Searches for post-quantum correlations depend on two distinct ingredients: how abundant such correlations are in the underlying correlation space and how efficiently the chosen representation preserves their distinguishability from quantum correlations. We quantify these effects by introducing the popularity rate and the projection acceptance, whose product defines the search performance. We apply these concepts in binary-outcome Bell scenarios with symmetry-related sites, using the CHSH and pyramid representations. Particular attention is paid to the distinction between exchange-symmetric synchronous correlations and correlations satisfying only exchange symmetry. In the synchronous quantum sector, the common-vector representation follows automatically from synchrony, whereas exchange symmetry alone defines a substantially larger correlation space. We show that this distinction can qualitatively change the effectiveness of a projected search: the pyramid representation, which retains substantial discriminatory power in the synchronous sector, has no discriminatory power when only exchange symmetry is imposed. Our results provide a simple framework for assessing and comparing searches for post-quantum correlations.
Searches for post-quantum correlations depend on two distinct ingredients: how abundant such correlations are in the underlying correlation space and how efficiently the chosen representation preserves their distinguishability from quantum correlations. We quantify these effects by introducing the popularity rate and the projection acceptance, whose product defines the search performance. We apply these concepts in binary-outcome Bell scenarios with symmetry-related sites, using the CHSH and pyramid representations. Particular attention is paid to the distinction between exchange-symmetric synchronous correlations and correlations satisfying only exchange symmetry. In the synchronous quantum sector, the common-vector representation follows automatically from synchrony, whereas exchange symmetry alone defines a substantially larger correlation space. We show that this distinction can qualitatively change the effectiveness of a projected search: the pyramid representation, which retains substantial discriminatory power in the synchronous sector, has no discriminatory power when only exchange symmetry is imposed. Our results provide a simple framework for assessing and comparing searches for post-quantum correlations.
We present a geometric characterisation of local models in the bipartite Bell scenario with three measurement settings per site and binary outcomes, i.e. the (3,3,2,2) case. Restricting attention to indistinguishable sites, thus resulting in a `symmetric local' scenario, we introduce a three-dimensional mixed-moment space in which the mixed moments are calculated under off-diagonal measurement settings.
In this reduced representation, the symmetric local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three inequalities are required to characterise the interior of this region. We call them the pyramid inequalities that separate symmetric local (SL) models from their complement, non-symmetric local (SL) models. Remarkably, in this representation, the 'symmetric quantum' region forms an elliptope defined by a single inequality. We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope in the 36-dimensional conditional-probability space, which possesses 684 facet-defining inequalities. The reduction from 684 to three reflects normalisation, symmetry reduction, and projection to the mixed-moment space.
In the pyramid representation, the hierarchy SL⊊SQ⊊NS appears geometrically as a tetrahedron embedded in a somewhat larger curved body of symmetric quantum models, SQ, which in turn is embedded in a cube of no-signalling models, NS. This provides a unified three-dimensional visualisation of symmetric local, symmetric quantum, and symmetric post-quantum correlations. The qualitative advantages of the pyramid representation over the standard one-dimensional CSHS representation of the (2,2,2,2) case are also discussed.
Violations of Bell inequalities are a hallmark of entanglement, with only entangled states capable of exceeding classical bounds in standard Bell tests. Here we analyze anomalous weak values of the CHSH-Bell operator in pre- and post-selected (PPS) quantum ensembles, using them to define separability-constrained bounds on Bell-type nonlocal correlations in the presence of post-selection. Fixing the overlap between the pre- and post-selected states, we compare three scenarios: unrestricted boundary states, one separable boundary state, and both boundary states separable. For each case, we derive both the maximal weak value for a fixed Bell operator and the maximal bound obtained by further optimizing over all CHSH operators. Our results show that post-selection and entanglement are distinct operational resources: post-selection alone can enhance correlations, but entanglement is necessary to exceed the corresponding separable PPS bounds, and their combination yields the strongest attainable correlations. Thus the separable PPS bound plays the role of a post-selected separability benchmark, distinct from the standard Bell bound. We further show that the enhancement beyond the separable bound closely tracks the concurrence of the states that optimize the bounds, identifying entanglement as the source of the additional correlation strength. Finally, we show that nonlocal weak values provide post-selected entanglement witnesses, and we give a constructive protocol that detects every pure two-qubit source state with nonzero concurrence in the ideal state-adapted setting, even in regimes where the corresponding standard CHSH entanglement test is inconclusive. More broadly, our results motivate hybrid protocols that combine post-selection and entanglement, with possible applications to improved quantum sensing, weak-value amplification, and quantum information processing.
Abstract We present a unified analysis showing that the dominant numerical and qualitative structure of Bell–CHSH correlations—including the ubiquitous appearance of √2, sinusoidal correlation curves, and violations of the CHSH bound—arises generically from Euclidean projection plus measurement-induced dimensional reduction. The argument is developed in three interlocking layers. A geometry-first layer uses macroscopic analogues (tuned guitar strings with threshold detectors) to make the projection mechanism inspectable: a shared relational state, local measurement axes, and a recording map that collapses continuous amplitudes to binary outcomes. A threshold-scan (“dial”) test shows that conditioned CHSH values rise smoothly above 2 as the acceptance criterion tightens. A Fourier-analytic layer demonstrates that the sign function—the operator that converts continuous amplitudes to ±1—injects an infinite series of odd harmonics into a pure fundamental signal. The classical triangle-wave correlation E(θ) = 2θ/π − 1 is the fundamental plus harmonic distortion; the quantum correlation E(θ) = −cos θ shares the same fundamental mode but without higher-harmonic content and with unit amplitude. A detection threshold acts as a conditioning map that progressively strips the distortion harmonics. At threshold τ ≈ 0.23, total harmonic distortion reaches near zero, the correlation matches −cos θ, and |S| = 2√2. A combinatorial layer introduces the Q/R/x ontological framework—where Q is the object, R the definite binary relationship, and x the genuinely ambiguous state—and connects it to the formula (n2 − n)/2. Binary measurement (n = 2) supports 1 pairwise relationship and a linear correlation. Acknowledging x as a third state (n = 3) yields 3 pairs—exactly the minimum for planar angular geometry and the cosine. These three layers converge: the gap between |S| = 2 and |S| = 2√2 is the total harmonic distortion introduced by binary quantisation of a continuous angular signal—equivalently, the information-theoretic cost of projecting a ternary relational reality onto a binary measurement space. We show that the hard-threshold model produces CHSH violations only through post-selection on coincidences; when no-detection events are properly accounted for, the model respects all Bell-type inequalities, including the Eberhard/CH inequality used in modern “loophole-free” experiments. This honestly limits the model’s scope but sharpens the open question: whether the trial-definition process upstream of the detectors (heralding, phase-matching) constitutes a form of selection beyond the hard threshold that could affect 1 full-accounting statistics. We propose the “dial test”—sweeping acceptance criteria and plotting violation strength versus acceptance fraction—as a concrete, falsifiable protocol for probing ensemble neutrality in any Bell experiment. Nothing in this work challenges the empirical validity of quantum mechanics or the logical structure of Bell’s theorem. Rather, it challenges the necessity of interpreting Bell-type correlations as evidence for an ontological division of reality into “quantum” and “classical” domains. The analysis suggests that what distinguishes the so-called quantum regime is not a change in physical law, but a change in how much of the underlying relational structure survives measurement and recording. If this reading is correct, its most consequential implication is that the correlations which motivated the inference of “action at a distance” have a local geometric origin: they are properties of a shared state established in the past, not signals transmitted in the present, and their statistical character is determined by how severely the observation compresses what was already there.
Bell’s inequality relies on a structural feature of the standard Bell-local model: the potential outcomes for all relevant measurement settings are jointly represented on a single hidden-variable space. This joint representation is not enforced by locality alone, and its status becomes nontrivial once the physical realization of measurement events and their associated symmetry structures are taken into account. We develop a symmetry-sensitive framework in which each measurement event is associated with an event-dependent symmetry action on the hidden-variable space. Within this framework, symmetry-compatible joint representability requires that these actions be compatible with a single equivariant hidden-variable model. We show that, for the symmetry structures considered here, no such common realization exists. Consequently, the standard jointly represented CHSH construction cannot be carried out within this symmetry-restricted class. The obstruction is algebraic rather than causal: spacelike separation rules out direct causal influence between the measurement events, but does not ensure compatibility of their symmetry actions. The analysis brings into view a representational structure beneath the standard Bell construction: joint representability is not only a probabilistic requirement, but also a question of symmetry compatibility. This perspective clarifies the assumptions underlying Bell’s theorem and highlights the role of symmetry in determining which hidden-variable models are physically realizable. This is version 1.0 of the manuscript.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
It is shown that a standard one-dimensional coined discrete-time quantum walk can generate operationally admissible post-quantum correlations in a coin-position Bell scenario, without any modification of its unitary nearest-neighbor dynamics. Post-quantum features enter exclusively through an extended operational preparation of the coin, described by a complementarity-violating Hermitian trace-one operator, while physical consistency is enforced solely at the level of observable statistics via admissibility and no-signaling. The extended preparation admits an experimental emulation through a two-component quasiprobability reconstruction over physical coin states, at the price of an increased sampling overhead. The walk-generated coin-position entanglement can support CHSH values exceeding Tsirelson's bound, even though the walk dynamics remains fully standard. We also show that physically natural coarse-grained position measurements can render such post-quantum correlations operationally inaccessible, strongly suppressing observable Bell violations. The purpose here is to contrast the separation between the existence of post-quantum behavior and its accessibility under realistic measurement constraints.