# 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

${\displaystyle \left|\sum _{i,j}a_{ij}s_{i}t_{j}\right|\leq 1}$

for all (real or complex) numbers si, tj of absolute value at most 1, then

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

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.[1]

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.[2]

## 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,[2][3] so they have limits.

With kR defined to be supn kR(n)[4] then Grothendieck proved that: ${\displaystyle 1.57\approx {\frac {\pi }{2}}\leq k_{\mathbb {R} }\leq \mathrm {sinh} ({\frac {\pi }{2}})\approx 2.3}$.

Krivine (1979)[5] improved the result by proving: kR ≤ 1.7822139781...=${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}}$, conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]

## 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. [7]

### 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[2] Clauser et al., 1969[8] Davie, 1984[9] Fishburn et al., 1994[10] Vértesi, 2008[11] Briët et al., 2011[12] Hua et al., 2015[13] Diviánszky et al., 2017[14]
2 ${\displaystyle {\sqrt {2}}}$ ≈ 1.41421
3 1.41421 1.41724 1.41758 1.4359
4 1.44521 1.44566 1.4841
5 ${\displaystyle {\frac {10}{7}}}$ ≈ 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
...
${\displaystyle {\frac {\pi }{2}}}$ ≈ 1.57079 1.67696

### Upper bounds

Some historical data on best known upper bounds of kR(∞, d):

d Grothendieck, 1953[2] Rietz, 1974[15] Krivine, 1979[5] Braverman et al., 2011[6] Hirsch et al., 2016[16]
2 ${\displaystyle {\sqrt {2}}}$ ≈ 1.41421
3 1.5163 1.4663
4 ${\displaystyle {\frac {\pi }{2}}}$ ≈ 1.5708
...
8 1.6641
...
${\displaystyle \mathrm {sinh} \left({\frac {\pi }{2}}\right)}$ ≈ 2.30130 2.261 ${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}}$ ≈ 1.78221 ${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon }$

## References

1. ^ Pisier, Gilles (April 2012), "Grothendieck's Theorem, Past and Present", Bulletin of the American Mathematical Society, 49 (2): 237–323, arXiv:, doi:10.1090/S0273-0979-2011-01348-9.
2. ^ a b c d Grothendieck, Alexander (1953), "Résumé de la théorie métrique des produits tensoriels topologiques", Bol. Soc. Mat. Sao Paulo, 8: 1–79, MR 0094682
3. ^ Blei, Ron C. (1987), "An elementary proof of the Grothendieck inequality", Proceedings of the American Mathematical Society, American Mathematical Society, 100 (1): 58–60, doi:10.2307/2046119, ISSN 0002-9939, JSTOR 2046119, MR 0883401
4. ^ Finch, Steven R. (2003), Mathematical constants, Cambridge University Press, ISBN 978-0-521-81805-6
5. ^ a b Krivine, J.-L. (1979), "Constantes de Grothendieck et fonctions de type positif sur les sphères", Advances in Mathematics, 31 (1): 16–30, doi:10.1016/0001-8708(79)90017-3, ISSN 0001-8708, MR 0521464
6. ^ a b Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf (2011), "The Grothendieck Constant is Strictly Smaller than Krivine's Bound", 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 453–462, arXiv:, doi:10.1109/FOCS.2011.77
7. ^ Acín, Antonio; Gisin, Nicolas; Toner, Benjamin (2006), Grothendieck’s constant and local models for noisy entangled quantum states, Physical Review A
8. ^ Clauser, John F.; Horne, Michael A.; Shimony, Abner; Holt, Richard A. (1969), Proposed Experiment to Test Local Hidden-Variable Theories, 23, Physical Review Letters, p. 880
9. ^ Davie, A. M. (1984), Unpublished
10. ^ Fishburn, P. C.; Reeds, J. A. (1994), Bell Inequalities, Grothendieck’s Constant, and Root Two, 7 (1), SIAM Journal on Discrete Mathematics, pp. 48–56, doi:10.1137/S0895480191219350
11. ^ Vértesi, Tamás (2008), More efficient Bell inequalities for Werner states, Physical Review A
12. ^ Briët, Jop; Buhrman, Harry; Toner, Ben (2011), A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement, Communications in Mathematical Physics
13. ^ Hua, Bobo; Li, Ming; Zhang, Tinggui; Zhou, Chunqin; Li-Jost, Xianqing; Fei, Shao-Ming (2015), Towards Grothendieck Constants and LHV Models in Quantum Mechanics, 48 (6), Journal of Physics A, p. 065302, arXiv:, Bibcode:2015JPhA...48f5302H, doi:10.1088/1751-8113/48/6/065302
14. ^ Diviánszky, Péter; Bene, Erika; Vértesi, Tamás (2017), Qutrit witness from the Grothendieck constant of order four, 96, Physical Review A, p. 012113
15. ^ Rietz, Ronald E. (1974), A proof of the Grothendieck inequality, 19 (3), Israel Journal of Mathematics, pp. 271–276, doi:10.1007/BF02757725
16. ^ Hirsch, Flavien; Quintino, Marco Túlio; Vértesi, Tamás; Navascués, Miguel; Brunner, Nicolas, Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant (PDF), arXiv:, Bibcode:2016arXiv160906114H