Grothendieck inequality

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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

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


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[edit]

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: .

Krivine (1979)[5] improved the result by proving: kR ≤ 1.7822139781...=, conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]

Grothendieck constant of order d[edit]

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[edit]

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 ≈ 1.41421
3 1.41421 1.41724 1.41758 1.4359
4 1.44521 1.44566 1.4841
5 ≈ 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
≈ 1.57079 1.67696

Upper bounds[edit]

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 ≈ 1.41421
3 1.5163 1.4644
4 ≈ 1.5708
8 1.6641
≈ 2.30130 2.261 ≈ 1.78221

See also[edit]


  1. ^ Pisier, Gilles (April 2012), "Grothendieck's Theorem, Past and Present", Bulletin of the American Mathematical Society, 49 (2): 237–323, arXiv:1101.4195, 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:1103.6161, 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, arXiv:quant-ph/0606138
  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, 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, arXiv:0806.0096
  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, Journal of Physics A, p. 065302, arXiv:1501.05507, 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, 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:1609.06114, Bibcode:2016arXiv160906114H

External links[edit]