arXiv:2609.25462

Sharper Bounds for the Complex Grothendieck Constant

Steven Heilman, Chris Jones, Giulio Malavolta

math.FAcs.DSquant-phSubmitted Sep 21, 2026• Announced Sep 23, 2026

Abstract

We show that $1.4<K_G^{\mathbb{C}}<1.404898554746$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of $1.40490913\ldots$ from 1987, and the lower bound improves on Davie's bound of $1.33807$ from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.

Comments: 21 pages

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arXiv:2609.25462

Sharper Bounds for the Complex Grothendieck Constant

Steven Heilman, Chris Jones, Giulio Malavolta

math.FAcs.DSquant-phSubmitted Sep 21, 2026• Announced Sep 23, 2026

arXiv HTML PDF

Abstract

We show that

1.4<K_G^{\mathbb{C}}<1.404898554746

, where

K_G^{\mathbb{C}}

is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of

1.40490913\ldots

from 1987, and the lower bound improves on Davie's bound of

1.33807

from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.

Comments: 21 pages

Related papers

Better bounds on finite-order Grothendieck constants

Sébastien Designolle, Tamás Vértesi, Sebastian Pokutta

Grothendieck constants $K_G(d)$ bound the advantage of $d$ -dimensional strategies over $1$ -dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$ , their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in\{3,4,5\}$ , we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$ . For $d\in\{4,7,8\}$ , we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex $d$ -dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$ .Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in\{3,4,5\}$, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$. For $d\in\{4,7,8\}$, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex $d$-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$.

arXiv:2409.03739Sep 5, 20240 citations

The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

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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

Bell Inequalities, Grothendieck’s Constant, and Root Two

P. C. Fishburn, James A. Reeds

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

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Apostol Vourdas

Abstract The Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA ${\mathcal{C}}$ ?> C that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form <?CDATA ${\mathcal{Q}}$ ?> Q that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when <?CDATA ${\mathcal{C}}\le 1$ ?> C ≤ 1 then <?CDATA ${\mathcal{Q}}$ ?> Q might take values greater than 1, up to the complex Grothendieck constant kG . Previous work in a quantum context, used Grothendieck’s theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with <?CDATA ${\mathcal{Q}}\in (1,{k}_{G})$ ?> Q ∈ ( 1 , k G ) , which is a classically forbidden region in the sense that <?CDATA ${\mathcal{C}}$ ?> C cannot take values in it.Abstract The Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA ${\mathcal{C}}$?> C that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form <?CDATA ${\mathcal{Q}}$?> Q that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when <?CDATA ${\mathcal{C}}\le 1$?> C ≤ 1 then <?CDATA ${\mathcal{Q}}$?> Q might take values greater than 1, up to the complex Grothendieck constant kG . Previous work in a quantum context, used Grothendieck’s theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with <?CDATA ${\mathcal{Q}}\in (1,{k}_{G})$?> Q ∈ ( 1 , k G ) , which is a classically forbidden region in the sense that <?CDATA ${\mathcal{C}}$?> C cannot take values in it.

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Heuristic lower bounds on real Grothendieck constants of finite and infinite order

Erika Bene, Tamás Vértesi

We propose candidate lower bounds on the real Grothendieck constants $K_{\mathrm G}(d)$ and their infinite-dimensional limit $K_{\mathrm G}$ using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply $K_{\mathrm G}\geq9π/16\simeq1.767146$ , improving the rigorous lower bound $6π/11\simeq1.713596$ . We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$ . Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on $K_{\mathrm G}(d)$ in the studied dimensions from $4$ to $24$ , conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.We propose candidate lower bounds on the real Grothendieck constants $K_{\mathrm G}(d)$ and their infinite-dimensional limit $K_{\mathrm G}$ using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply $K_{\mathrm G}\geq9π/16\simeq1.767146$, improving the rigorous lower bound $6π/11\simeq1.713596$. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on $K_{\mathrm G}(d)$ in the studied dimensions from $4$ to $24$, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.

quant-pharXiv:2609.35399Sep 29, 20260 citations

We show that

1.4<K_G^{\mathbb{C}}<1.404898554746

, where

K_G^{\mathbb{C}}

is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of

1.40490913\ldots

from 1987, and the lower bound improves on Davie's bound of

1.33807

from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.

Grothendieck constants

K_G(d)

bound the advantage of

d

-dimensional strategies over

1

-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from

d=2

, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For

d\in\{3,4,5\}

, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for

d\leqslant9

. For

d\in\{4,7,8\}

, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants

K_G(d\mapsto2)

as the advantage of complex

d

-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on

K_G(d\mapsto2)

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.

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 $.

Abstract The Grothendieck theorem considers a ‘classical’ quadratic form <?CDATA

{\mathcal{C}}

?> C that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form <?CDATA

{\mathcal{Q}}

?> Q that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when <?CDATA

{\mathcal{C}}\le 1

?> C ≤ 1 then <?CDATA

{\mathcal{Q}}

?> Q might take values greater than 1, up to the complex Grothendieck constant kG . Previous work in a quantum context, used Grothendieck’s theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with <?CDATA

{\mathcal{Q}}\in (1,{k}_{G})

?> Q ∈ ( 1 , k G ) , which is a classically forbidden region in the sense that <?CDATA

{\mathcal{C}}

?> C cannot take values in it.

We propose candidate lower bounds on the real Grothendieck constants

K_{\mathrm G}(d)

and their infinite-dimensional limit

K_{\mathrm G}

using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply

K_{\mathrm G}\geq9π/16\simeq1.767146

, improving the rigorous lower bound

6π/11\simeq1.713596

. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to

24

. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on

K_{\mathrm G}(d)

in the studied dimensions from

4

24

, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.