Quantum Optimality in the Odd-Cycle game: the topological odd-blocker, marked connected components of the giant, consistency of pearls, vanishing homotopy

TL;DR

The paper develops a topological framework for understanding quantum optimality in the Odd-Cycle game, linking parallel repetition to a foam-minimization problem on the torus. It introduces the topological odd-blocker, the marked giant connected component, pearls, and consistent regions as geometric-analytic tools to bound quantum win probabilities. Through a sequence of definitions and theorems, it connects contraction-map probabilities under parallel repetition to foam surface-area measures, and shows that certain giant-component/topology conditions imply strong bounds on quantum advantage (up to constants, including an O(n^2) foam bound). This work extends previous XOR-game analyses to the Odd-Cycle setting, blending quantum information, combinatorics, and topology to reveal dualities between game-theoretic optimality and geometric minimization problems with potential implications for multi-prover nonlocal games and foam-like optimizations.

Abstract

We characterize optimality of Quantum strategies for the Odd-Cycle game. Separate from other game-theoretic settings, parallel repetition for the Odd-Cycle game is related to the foam problem, which can be formulated through a minimization of the surface area. In comparison to previous works on minimizing the surface area, we quantify how properties of the marked giant connected component can be related to the maximum winning probability using Quantum strategies. Objects that we introduce to formulate such connections include the topological odd-blocker, previous examples of error bounds for other Quantum games that have been formulated by the author, pearls, consistent regions, and the cycle elimination problem.