as a non-classical phenomenon, and from an applied perspective as a resource for quantum advantage. It is commonly defined in terms of hidden variables, for which it forces a contradiction with the assumptions of parameter-independence and determinism. The former can be justified by the empirical property of non-signalling or non-disturbance, and the latter by the empirical property of measurement sharpness. However, in realistic experiments neither empirical property holds exactly, which leads to possible objections to contextuality as a form of non-classicality, and potential vulnerabilities for supposed quantum advantages. We introduce measures to quantify both properties, and introduce quantified relaxations of the corresponding assumptions. We prove the continuity of a known measure of contextuality, the contextual fraction, which ensures its robustness to noise. We then bound the extent to which these relaxations can account for contextuality, via corrections terms to the contextual fraction (or to any non-contextuality inequality), culminating in a notion of genuine contextuality, which is robust to experimental imperfections. We then show that our result is general enough to apply or relate to a variety of established results and experimental set-ups. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.