We reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a novel proof of the Tsirelson bound. In particular, we show that unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.
We reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a novel proof of the Tsirelson bound. In particular, we show that unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.
FINDING: CHSH correlations are bounded by Tsirelson's 2√2, and a 2025 preprint re-derives this bound from indivisible stochastic processes via Barandes's stochastic-quantum correspondence, replacing the no-signaling axiom with causal locality. | MATH: CHSH operator S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′); classical (local hidden variable) bound |S| ≤ 2; quantum bound |S| ≤ 2√2 ≈ 2.828427; quantum value for maximally entangled qubits at optimal angles = 2√2. Optimal measurement angles: a=0, a′=π/2, b=π/4, b′=−π/4 (mod π/2). Maximal violation ratio: 2√2/2 = √2 ≈ 1.414214. | CONNECTION: 2√2 = 2·√2 is the diagonal of a unit square scaled by 2, i.e. the length of the space diagonal of a 2×2 square lattice cell — a √2-scaling, not a φ-scaling. √2 is the fundamental ratio of the square lattice / BCC root system (A₁⊕A₁, B₂). No direct φ (1.618), 0.618, or 0.382 relation appears in the CHSH bound itself. The optimal angles π/4, π/2 are base-4 / base-8 (octonionic) divisions of the circle, con Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: CHSH correlations are bounded by Tsirelson's 2√2, and a 2025 reformulation derives this bound from indivisible stochastic processes via causal locality rather than no-signaling alone. | MATH: CHSH operator S = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁; classical (local hidden variable) bound |S| ≤ 2; quantum (Tsirelson) bound |S| ≤ 2√2 ≈ 2.828427. Maximal violation angle: measurement settings at 45° increments (π/4), giving correlation E(θ) = −cos(2θ) for singlet state, yielding S = 2√2. | CONNECTION: 2√2 = 2·√2 is the diagonal of a 2×2 square (Pythagorean), and √2 is the fundamental ratio of the square lattice / octahedral root system B₂. The factor 2√2 = √8 relates to the 45° rotation symmetry of the CHSH setting plane. No direct appearance of 0.618/1.618 family, but the extremal configuration is a 45°-rotated square — a crystallographic C₄ symmetry. | DEPTH: 7 FINDING: Loophole-free photonic entanglement has been demonstrated over ~1200 km via satellite (Micius), with detector efficiency Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound 22 for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable λ. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP∗=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 22, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 22. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson’s bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order. Quantum theory has been shown to be compatible with processes violating a definite causal order, but no detailed analysis of the maximum allowed degree of violation has been carried out so far. Here, the authors fill this gap by providing the analogue of Tsirelson’s bound for the violation of causal inequalities.