analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
Let be the unit ball of a separable Hilbert space over the real or complex numbers. Then the scalar product, has the following special property:
In other words, , as a function of two variables, is an element of the projective tensor product . 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.
Due to:
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:
Discussion of Grothendieck’s inequality in quantum physics, in relation to Bell's inequality, originates with:
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 [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]
Last revised on December 23, 2022 at 10:44:50. See the history of this page for a list of all contributions to it.