Finite de Finetti for convex bodies and Polynomial Optimization

TL;DR

This work addresses polynomial optimization over convex state spaces arising in general probabilistic theories by developing a finite de Finetti representation theorem for convex bodies and a uniform monogamy-of-entanglement bound. Building on Frenkel's integral representation of relative entropy, the authors define a GPT-adapted mutual information and prove a bound that yields a quantitative de Finetti theorem with a rate of $\frac{2 c(A,B)}{\sqrt{n}}$. This leads to a convergent outer hierarchy for problems with both equality and inequality constraints, together with a constructive rounding scheme that outputs certified interior points with explicit error control. The approach also provides a practical link to conic lifts, enabling lifted-cone reformulations and finite-convergence guarantees, and applies to expressing the optimal GPT value of two-player non-local games as a polynomial optimization problem. Overall, the results offer finite-level convergence guarantees, constraint-compatible de Finetti approximations, and interior-point certifiability for broad classes of GPT-inspired polynomial optimization problems.

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.

Figures (3)

• Figure 1: The figure schematically shows how the correspondence between affine maps $f$ and linear maps $\hat{F}_{\hookrightarrow}$ works. We start with an affine map $f$ and build out of it a bilinear map $F$ via an image identification construction. From this we use the universal property of the tensor product to get a unique linear map $\hat{F}$. Using the embedding $\iota$ of $K_A' \dot{\otimes} K_B'$ into its bidual yields a map $\hat{F}_{\hookrightarrow} = \hat{F} \circ \iota$.
• Figure 2: We compare the four optimization problems $p_1, p_2, p_3, p_4$. Because the optimization variable in $p_1$ is restricted to product states, we obtain $p_1=p_2$ immediately. Moreover, every state feasible for $p_1$ is also feasible for $p_3$, whereas the converse does not hold; hence $p_3 \leq p_1$. Concretely, any state feasible for $p_3$ can be expressed as $x_{AB}=\sum_{\lambda}p_{\lambda} x_A^{\lambda}\otimes x_B^{\lambda}$. Hence, the condition $\hat{F}_{\hookrightarrow}\left(x_{AB}\right)=0$ implies $\sum_{\lambda}p_{\lambda} \hat{F}_{\hookrightarrow}\left(x_A^{\lambda}\otimes x_B^{\lambda}\right)=0$. However, in contrast to the case $p_1$, this does not generally imply that $\hat{F}_{\hookrightarrow}\left(x_A^{\lambda}\otimes x_B^{\lambda}\right)=0$ for all $\lambda$ as in $p_1$. As it turns out, this subtle but essential difference poses a significant challenge for approximation schemes. Since $K_A\dot{\otimes} K_B\subseteq K_A\hat{\otimes} K_B$, we also have $p_4\leq p_3$.
• Figure 3: $\Phi$ and $\Psi$ are isomorphisms of vector spaces (cf. Pl_vala_2023).

Theorems & Definitions (43)

• Theorem : Finite de Finetti for convex bodies, informal
• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5