B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite $n \times n$ real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of $n \times n$ matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite $n \times n$ real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of $n \times n$ matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.

SIAM Journal on Discrete MathematicsFeb 1, 199450 citations

Bell inequalities for arbitrary situations

Mohammad Heshmatpour

Physics Letters AJan 19, 20153 citations

Quantifying quantum nonlocality

Ben Toner

Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.

Aug 12, 20132 citations

All the Bell Inequalities

Asher Peres

Foundations of PhysicsApr 1, 1999222 citations

Unifying Bell Inequalities via Polytopes and Quantum Communication Advantages — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Buscemi unifies Bell inequalities via a generalized framework where quantum entanglement and nonlocality emerge from a single polytope structure, with violations tied to communication complexity advantages. | MATH: Bell inequalities are facets of the local polytope $\mathcal{P}_{\text{loc}} = \text{conv}\{p(a,b|x,y) = p(a|x)p(b|y)\}$ . Buscemi's unification introduces "nonlocal games" with quantum strategies yielding success probabilities $\omega_q(G) > \omega_c(G)$ , where the gap is bounded by the Grothendieck constant $K_G$ (1.676 ≤ $K_G$ ≤ 1.782) for correlation-type inequalities. The facet structure is characterized by the dimension of the probability simplex and the rank of the correlation matrix $C_{ab} = \langle A_x B_y \rangle$ , with Tsirelson bounds $\langle B \rangle_q \leq 2\sqrt{2}$ for CHSH. | CONNECTION: The polytope facets correspond to root systems of type $A_n$ (for n-party correlations) and the CHSH inequality's maximal violation $ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Buscemi unifies Bell inequalities via a generalized framework where quantum entanglement and nonlocality emerge from a single polytope structure, with violations tied to communication complexity advantages. | MATH: Bell inequalities are facets of the local polytope \( \mathcal{P}_{\text{loc}} = \text{conv}\{p(a,b|x,y) = p(a|x)p(b|y)\} $. Buscemi's unification introduces "nonlocal games" with quantum strategies yielding success probabilities $ \omega_q(G) > \omega_c(G) $, where the gap is bounded by the Grothendieck constant $ K_G $ (1.676 ≤ $ K_G $ ≤ 1.782) for correlation-type inequalities. The facet structure is characterized by the dimension of the probability simplex and the rank of the correlation matrix $ C_{ab} = \langle A_x B_y \rangle $, with Tsirelson bounds $ \langle B \rangle_q \leq 2\sqrt{2} $ for CHSH. | CONNECTION: The polytope facets correspond to root systems of type $ A_n $ (for n-party correlations) and the CHSH inequality's maximal violation \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 6, 20260 citations

B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite

n \times n

real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of

n \times n

matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.

\mathcal{P}_{\text{loc}} = \text{conv}\{p(a,b|x,y) = p(a|x)p(b|y)\}

. Buscemi's unification introduces "nonlocal games" with quantum strategies yielding success probabilities

\omega_q(G) > \omega_c(G)

, where the gap is bounded by the Grothendieck constant

K_G

(1.676 ≤

K_G

≤ 1.782) for correlation-type inequalities. The facet structure is characterized by the dimension of the probability simplex and the rank of the correlation matrix

C_{ab} = \langle A_x B_y \rangle

, with Tsirelson bounds

\langle B \rangle_q \leq 2\sqrt{2}

for CHSH. | CONNECTION: The polytope facets correspond to root systems of type

A_n

(for n-party correlations) and the CHSH inequality's maximal violation \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com