nLab Grothendieck inequality

Statement

Let $B$ be the unit ball of a separable Hilbert space over the real or complex numbers. Then the scalar product, $\langle\cdot,\cdot\rangle : B \times B \to \mathbb{C}$ has the following special property:

Theorem

There exist sequences $f_n,g_n: B \to \mathbb{C}$ of norm-continuous functions, such that

• $\langle x, y \rangle = \sum_n f_n(x) g_n(y)$ for all $x, y \in B$

• $\sum_n \, \sup_B \left| f_n \right| \, \sup_B \left| g_n \right| \, < \, \infty$

In other words, $\langle\cdot,\cdot\rangle$, as a function of two variables, is an element of the projective tensor product $C(B) {\displaystyle\hat{\otimes}} C(B)$. Its projective tensor norm is known as Grothendieck’s constant. The precise value of this constant is different in the real and complex case, and neither one is known exactly.

Related concepts

• inequality

• Cauchy-Schwarz inequality

• Bell's inequality

References

General

Due to:

• Alexander Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Annales de l’Institut Fourier, 4 (1952) 73-112 [numdam:AIF_1952__4__73_0]

Review:

• Leqi Zhu, Grothendieck’s inequality (2018) [pdf]

• Ron Blei, Analysis in integer and fractional dimensions, Cambridge University Press (2009) [doi:10.1017/CBO9780511543012]

See also:

• Wikipedia, Grothendieck inequality

In quantum physics

Discussion of Grothendieck’s inequality in quantum physics, in relation to Bell's inequality, originates with:

• Boris S. Tsirelson, Quantum analogues of the Bell inequalities. The case of two spatially separated domains, Journal of Soviet Mathematics 36 (1987) 557–570 [doi:10.1007/BF01663472]

reviewed in

• Boris S. Tsirelson, Some results and problems on quantum Bell-type inequalities Hadronic Journal Supplement 8 4 (1993) 329-345 [pdf, pdf web]

  (but see the erratum here)

• Wikipedia, Tsirelson’s bound

Further discussion:

• Antonio Acín, Nicolas Gisin, and Benjamin Toner, Grothendieck’s constant and local models for noisy entangled quantum states, Phys. Rev. A 73 (2006) 062105 [doi:10.1103/PhysRevA.73.062105, arXiv:quant-ph/0606138]

• Hoshang Heydari, Quantum Violation: Beyond Clauser-Horne-Shimony-Holt Inequality, J. Phys. A: Math. Gen. 39 (2006) 11869-11875 [arXiv:quant-ph/0603050, doi:10.1088/0305-4470/39/38/012]

• Jop Briët, Harry Buhrman & Ben Toner, A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement, Comm. Math. Phys. 305 (2011) 827–843 [doi:10.1007/s00220-011-1280-3]

• Flavien Hirsch, Marco Túlio Quintino, Tamás Vértesi, Miguel Navascués, Nicolas Brunner, Quantum 1 (2017) 3 Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$ [doi:10.22331/q-2017-04-25-3, arXiv:1609.06114]

• A. Vourdas, Grothendieck bound in a single quantum system, J. Phys. A: Math. Theor. 55 (2022) 435206 [arXiv:2212.11663, doi:10.1088/1751-8121/ac9dcf]