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Upper bounds for Grothendieck constants, quantum correlation matrices and CCP functions

Frank Oertel

TL;DR

The paper advances a unified, function-theoretic framework for bounding the Grothendieck constants in both the real and complex settings by embedding correlation matrices into a theory of Hadamard-transform maps and Gaussian structures. It develops a quantum-correlation perspective, introduces a Gaussian inner-product splitting property, and studies completely correlation-preserving (CCP) functions via Schoenberg-type results to derive and compare upper bounds, including recoveries of classical bounds such as $K_G^\mathbb{R} \le \sinh(\pi/2)$, $K_G^\mathbb{R} < \frac{\pi}{2\ln(1+\sqrt{2})}$, and $K_G^\mathbb{C} \le 1.40491$. The approach unifies real and complex cases, connects to Bell inequalities and the Walsh–Hadamard transform, and provides an algorithmic scheme that underpins the estimation of Grothendieck constants via operator-ideal and SDP viewpoints. By leveraging Gaussian analysis, Hermite polynomials, and hypergeometric representations, the work offers short proofs for strong bounds and a structured path toward addressing open problems in the exact values of $K_G^\mathbb{F}$.

Abstract

Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes either the real or the complex field, we concentrate our search on their smallest upper bound. To this end, we establish a basic framework, built on functions which map correlation matrices to correlation matrices entrywise by means of the Hadamard product, such as the Krivine function in the real case or the Haagerup function in the complex case. By making use of multivariate real and complex Gaussian analysis, higher transcendental functions, integration over spheres and combinatorics of the inversion of Maclaurin series, we provide an approach by which we also recover all famous upper bounds of Grothendieck himself ($K_G^\mathbb{R} \leq \sinh(π/2) \approx 2.301$), Krivine ($K_G^\mathbb{R} \leq \fracπ{2 \ln(1 + \sqrt{2})} \approx 1,782$) and Haagerup ($K_G^\mathbb{C} \leq 1.405$, numerically approximated); each of them as a special case. In doing so, we aim to unify the real and complex case as much as possible and apply our results to several concrete examples, including the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian copula - with foundations of quantum theory and quantum information theory in mind. Moreover, we offer a shortening and a simplification of the proof of the strongest estimation until now; namely that $K_G^\mathbb{R} < \fracπ{2 \ln(1 + \sqrt{2})}$. We summarise our key results in form of an algorithmic scheme and shed light on related open problems and topics for future research.

Upper bounds for Grothendieck constants, quantum correlation matrices and CCP functions

TL;DR

The paper advances a unified, function-theoretic framework for bounding the Grothendieck constants in both the real and complex settings by embedding correlation matrices into a theory of Hadamard-transform maps and Gaussian structures. It develops a quantum-correlation perspective, introduces a Gaussian inner-product splitting property, and studies completely correlation-preserving (CCP) functions via Schoenberg-type results to derive and compare upper bounds, including recoveries of classical bounds such as , , and . The approach unifies real and complex cases, connects to Bell inequalities and the Walsh–Hadamard transform, and provides an algorithmic scheme that underpins the estimation of Grothendieck constants via operator-ideal and SDP viewpoints. By leveraging Gaussian analysis, Hermite polynomials, and hypergeometric representations, the work offers short proofs for strong bounds and a structured path toward addressing open problems in the exact values of .

Abstract

Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant in the famous Grothendieck inequality (unsolved since 1953), where denotes either the real or the complex field, we concentrate our search on their smallest upper bound. To this end, we establish a basic framework, built on functions which map correlation matrices to correlation matrices entrywise by means of the Hadamard product, such as the Krivine function in the real case or the Haagerup function in the complex case. By making use of multivariate real and complex Gaussian analysis, higher transcendental functions, integration over spheres and combinatorics of the inversion of Maclaurin series, we provide an approach by which we also recover all famous upper bounds of Grothendieck himself (), Krivine () and Haagerup (, numerically approximated); each of them as a special case. In doing so, we aim to unify the real and complex case as much as possible and apply our results to several concrete examples, including the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian copula - with foundations of quantum theory and quantum information theory in mind. Moreover, we offer a shortening and a simplification of the proof of the strongest estimation until now; namely that . We summarise our key results in form of an algorithmic scheme and shed light on related open problems and topics for future research.
Paper Structure (23 sections, 80 theorems, 369 equations, 1 figure)

This paper contains 23 sections, 80 theorems, 369 equations, 1 figure.

Table of Contents

  1. Introduction and motivation: the outstanding story of Grothendieck's theorem
  2. Historical perspective and theoretical framework
  3. Preliminaries, terminology and notation
  4. Complex Gaussian random vectors and the probability law $\mathbb C N_{2n}(0, \Sigma_{2n}(\zeta))$
  5. General complex Gaussian random vectors in $\mathbb C^n$ and their probability distribution
  6. Partitioned complex Gaussian random vectors in $\mathbb C^{2n}$ and the probability law ${\mathbb C}N_{2n}(0, \Sigma_{2n}(\zeta))$
  7. A quantum correlation matrix version of the Grothendieck inequality
  8. Gram matrices, quantum correlation, and beyond
  9. The Grothendieck inequality, correlation matrices and the matrix norm $\Vert \cdot \Vert_{\infty, 1}^\mathbb F$
  10. Characterisation of $K_G^\mathbb F$ through operator ideals and violation of Bell inequalities (a brief digression)
  11. $K_G^\mathbb R(2)$ and the Walsh-Hadamard transform: Krivine's approach revisited
  12. The Gaussian inner product splitting property
  13. Powers of inner products of random vectors, uniformly distributed on the sphere
  14. Gaussian sign-correlation
  15. Integration over $\mathbb{S}^{n-1}$ and the Gamma function
  16. ...and 8 more sections

Key Result

Theorem 1.1.1

Let $\mathbb F \in \{\mathbb R, \mathbb C\}$. There is an absolute constant $K>0$ such that for any $m, n \in \mathbb N$, for any $A = (a_{ij}) \in \mathbb M_{m,n}(\mathbb F)$, any $\mathbb F$-Hilbert space $H$, and any $(u_1, \ldots, u_m) \in B_H^m$, $(v_1, \ldots, v_n) \in B_H^n$, the following in

Figures (1)

  • Figure 1: $R_2(A) = J_2 \circ A \circ J_2^{-1}$

Theorems & Definitions (127)

  • Theorem 1.1.1: Grothendieck inequality in matrix form
  • Proposition 1.1.2
  • Theorem 1.1.3
  • Lemma 1.2.1
  • Proposition 1.2.2
  • Corollary 1.2.3
  • Lemma 1.2.4
  • Remark 1.2.5
  • Proposition 2.1.1
  • Definition 2.1.2: Complex Gaussian random vector
  • ...and 117 more