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

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 $\big( A_k \otimes \textbf{I} \big) \ket{\psi}$, and $\big( \textbf{I} \otimes \big( \frac{\pm B_{kl} + B_{lk}}{\sqrt{2}} \big) \big) \ket{\psi}$, (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.