Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant k with the following property. If a_i,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 s_i, t_j 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 S_i, T_j in the unit ball B(H) of a (real or complex) Hilbert space H. The smallest constant k 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_R(n) and k_C(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 k_R(n) and k_C(n) are easily seen to be increasing, and Grothendieck's result states that they are bounded,^[2]^[3] so they have limits.

With k_R defined to be sup_n k_R(n)^[4] 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)^[5] improved the result by proving: k_R ≤ 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).^[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 k_R(n, d) and k_C(n, d) for the real and complex case, respectively. With increasing d these constants are monotone increasing and their limit is k_R(n) and k_C(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_R(∞, d) and k_C(∞, d), respectively.

The Grothendieck constant k_R(∞, 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 k_R(∞, 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]
2		${\sqrt {2}}$ ≈ 1.41421
3		1.41421			1.41724		1.41758
4					1.44521		1.44566
5				${\frac {10}{7}}$ ≈ 1.42857	1.46007		1.46112
6					1.46007		1.47017
7						1.46286	1.47583
8						1.47586	1.47972
9						1.48608
...
∞	${\frac {\pi }{2}}$ ≈ 1.57079		1.67696

Upper bounds

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

d	Grothendieck, 1953^[2]	Rietz, 1974^[14]	Krivine, 1979^[5]	Braverman et al., 2011^[6]	Hirsch et al., 2016^[15]
2			${\sqrt {2}}$ ≈ 1.41421
3			1.5163		1.4663
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	${\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon$

References

↑ Pisier, Gilles (April 2012), "Grothendieck's Theorem, Past and Present", Bulletin of the American Mathematical Society, 49 (2): 237–323, doi:10.1090/S0273-0979-2011-01348-9 .
1 2 3 4 Grothendieck, Alexander (1953), "Résumé de la théorie métrique des produits tensoriels topologiques", Bol. Soc. Mat. Sao Paulo, 8: 1–79, MR 0094682
↑ 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 883401
↑ Finch, Steven R. (2003), Mathematical constants, Cambridge University Press, ISBN 978-0-521-81805-6
1 2 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 521464
1 2 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
↑ Acín, Antonio; Gisin, Nicolas; Toner, Benjamin (2006), Grothendieck’s constant and local models for noisy entangled quantum states, Physical Review A
↑ 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
↑ Davie, A. M. (1984), Unpublished
↑ 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
↑ Vértesi, Tamás (2008), More efficient Bell inequalities for Werner states, Physical Review A
↑ Briët, Jop; Buhrman, Harry; Toner, Ben (2011), A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement, Communications in Mathematical Physics
↑ 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, doi:10.1088/1751-8113/48/6/065302
↑ Rietz, Ronald E. (1974), A proof of the Grothendieck inequality, 19 (3), Israel Journal of Mathematics, pp. 271–276, doi:10.1007/BF02757725
↑ 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

External links

Weisstein, Eric W. "Grothendieck's Constant". MathWorld.

(NB: the historical part is not exact there.)

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