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

Bell inequalities and the Grothendieck-Tsirelson inequality

Guillaume Aubrun, Stanisław Szarek

Mathematical surveys and monographsAug 30, 20170 citations

Approximating the little Grothendieck problem over the orthogonal and unitary groups

Christopher Kennedy, Amit Singer, Afonso S. Bandeira

Springer eBooksMar 1, 20160 citations

Grothendieck’s constant and local models for noisy entangled quantum states

Antonio Acín, Nicolas Gisin, Ben Toner

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ${\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4$, we show that there is a local model for projective measurements if and only if $p\ensuremath{\leqslant}1∕{K}_{G}(3)$, where ${K}_{G}(3)$ is Grothendieck's constant of order 3. Known bounds on ${K}_{G}(3)$ prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary quantum states.We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ${\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4$, we show that there is a local model for projective measurements if and only if $p\ensuremath{\leqslant}1∕{K}_{G}(3)$, where ${K}_{G}(3)$ is Grothendieck's constant of order 3. Known bounds on ${K}_{G}(3)$ prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary quantum states.

arXiv:quant-ph/0606138Physical Review AJun 9, 2006200 citations

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

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)

{K}_{G}(3)

is Grothendieck's constant of order 3. Known bounds on

{K}_{G}(3)

prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary quantum states.

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.