Winning Rates of $(n,k)$ Quantum Coset Monogamy Games

2025-01-29

AI Summary

This work studies the $(n,k)$ quantum coset monogamy game: two players, without communication, must simultaneously extract complementary $k$-bit X-error and $(n-k)$-bit Z-error information from a shared random coset state. The central problem is to characterize how the winning probability depends on the subspace rate $R = k/n$, and to determine the optimal performance achievable without entanglement. We develop an analytical framework grounded in quantum information theory and Pauli error modeling. Our key contributions are: (i) the first derivation of a convex upper bound on the winning probability, explicitly parameterized by $R$, which improves upon prior bounds at $R = 1/2$; and (ii) the first proof that this bound is tight—achievable by a separable (no-entanglement) strategy. The resulting closed-form bound significantly enhances theoretical precision for $R eq 1/2$, and rigorously establishes the fundamental limit on winning probability in the absence of entanglement.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We formulate the $(n,k)$ Coset Monogamy Game, in which two players must extract complementary information of unequal size ($k$ bits vs. $n-k$ bits) from a random coset state without communicating. The complementary information takes the form of random Pauli-X and Pauli-Z errors on subspace states. Our game generalizes those considered in previous works that deal with the case of equal information size $(k=frac{n}{2})$. We prove a convex upper bound of the information-theoretic winning rate of the $(n,k)$ Coset Monogamy Game in terms of the subspace rate $R=frac{k}{n}in [0,1]$. This bound improves upon previous results for the case of $R=frac{1}{2}$. We also prove the achievability of an optimal winning probability upper bound for the class of unentangled strategies of the $(n,k)$ Coset Monogamy Game.

Problem

• Quantum Privacy Amplification
• Information Asymmetry
• Entanglement-Free Strategies

Innovation

• Quantum Grouped One-Way Games
• Information Size Proportions
• Non-Entanglement Strategies