Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant ${\displaystyle K_{G}}$ with the following property. If M_ij is an n by n (real or complex) matrix with

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

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

${\displaystyle \left|\sum _{i,j}M_{ij}\langle S_{i},T_{j}\rangle \right|\leq K_{G}}$,

for all vectors S_i, T_j in the unit ball B(H) of a (real or complex) Hilbert space H, the constant ${\displaystyle K_{G}}$ 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 ${\displaystyle K_{G}(d)}$. In fact there are two Grothendieck constants, ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ and ${\displaystyle K_{G}^{\mathbb {C} }(d)}$, 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 existence of the constants in a paper published in 1953.[2]

Bounds on the constants

The sequences ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ and ${\displaystyle K_{G}^{\mathbb {C} }(d)}$ are easily seen to be increasing, and Grothendieck's result states that they are bounded,[2][3] so they have limits.

With ${\displaystyle K_{G}^{\mathbb {R} }}$ defined to be ${\displaystyle \textstyle \sup _{d}K_{G}^{\mathbb {R} }(d)}$[4] then Grothendieck proved that: ${\displaystyle 1.57\approx {\frac {\pi }{2}}\leq K_{G}^{\mathbb {R} }\leq \mathrm {sinh} ({\frac {\pi }{2}})\approx 2.3}$.

Krivine (1979)[5] improved the result by proving: ${\displaystyle K_{G}^{\mathbb {R} }\leq {\frac {\pi }{2\ln(1+{\sqrt {2}})}}\approx 1.7822}$, conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]

Grothendieck constant of order d

Boris Tsirelson showed that the Grothendieck constants ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ 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 ${\displaystyle K_{G}^{\mathbb {R} }(2d^{2})}$.[7][8]

Lower bounds

Some historical data on best known lower bounds of ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ is summarized in the following table.

d	Grothendieck, 1953[2]	Krivine, 1979[5]	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					1.41724		1.41758	1.4359
4					1.44521		1.44566	1.4841
5				${\displaystyle {\frac {10}{7}}}$ ≈ 1.42857	1.46007		1.46112
6							1.47017
7						1.46286	1.47583
8						1.47586	1.47972
9						1.48608
...
∞	${\displaystyle {\frac {\pi }{2}}}$ ≈ 1.57079		1.67696

Upper bounds

Some historical data on best known upper bounds of ${\displaystyle K_{G}^{\mathbb {R} }(d)}$:

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		1.4644
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	${\displaystyle {\frac {\pi }{2\ln(1+{\sqrt {2}})}}-\varepsilon }$

See also

• Pisier–Ringrose inequality

References

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. 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. 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. 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. Boris Tsirelson (1987). "Quantum analogues of the Bell inequalities. The case of two spatially separated domains" (PDF). Journal of Soviet Mathematics. 36 (4): 557–570. doi:10.1007/BF01663472.
8. Acín, Antonio; Gisin, Nicolas; Toner, Benjamin (2006), "Grothendieck's constant and local models for noisy entangled quantum states", Physical Review A, 73 (6): 062105, arXiv:quant-ph/0606138, Bibcode:2006PhRvA..73f2105A, doi:10.1103/PhysRevA.73.062105
9. Davie, A. M. (1984), Unpublished
10. Fishburn, P. C.; Reeds, J. A. (1994), "Bell Inequalities, Grothendieck's Constant, and Root Two", SIAM Journal on Discrete Mathematics, 7 (1): 48–56, doi:10.1137/S0895480191219350
11. Vértesi, Tamás (2008), "More efficient Bell inequalities for Werner states", Physical Review A, 78 (3): 032112, arXiv:0806.0096, Bibcode:2008PhRvA..78c2112V, doi:10.1103/PhysRevA.78.032112
12. Briët, Jop; Buhrman, Harry; Toner, Ben (2011), "A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement", Communications in Mathematical Physics, 305 (3): 827, Bibcode:2011CMaPh.305..827B, doi:10.1007/s00220-011-1280-3
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", Journal of Physics A: Mathematical and Theoretical, Journal of Physics A, 48 (6): 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", Physical Review A, 96 (1): 012113, arXiv:1707.04719, Bibcode:2017PhRvA..96a2113D, doi:10.1103/PhysRevA.96.012113
15. Rietz, Ronald E. (1974), "A proof of the Grothendieck inequality", Israel Journal of Mathematics, 19 (3): 271–276, doi:10.1007/BF02757725
16. Hirsch, Flavien; Quintino, Marco Túlio; Vértesi, Tamás; Navascués, Miguel; Brunner, Nicolas (2017), "Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant", Quantum, 1: 3, arXiv:1609.06114, Bibcode:2016arXiv160906114H, doi:10.22331/q-2017-04-25-3

External links

• Weisstein, Eric W. "Grothendieck's Constant". MathWorld. (NB: the historical part is not exact there.)