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In [[mathematics]], the '''Grothendieck inequality''' states that there is a universal constant ''k'' with the following property. If ''a''<sub>''i'',''j''</sub> is an ''n'' by ''n'' ([[real number|real]] or [[complex number|complex]]) [[matrix (mathematics)|matrix]] with
:<math> \left| \sum_{i,j} a_{ij} s_i t_j \right|\le 1</math>

for all (real or complex) numbers ''s''<sub>''i''</sub>, ''t''<sub>''j''</sub> of absolute value at most 1, then
:<math> \left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|\le k</math>,

for all [[vector (mathematics)|vectors]] ''S''<sub>''i''</sub>, ''T''<sub>''j''</sub> 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 ''k''<sub>'''R'''</sub>(''n'') and ''k''<sub>'''C'''</sub>(''n'') for each ''n'' depending on whether one works with real or complex numbers, respectively.<ref>{{Citation | last = Pisier  | first = Gilles  | authorlink = Gilles Pisier |date=April 2012 | title = Grothendieck's Theorem, Past and Present  | journal = [[Bulletin of the American Mathematical Society]]  | volume = 49 | issue = 2  | pages = 237–323  | doi = 10.1090/S0273-0979-2011-01348-9 | arxiv = 1101.4195}}.</ref>

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.<ref name="a"/>

==Bounds on the constants==
The sequences ''k''<sub>'''R'''</sub>(''n'') and ''k''<sub>'''C'''</sub>(''n'') are easily seen to be increasing, and Grothendieck's result states that they are [[bounded sequence|bounded]],<ref name="a">{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Résumé de la théorie métrique des produits tensoriels topologiques | mr=0094682  | year=1953 | journal=Bol. Soc. Mat. Sao Paulo | volume=8 | pages=1–79}}</ref><ref>{{Citation | last1=Blei | first1=Ron C. | title=An elementary proof of the Grothendieck inequality | jstor=2046119 | mr=883401  | year=1987 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=100 | issue=1 | pages=58–60 | doi=10.2307/2046119 | publisher=American Mathematical Society}}</ref> so they have [[limit of a sequence|limits]].

With ''k''<sub>'''R'''</sub> defined to be sup<sub>''n''</sub> ''k''<sub>'''R'''</sub>(''n'')<ref>{{Citation | last1=Finch | first1=Steven R. | title=Mathematical constants | publisher=[[Cambridge University Press]] | isbn=978-0-521-81805-6 | year=2003}}</ref> then Grothendieck proved that: <math> 1.57 \approx \frac{\pi}{2} \leq k_{\R} \leq \mathrm{sinh}(\frac{\pi}{2}) \approx 2.3</math>.

{{harvtxt|Krivine|1979}}<ref name="krivine">{{Citation | last1=Krivine | first1=J.-L. | title=Constantes de Grothendieck et fonctions de type positif sur les sphères | doi=10.1016/0001-8708(79)90017-3 | mr=521464  | year=1979 | journal=Advances in Mathematics | issn=0001-8708 | volume=31 | issue=1 | pages=16–30}}</ref>  improved the result by proving: ''k''<sub>'''R'''</sub> ≤ 1.7822139781...=<math> \frac{\pi}{2 \ln(1+\sqrt{2})}</math>, conjecturing that the upper bound is tight. However, this conjecture was disproved by {{harvtxt|Braverman|Makarychev|Makarychev|Naor|2011}}.<ref name="braverman">{{citation
 | last1 = Braverman | first1 = Mark
 | last2 = Makarychev | first2 = Konstantin
 | last3 = Makarychev | first3 = Yury
 | last4 = Naor | first4 = Assaf
 | arxiv = 1103.6161
 | contribution = The Grothendieck Constant is Strictly Smaller than Krivine's Bound
 | doi = 10.1109/FOCS.2011.77
 | pages = 453–462
 | title = [[Symposium on Foundations of Computer Science|52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS)]]
 | year = 2011}}</ref>

==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 ''k''<sub>'''R'''</sub>(''n'', ''d'') and ''k''<sub>'''C'''</sub>(''n'', ''d'') for the real and complex case, respectively. With increasing ''d'' these constants are monotone increasing and their limit is ''k''<sub>'''R'''</sub>(''n'') and ''k''<sub>'''C'''</sub>(''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 ''k''<sub>'''R'''</sub>(∞, ''d'') and ''k''<sub>'''C'''</sub>(∞, ''d''), respectively.

The Grothendieck constant ''k''<sub>'''R'''</sub>(∞, 3) plays an essential role in the [[quantum nonlocality]] problem of the two-qubit [[Werner state]]s.
<ref>{{Citation | last1=Acín | first1=Antonio | last2=Gisin | first2=Nicolas | last3=Toner | first3=Benjamin  | title=Grothendieck’s constant and local models for noisy entangled quantum states | publisher=[[Physical Review A]] | year=2006| arxiv=quant-ph/0606138 }}</ref>

===Lower bounds===

Some historical data on best known lower bounds of ''k''<sub>'''R'''</sub>(∞, ''d'') is summarized in the following table.
Implied bounds are shown in italics.

{| class="wikitable"
! ''d''
! Grothendieck, 1953<ref name="a"/>
! Clauser et al., 1969<ref>{{Citation | last1=Clauser | first1=John F. | last2=Horne | first2=Michael A. | last3=Shimony | first3=Abner | last4=Holt | first4=Richard A. | title=Proposed Experiment to Test Local Hidden-Variable Theories | publisher=[[Physical Review Letters]] | volume = 23 | pages = 880 | year=1969}}</ref>
! Davie, 1984<ref>{{Citation | last1=Davie | first1=A. M. | title=Unpublished | year=1984}}</ref>
! Fishburn et al., 1994<ref>{{Citation | last1=Fishburn | first1=P. C. | last2=Reeds | first2=J. A. | title=Bell Inequalities, Grothendieck’s Constant, and Root Two | publisher=[[SIAM Journal on Discrete Mathematics]]  | volume = 7 | issue = 1 | pages = 48–56 | year=1994 | doi = 10.1137/S0895480191219350}}</ref>
! Vértesi, 2008<ref>{{Citation | last1=Vértesi | first1=Tamás | title=More efficient Bell inequalities for Werner states | publisher=[[Physical Review A]] | year=2008| arxiv=0806.0096 }}</ref>
! Briët et al., 2011<ref>{{Citation | last1=Briët | first1=Jop | last2=Buhrman | first2=Harry | last3=Toner | first3=Ben  | title=A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement | publisher=[[Communications in Mathematical Physics]] | year=2011}}</ref>
! Hua et al., 2015<ref>{{Citation | last1=Hua | first1=Bobo | last2=Li | first2=Ming | last3=Zhang | first3=Tinggui | last4=Zhou | first4=Chunqin | last5=Li-Jost | first5=Xianqing | last6=Fei | first6=Shao-Ming  | title=Towards Grothendieck Constants and LHV Models in Quantum Mechanics | publisher=[[Journal of Physics A]] | volume=48 | issue=6 | pages=065302 | doi=10.1088/1751-8113/48/6/065302 | year=2015| arxiv=1501.05507 | bibcode=2015JPhA...48f5302H }}</ref>
! Diviánszky et al., 2017<ref>{{Citation | last1=Diviánszky | first1=Péter | last2=Bene | first2=Erika | last3=Vértesi | first3=Tamás | title=Qutrit witness from the Grothendieck constant of order four | publisher=[[Physical Review A]] | volume=96 | pages=012113 | year=2017}}</ref>
|-
! 2
| || <math>\sqrt{2}</math> ≈ 1.41421 || || || || || ||
|-
! 3
| || ''1.41421'' || || || 1.41724 ||  || 1.41758 || 1.4359
|-
! 4
| || || || || 1.44521 ||  || 1.44566 || 1.4841
|-
! 5
| || || || <math>\frac{10}{7}</math> ≈ 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 || ||
|-
! ...
|-
! ∞
| <math>\frac{\pi}{2}</math> ≈ 1.57079 || || 1.67696 || || || || ||
|}

===Upper bounds===

Some historical data on best known upper bounds of ''k''<sub>'''R'''</sub>(∞, ''d''):

{| class="wikitable"
! ''d''
! Grothendieck, 1953<ref name="a"/>
! Rietz, 1974<ref>{{Citation | last1=Rietz | first1=Ronald E. | title=A proof of the Grothendieck inequality | publisher=[[Israel Journal of Mathematics]]  | volume = 19 | issue = 3 | pages = 271–276 | year=1974 | doi = 10.1007/BF02757725}}</ref>
! Krivine, 1979<ref name="krivine"/>
! Braverman et al., 2011<ref name="braverman"/>
! Hirsch et al., 2016<ref>{{Citation | last1=Hirsch | first1=Flavien | last2=Quintino | first2=Marco Túlio | last3=Vértesi | first3=Tamás | last4=Navascués | first4=Miguel | last5=Brunner | first5=Nicolas | title=Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant | arxiv = 1609.06114 | url=https://arxiv.org/pdf/1609.06114.pdf| bibcode=2016arXiv160906114H }}</ref>
|-
! 2
| || || <math>\sqrt{2}</math> ≈ 1.41421 || ||
|-
! 3
| || || 1.5163 || || 1.4644
|-
! 4
| || || <math>\frac{\pi}{2}</math> ≈ 1.5708 || ||
|-
! ...
|-
! 8
| || || 1.6641 || ||
|-
! ...
|-
! ∞
| <math>\mathrm{sinh}\left(\frac{\pi}{2}\right)</math> ≈ 2.30130 || 2.261 || <math>\frac{\pi}{2\ln(1+\sqrt{2})}</math> ≈ 1.78221 || <math>\frac{\pi}{2\ln(1+\sqrt{2})}-\varepsilon</math> ||
|}

==See also==
*[[Pisier–Ringrose inequality]]

==References==
{{Reflist}}

==External links==
*{{MathWorld|urlname=GrothendiecksConstant|title=Grothendieck's Constant}} (NB: the historical part is not exact there.)

[[Category:Theorems in functional analysis]]
[[Category:Inequalities]]