# Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant $K_{G}$ with the following property. If Mi,j is an n by n (real or complex) matrix with

$\left|\sum _{i,j}M_{ij}s_{i}t_{j}\right|\leq 1$ for all (real or complex) numbers si, tj of absolute value at most 1, then

$\left|\sum _{i,j}M_{ij}\langle S_{i},T_{j}\rangle \right|\leq K_{G}$ ,

for all vectors Si, Tj in the unit ball B(H) of a (real or complex) Hilbert space H, the constant $K_{G}$ being independent of n. For a fixed Hilbert space dimension d, the smallest constant which satisfies this property for all n by n matrices is called a Grothendieck constant and denoted $K_{G}(d)$ . In fact there are two Grothendieck constants, $K_{G}^{\mathbb {R} }(d)$ and $K_{G}^{\mathbb {C} }(d)$ , depending on whether one works with real or complex numbers, respectively.

The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the existence of the constants in a paper published in 1953.

## Bounds on the constants

The sequences $K_{G}^{\mathbb {R} }(d)$ and $K_{G}^{\mathbb {C} }(d)$ are easily seen to be increasing, and Grothendieck's result states that they are bounded, so they have limits.

With $K_{G}^{\mathbb {R} }$ defined to be $\textstyle \sup _{d}K_{G}^{\mathbb {R} }(d)$ then Grothendieck proved that: $1.57\approx {\frac {\pi }{2}}\leq K_{G}^{\mathbb {R} }\leq \mathrm {sinh} ({\frac {\pi }{2}})\approx 2.3$ .

Krivine (1979) improved the result by proving: $K_{G}^{\mathbb {R} }\leq {\frac {\pi }{2\ln(1+{\sqrt {2}})}}\approx 1.7822$ , conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).

## Grothendieck constant of order d

Boris Tsirelson showed that the Grothendieck constants $K_{G}^{\mathbb {R} }(d)$ play an essential role in the problem of quantum nonlocality: the Tsirelson bound of any full correlation bipartite Bell inequality for a quantum system of dimension d is upperbounded by $K_{G}^{\mathbb {R} }(2d^{2})$ .

### Lower bounds

Some historical data on best known lower bounds of $K_{G}^{\mathbb {R} }(d)$ is summarized in the following table.

d Grothendieck, 1953 Krivine, 1979 Davie, 1984 Fishburn et al., 1994 Vértesi, 2008 Briët et al., 2011 Hua et al., 2015 Diviánszky et al., 2017
2 ${\sqrt {2}}$ ≈ 1.41421
3 1.41724 1.41758 1.4359
4 1.44521 1.44566 1.4841
5 ${\frac {10}{7}}$ ≈ 1.42857 1.46007 1.46112
6 1.47017
7 1.46286 1.47583
8 1.47586 1.47972
9 1.48608
...
${\frac {\pi }{2}}$ ≈ 1.57079 1.67696

### Upper bounds

Some historical data on best known upper bounds of $K_{G}^{\mathbb {R} }(d)$ :

d Grothendieck, 1953 Rietz, 1974 Krivine, 1979 Braverman et al., 2011 Hirsch et al., 2016
2 ${\sqrt {2}}$ ≈ 1.41421
3 1.5163 1.4644
4 ${\frac {\pi }{2}}$ ≈ 1.5708
...
8 1.6641
...
$\mathrm {sinh} \left({\frac {\pi }{2}}\right)$ ≈ 2.30130 2.261 ${\frac {\pi }{2\ln(1+{\sqrt {2}})}}$ ≈ 1.78221 ${\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon$ 