The perturbative method for quantum correlations

TL;DR

This paper introduces a Lie-theoretic perturbative approach to analyze quantum correlations, revealing structural properties near classical points and implications for quantum strategy optimization.

Contribution

It develops a novel perturbative method using Lie groups to study quantum correlations and decomposes Bell operators to understand local optimality and resource requirements.

Findings

Near classical points, Bell operators decompose into subset games.

Classically optimal points remain locally optimal in 2D quantum strategies.

The approach provides insights into the dimension as a resource for quantum learning.

Abstract

The set $\mathcal{Q}$ of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an $(n, 2, d)$ Bell operator decomposes into a direct sum of $(k, 2, d-1)$ Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the $(n, 2, 2)$ case, if $p_0$ is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of $\mathcal{Q}$ is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in $\mathcal{Q}(D)$ unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation.