Frank Oertel (CPNSS, LSE): “A statistical interpretation of Grothendieck’s inequality and its relation to the size of non-locality of quantum mechanics”

Abstract: In 1953 A. Grothendieck proved a theorem that he called The Fundamental Theorem on the Metric Theory of Tensor Products. This result is known today as Grothendieck’s inequality (or Grothendieck’s theorem). Originally, it is recognised as one of the major results of Banach space theory.

Grothendieck formulated his deep result in the language of tensor norms on tensor products of Banach spaces. To this end he described how to generate new tensor norms from known ones and unfolded a powerful duality theory between tensor norms. Only in 1968, thanks to J. Lindenstrauss and A. Pelczynski Grothendieck’s inequality was decoded and equivalently rewritten – in matrix form – which lead to its global breakthrough.

However, since the appearance of Grothendieck’s paper in 1953 there exists the (still) open problem to determine the smallest possible constant – called the Grothendieck constant – which can be used in Grothendieck’s inequality.

In addition to its multifaceted representations in functional analysis the Grothendieck inequality admits further equivalent formulations – each one of them reflecting deep and surprising links with different scientific branches, such as semidefinite programming in convex optimisation, NP-hard combinatorial optimisation, graph theory, communication complexity, private data analysis, geomathematics and – due to the pioneering work of B. S. Tsirelson in 1985 – even foundations and philosophy of quantum mechanics.

Based on matrix analysis and a few techniques from multivariate statistics we will present a further equivalent representation of Grothendieck’s inequality (over the reals) which reveals also its deep underlying statistical nature. Using this representation, we will revisit Tsirelson’s approach and sketch how Grothendieck’s inequality is intertwined with the violation of a Bell inequality, based on Tsirelson’s observation that the Grothendieck constant – which is strictly larger than 1 – gives an upper bound of the deviation of the quantum mechanical correlation model from the “classic” Kolmogorovian statistical correlation model.

Frank Oertel is a mathematician with interests in the philosophy of physics, as well as in abstract analysis, operator theory, stochastic analysis, and mathematical finance.