The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$ , where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

math.FACross-listedarXiv:2603.30039Apr 1, 2026

Hadamard’s Matrices, Grothendieck’s Constant, and Root Two

Dominique Fortin

Springer optimization and its applicationsJan 1, 20100 citations

Approximating the little Grothendieck problem over the orthogonal and unitary groups

Afonso S. Bandeira, Christopher Kennedy, Amit Singer

arXiv:1308.5207Mathematical ProgrammingMar 25, 201637 citations

Simulating Quantum Correlations with Finite Communication

Oded Regev, Ben Toner

Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.

Annual Symposium on Foundations of Computer ScienceOct 1, 200714 citations

Approximating the little Grothendieck problem over the orthogonal and unitary groups

Christopher Kennedy, Amit Singer, Afonso S. Bandeira

Springer eBooksMar 1, 20160 citations

The Grothendieck constant

K_{G}

is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of

K_{G}

is unknown. The best known lower bound on

K_{G}

was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that

K_{G} \ge K_{DR} + 10^{-12}

, where

K_{DR}

denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has

Ω(1)

weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.