Grothendieck inequality


In mathematics, the Grothendieck inequality states that there is a universal constant with the following property. If Mi,j is an n by n matrix with
for all numbers si, tj of absolute value at most 1, then
for all vectors Si, Tj in the unit ball B of a Hilbert space H, the constant 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. In fact there are two Grothendieck constants, and, 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 and are easily seen to be increasing, and Grothendieck's result states that they are bounded, so they have limits.
With defined to be then Grothendieck proved that:.
improved the result by proving:, conjecturing that the upper bound is tight. However, this conjecture was disproved by.

Grothendieck constant of order ''d''

showed that the Grothendieck constants 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.

Lower bounds

Some historical data on best known lower bounds of is summarized in the following table.
Implied bounds are shown in italics.
dGrothendieck, 1953Clauser et al., 1969Davie, 1984Fishburn et al., 1994Vértesi, 2008Briët et al., 2011Hua et al., 2015Diviánszky et al., 2017
2≈ 1.41421
31.414211.417241.417581.4359
41.445211.445661.4841
5≈ 1.428571.460071.461121.4841
61.460071.470171.4841
71.462861.475831.4841
81.475861.479721.4841
91.48608
...--------
∞≈ 1.570791.67696

Upper bounds

Some historical data on best known upper bounds of :
dGrothendieck, 1953Rietz, 1974Krivine, 1979Braverman et al., 2011Hirsch et al., 2016
2≈ 1.41421
31.51631.4644
4≈ 1.5708
...-----
81.6641
...-----
∞≈ 2.301302.261≈ 1.78221