# Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant k with the following property. If ai,j is an n by n (real or complex) matrix with

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

$\left|\sum _{i,j}a_{ij}\langle S_{i},T_{j}\rangle \right|\leq k$ ,

for all vectors Si, Tj in the unit ball B(H) of a (real or complex) Hilbert space H, the constant k being independent of n. For a fixed n, the smallest constant which satisfies this property for all n by n matrices is called a Grothendieck constant and denoted k(n). In fact there are two Grothendieck constants kR(n) and kC(n) for each n 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 inequality and the existence of the constants in a paper published in 1953.

## Bounds on the constants

The sequences kR(n) and kC(n) are easily seen to be increasing, and Grothendieck's result states that they are bounded, so they have limits.

With kR defined to be supn kR(n) then Grothendieck proved that: $1.57\approx {\frac {\pi }{2}}\leq k_{\mathbb {R} }\leq \mathrm {sinh} ({\frac {\pi }{2}})\approx 2.3$ .

Krivine (1979) improved the result by proving: kR ≤ 1.7822139781...=${\frac {\pi }{2\ln(1+{\sqrt {2}})}}$ , conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).

## Grothendieck constant of order d

If we replace the (real or complex) Hilbert space H in the above definition with a (real or complex) d-dimensional Euclidean space, we get the constants kR(n, d) and kC(n, d) for the real and complex case, respectively. With increasing d these constants are monotone increasing and their limit is kR(n) and kC(n), respectively. For each d, with increasing n the constants are also increasing and their limit is the Grothendieck constant of order d which can be denoted as kR(∞, d) and kC(∞, d), respectively.

The Grothendieck constant kR(∞, 3) plays an essential role in the quantum nonlocality problem of the two-qubit Werner states. 

### Lower bounds

Some historical data on best known lower bounds of kR(∞, d) is summarized in the following table. Implied bounds are shown in italics.

d Grothendieck, 1953 Clauser et al., 1969 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 1.41421 1.41724 1.41758 1.4359
4 1.44521 1.44566 1.4841
5 ${\frac {10}{7}}$ ≈ 1.42857 1.46007 1.46112 1.4841
6 1.46007 1.47017 1.4841
7 1.46286 1.47583 1.4841
8 1.47586 1.47972 1.4841
9 1.48608
...
${\frac {\pi }{2}}$ ≈ 1.57079 1.67696

### Upper bounds

Some historical data on best known upper bounds of kR(∞, 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 ${\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon$ 