Finite de Finetti for convex bodies and Polynomial Optimization

TL;DR

This paper establishes a finite de Finetti theorem for convex bodies using a novel relative entropy approach, enabling convergent hierarchies for polynomial optimization problems with constraints and applications to non-local games.

Contribution

It introduces a finite de Finetti theorem for convex bodies based on relative entropy, extending quantum techniques to general convex sets, and develops convergent hierarchies for polynomial optimization.

Findings

Proves a finite de Finetti theorem for convex bodies.

Develops a convergent hierarchy for polynomial optimization with constraints.

Provides a constructive scheme for certified interior points.

Abstract

Leveraging a recently proposed notion of relative entropy in general probabilistic theories (GPT), we prove a finite de Finetti representation theorem for general convex bodies. We apply this result to address a fundamental question in polynomial optimization: the existence of a convergent outer hierarchy for problems with inequality constraints and analytical convergence guarantees. Our strategy generalizes a quantitative monogamy-of-entanglement argument from quantum theory to arbitrary convex bodies, establishing a uniform upper bound on mutual information in multipartite extensions. This leads to a finite de Finetti theorem and, subsequently, a convergent conic hierarchy for a wide class of polynomial optimization problems subject to both equality and inequality constraints. We further provide a constructive rounding scheme that yields certified interior points with controlled approximation error. As an application, we express the optimal GPT value of a two-player non-local game as a polynomial optimization problem, allowing our techniques to produce approximation schemes with finite convergence guarantees.