Approximating fixed size quantum correlations in polynomial time

TL;DR

This paper presents a polynomial-time method for approximating the entangled value of fixed-size two-player free games with fixed-dimensional entanglement, using novel quantum de Finetti theorems and SDP hierarchies.

Contribution

It introduces a new approach combining quantum de Finetti theorems, symmetry reduction, and SDP hierarchies to efficiently approximate quantum game values.

Findings

Polynomial-time algorithms for fixed-size quantum games

New quantum de Finetti theorems for constrained separability

Effective rounding schemes for entangled strategies

Abstract

We show that $\varepsilon$-additive approximations of the optimal value of fixed-size two-player free games with fixed-dimensional entanglement assistance can be computed in time $\mathrm{poly}(1/\varepsilon)$. This stands in contrast to previous analytic approaches, which focused on scaling with the number of questions and answers, but yielded only strict $\mathrm{exp}(1/\varepsilon)$ guarantees. Our main result is based on novel Bose-symmetric quantum de Finetti theorems tailored for constrained quantum separability problems. These results give rise to semidefinite programming (SDP) outer hierarchies for approximating the entangled value of such games. By employing representation-theoretic symmetry reduction techniques, we demonstrate that these SDPs can be formulated and solved with computational complexity $\mathrm{poly}(1/\varepsilon)$, thereby enabling efficient $\varepsilon$-additive approximations. In addition, we introduce a measurement-based rounding scheme that translates the resulting outer bounds into certifiably good inner sequences of entangled strategies. These strategies can, for instance, serve as warm starts for see-saw optimization methods. We believe that our techniques are of independent interest for broader classes of constrained separability problems in quantum information theory.