Better bounds on finite-order Grothendieck constants

Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in{3,4,5}$, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$. For $d\in{4,7,8}$, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex $d$-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$.

💡 Research Summary

The paper tackles the long‑standing problem of determining tighter lower bounds for finite‑order Grothendieck constants (K_G(d)), which quantify the maximal advantage that (d)-dimensional real vector strategies have over one‑dimensional (scalar) strategies in a canonical bilinear optimisation problem. While the exact value of (K_G(2)) is known, all higher‑order constants remain unknown, with only modest upper and lower bounds available in the literature.

The authors adopt a two‑step approach inspired by a recent Frank‑Wolfe (FW) projection technique. First, they seek a matrix (M) that makes the Grothendieck inequality as tight as possible (Problem 1). To this end they start from a point (P) inside the SDP set (\mathrm{SDP}_d) and iteratively project it onto the correlation polytope (\mathrm{SDP}_1) using a FW algorithm that exploits group symmetries. The projection yields a separating hyperplane (M) (or a facet (A) when symmetry forces the hyperplane to lie on a face of the symmetrised polytope). Second, they need to evaluate (\mathrm{SDP}_1(M)), which is equivalent to a Max‑Cut problem and therefore NP‑hard (Problem 2). For modest matrix sizes they solve this exactly with a state‑of‑the‑art binary quadratic optimiser; for larger, highly symmetric instances they resort to a “seesaw” heuristic that provides reliable lower bounds but no formal certification.

Using this pipeline the authors construct explicit rectangular instances for dimensions (d=3,4,5). By solving the associated binary quadratic programmes exactly they certify ratios (\mathrm{SDP}_d(M)/\mathrm{SDP}_1(M)) that improve all previously known lower bounds for (K_G(d)) up to (d=9) via the monotonicity property (K_G(d)\le K_G(d’)) for (d\le d’). For example, they raise the lower bound for (K_G(3)) from about 1.20 to roughly 1.28, and similarly improve the bounds for (K_G(4)) and (K_G(5)).

In parallel, the paper explores highly symmetric line‑packing configurations—root systems (D_d), the exceptional lattices (E_7) and (E_8), and related structures—for dimensions (d=4,7,8). These give rise to Gram matrices (P) that, after a diagonal shift of (\lambda=2/3), produce candidate facets (A) of the symmetrised correlation polytope. Although the authors cannot yet certify (\mathrm{SDP}_1(A)) exactly, their heuristic evaluations suggest that these symmetric instances would yield even larger lower bounds than the rectangular ones if solved to optimality.

Beyond the standard constants, the authors study the generalized Grothendieck constants (K_G(d!\rightarrow!2)), which compare (d)-dimensional complex quantum strategies with real‑qubit (2‑dimensional real) strategies. They reinterpret these constants as the quantitative advantage of complex (d)-dimensional quantum mechanics over real qubit mechanics, a perspective already hinted at in earlier works on Bell‑inequality violations. By refining existing Bell‑inequality based upper bounds and applying their FW‑based construction, they obtain a notably tighter upper bound for (K_G(3!\rightarrow!2)) and improve several other entries in the table of known bounds.

The paper’s contributions can be summarised as follows:

1. Introduction of a systematic FW‑plus‑symmetry framework for generating strong candidate matrices for Grothendieck‑inequality lower bounds.
2. Exact certification of new lower bounds for (K_G(3), K_G(4), K_G(5)) and, via monotonicity, for all (d\le 9).
3. Identification of highly symmetric line‑packing instances that promise even better bounds for (d=4,7,8), albeit currently only heuristically.
4. Extension of the analysis to generalized constants (K_G(d!\rightarrow!2)) with improved numerical bounds and a clear physical interpretation in terms of quantum non‑locality.

Methodologically, the work showcases the power of modern convex‑optimization tools (the FrankWolfe.jl package, blended pairwise conditional gradients, and sophisticated binary quadratic solvers) when combined with group‑theoretic symmetry reductions. Practically, the improved constants tighten the analysis of approximation algorithms that rely on Grothendieck’s inequality (e.g., cut‑norm approximation) and sharpen the quantitative understanding of quantum violations of Bell inequalities.

The authors conclude by highlighting open challenges: developing exact solvers for the large symmetric instances, extending the framework to other target dimensions (n>2) (i.e., (K_G(d!\rightarrow!n))), and exploring deeper connections between Grothendieck‑type inequalities and quantum information theory. Their results represent a significant step forward in both the combinatorial optimisation and quantum‑foundations communities.