Optimal, and approximately optimal, quantum strategies for $\mathrm{XOR}^{*}$ and $\mathrm{FFL}$ games

TL;DR

The paper addresses the problem of characterizing optimal and approximately optimal quantum strategies for non-local XOR games, notably including XOR, XOR$^*$, CHSH(n), and the Fortnow-Feige-Lovasz (FFL) game. Building on Ostrev's Frobenius-norm framework, it develops an intertwining operator $T$ with unit Frobenius norm to translate optimal XOR strategies into dual or related settings, deriving explicit error bounds for permuting observables and showing how the $2/3$ quantum/classical bias in FFL arises in the dual context. The main contributions include precise norm-based approximate optimality criteria (epsilon-approximate strategies), nontrivial bounds for operator permutivity, and a structured pathway to extend the methodology to less regular games beyond the XOR family. These results provide a rigorous, quantitative toolkit for analyzing quantum advantages in two-player non-local games and suggest a roadmap for constructing approximately optimal strategies in more complex game settings with potential cryptographic and foundational implications.

Abstract

We analyze optimal, and approximately optimal, quantum strategies for a variety of non-local XOR games. Building upon previous arguments due to Ostrev in 2016, which characterized approximately optimal, and optimal, strategies that players Alice and Bob can adopt for maximizing a linear functional to win non-local games after a Referee party examines each answer to a question drawn from some probability distribution, we identify additional applications of the framework for analyzing the performance of a broader class of quantum strategies in which it is possible for Alice and Bob to realize quantum advantage if the two players adopt strategies relying upon quantum entanglement, two-dimensional resource systems, and reversible transformations. For the Fortnow-Feige-Lovasz (FFL) game, the 2016 framework is directly applicable, which consists of five steps, including: (1) constructing a suitable, nonzero, linear transformation for the intertwining operations, (2) demonstrating that the operator has unit Frobenius norm, (3) constructing error bounds, and corresponding approximate operations, for $\big( A_k \otimes \textbf{I} \big) \ketψ$, and $\big( \textbf{I} \otimes \big( \frac{\pm B_{kl} + B_{lk}}{\sqrt{2}} \big) \big) \ketψ$, (4) constructing additional bounds for permuting the order in which $A^{j_i}_i$ operators are applied, (5) obtaining Frobenius norm upper bounds for Alice and Bob's strategies. We draw the attention of the reader to applications of this framework in other games with less regular structure.