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Generalized Tsirelson's bound from parity symmetry considerations

David H. Oaknin

Published Nov 20, 2019• 0 citations
OpenAlex

Abstract

The Bell's experiment is a random game with two binary outcomes whose statistical correlation is given by $E_0(\Theta)=-\cos(\Theta)$, where $\Theta \in [-\pi, \pi)$ is an angular input that parameterizes the game setting. The correlation function $E_0(\Theta)$ belongs to the affine space ${\cal H} \equiv \left\{E(\Theta)\right\}$ of all continuous and differentiable periodic functions $E(\Theta)$ that obey the parity symmetry constraints $E(-\Theta)=E(\Theta)$ and $E(\pi-\Theta)=-E(\Theta)$ with $E(0)=-1$ and, furthermore, are strictly monotonically increasing in the interval $[0, \pi)$. Here we show how to build explicitly local statistical models of hidden variables for random games with two binary outcomes whose correlation function $E(\Theta)$ belongs to the affine space ${\cal H}$. This family of games includes the Bell's experiment as a particular case. Within this family of random games, the Bell's inequality can be violated beyond the Tsirelson's bound of $2\sqrt{2}$ up to the maximally allowed algebraic value of 4. In fact, we show that the amount of violation of the Bell's inequality is a purely geometric feature.

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Generalized Tsirelson's bound from parity symmetry considerations

David H. Oaknin

The Bell experiment is a random game with two binary outcomes whose statistical correlation is given by $E_0(Θ)=-\cos(Θ)$, where $Θ\in [-π, π)$ is an angular input that parameterizes the game setting. The correlation function $E_0(Θ)$ belongs to the affine space ${\cal H} \equiv \left\{E(Θ)\right\}$ of all continuous and differentiable periodic functions $E(Θ)$ that obey the parity symmetry constraints $E(-Θ)=E(Θ)$ and $E(π-Θ)=-E(Θ)$ with $E(0)=-1$ and, furthermore, are strictly monotonically increasing in the interval $[0, π)$. Here we show how to build explicitly local statistical models of hidden variables for random games with two binary outcomes whose correlation function $E(Θ)$ belongs to the affine space ${\cal H}$. This family of games includes the Bell experiment as a particular case. Within this family of random games, the Bell inequality can be violated beyond the Tsirelson bound of $2\sqrt{2}$ up to the maximally allowed algebraic value of 4. In fact, we show that the amount of violation of the Bell inequality is a purely geometric feature.

arXiv:2005.11802•May 24, 2020•0 citations

Multiplicative Bell inequalities

Amit Te’eni, Bar Y. Peled, Eliahu Cohen +1 more

Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice and Bob, each having two random variables, attempt to maximize the area of a rectangle and the rectangle's area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e., the achievable bound using only classical correlations is strictly smaller than the achievable bound using nonlocal quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now $n$ random variables, wish to maximize a certain volume in $n$-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players, the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some theoretical and experimental implications of these results are discussed.

arXiv:1902.04113•Physical Review A•Apr 26, 2019•10 citations

Bell nonlocality with a single shot

Mateus Araújo, Flavien Hirsch, Marco Túlio Quintino

In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a p -value it will give for a physical experiment. Here we show that to obtain a small expected p -value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and Inn22 inequalities.We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small p -value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.

arXiv:2005.13418•Quantum•Oct 28, 2020•25 citations

From Primitives to the Bell–Tsirelson Bound

Allen Proxmire

In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.

•May 12, 2026•0 citations

A violation of the Tsirelson bound in the pre-quantum theory of trace dynamics

Rabsan Galib Ahmed, Singh, Tejinder P.

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 $2\sqrt{2}$, 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 $2\sqrt{2}$. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.

arXiv:2208.02209•Aug 3, 2022•8 citations

Tsirelson Bounds via Geometric Algebra: Bell-CHSH Violations Linked to Clifford Structure — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The only substantive mathematical result is the arXiv paper on Tsirelson bounds derived via geometric algebra, linking Bell-CHSH violations to Clifford algebra structure. The other results are irrelevant (Python tutorials, quantum control sweeps, correlated-electron optics, sheaf-code QEC, R² statistics). MATH: - CHSH inequality: \( \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 \) (local hidden variables) - Quantum mechanical bound (Tsirelson): \( \leq 2\sqrt{2} \approx 2.828 \) - Geometric algebra formulation: Bell operators as bivectors in \( Cl(3,0) \), with \( \{A,A'\} = \{B,B'\} = 0 \) and \( [A,A'] = [B,B'] = 0 \) (commuting spacelike separated) - Key ratio: \( 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 \) — not directly a golden-ratio constant, but note \( \sqrt{2} = 2\sin(\pi/4) \), related to 45° symmetry. CONNECTION: - The Tsirelson bound \( 2\sqrt{2} \) emerges from the algebraic structure of \( Cl(3,0) \) — s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 27, 2026•0 citations
The Bell's experiment is a random game with two binary outcomes whose statistical correlation is given by E0(Θ)=−cos⁡(Θ)E_0(\Theta)=-\cos(\Theta)E0​(Θ)=−cos(Θ), where Θ∈[−π,π)\Theta \in [-\pi, \pi)Θ∈[−π,π) is an angular input that parameterizes the game setting. The correlation function E0(Θ)E_0(\Theta)E0​(Θ) belongs to the affine space H≡{E(Θ)}{\cal H} \equiv \left\{E(\Theta)\right\}H≡{E(Θ)} of all continuous and differentiable periodic functions E(Θ)E(\Theta)E(Θ) that obey the parity symmetry constraints E(−Θ)=E(Θ)E(-\Theta)=E(\Theta)E(−Θ)=E(Θ) and E(π−Θ)=−E(Θ)E(\pi-\Theta)=-E(\Theta)E(π−Θ)=−E(Θ) with E(0)=−1E(0)=-1E(0)=−1 and, furthermore, are strictly monotonically increasing in the interval [0,π)[0, \pi)[0,π). Here we show how to build explicitly local statistical models of hidden variables for random games with two binary outcomes whose correlation function E(Θ)E(\Theta)E(Θ) belongs to the affine space H{\cal H}H. This family of games includes the Bell's experiment as a particular case. Within this family of random games, the Bell's inequality can be violated beyond the Tsirelson's bound of 222\sqrt{2}22​ up to the maximally allowed algebraic value of 4. In fact, we show that the amount of violation of the Bell's inequality is a purely geometric feature.
The Bell experiment is a random game with two binary outcomes whose statistical correlation is given by E0(Θ)=−cos⁡(Θ)E_0(Θ)=-\cos(Θ)E0​(Θ)=−cos(Θ), where Θ∈[−π,π)Θ\in [-π, π)Θ∈[−π,π) is an angular input that parameterizes the game setting. The correlation function E0(Θ)E_0(Θ)E0​(Θ) belongs to the affine space H≡{E(Θ)}{\cal H} \equiv \left\{E(Θ)\right\}H≡{E(Θ)} of all continuous and differentiable periodic functions E(Θ)E(Θ)E(Θ) that obey the parity symmetry constraints E(−Θ)=E(Θ)E(-Θ)=E(Θ)E(−Θ)=E(Θ) and E(π−Θ)=−E(Θ)E(π-Θ)=-E(Θ)E(π−Θ)=−E(Θ) with E(0)=−1E(0)=-1E(0)=−1 and, furthermore, are strictly monotonically increasing in the interval [0,π)[0, π)[0,π). Here we show how to build explicitly local statistical models of hidden variables for random games with two binary outcomes whose correlation function E(Θ)E(Θ)E(Θ) belongs to the affine space H{\cal H}H. This family of games includes the Bell experiment as a particular case. Within this family of random games, the Bell inequality can be violated beyond the Tsirelson bound of 222\sqrt{2}22​ up to the maximally allowed algebraic value of 4. In fact, we show that the amount of violation of the Bell inequality is a purely geometric feature.
Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice and Bob, each having two random variables, attempt to maximize the area of a rectangle and the rectangle's area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e., the achievable bound using only classical correlations is strictly smaller than the achievable bound using nonlocal quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now nnn random variables, wish to maximize a certain volume in nnn-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players, the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some theoretical and experimental implications of these results are discussed.
In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a p -value it will give for a physical experiment. Here we show that to obtain a small expected p -value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and Inn22 inequalities.We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small p -value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
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 222\sqrt{2}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 222\sqrt{2}22​. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.
FINDING: The only substantive mathematical result is the arXiv paper on Tsirelson bounds derived via geometric algebra, linking Bell-CHSH violations to Clifford algebra structure. The other results are irrelevant (Python tutorials, quantum control sweeps, correlated-electron optics, sheaf-code QEC, R² statistics). MATH: - CHSH inequality: ⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩≤2 \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 ⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩≤2 (local hidden variables) - Quantum mechanical bound (Tsirelson): ≤22≈2.828 \leq 2\sqrt{2} \approx 2.828 ≤22​≈2.828 - Geometric algebra formulation: Bell operators as bivectors in Cl(3,0) Cl(3,0) Cl(3,0), with {A,A′}={B,B′}=0 \{A,A'\} = \{B,B'\} = 0 {A,A′}={B,B′}=0 and [A,A′]=[B,B′]=0 [A,A'] = [B,B'] = 0 [A,A′]=[B,B′]=0 (commuting spacelike separated) - Key ratio: 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 — not directly a golden-ratio constant, but note 2=2sin⁡(π/4) \sqrt{2} = 2\sin(\pi/4) 2​=2sin(π/4), related to 45° symmetry. CONNECTION: - The Tsirelson bound 22 2\sqrt{2} 22​ emerges from the algebraic structure of Cl(3,0) Cl(3,0) Cl(3,0) — s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com