Complete characterization of perfect quantum strategies in quantum magic rectangle games

TL;DR

This paper fully characterizes perfect quantum strategies in quantum magic rectangle games, revealing the algebraic structure of solution states and conditions for strategy existence, with implications for quantum information.

Contribution

It provides a complete analytical framework for perfect strategies, including necessary and sufficient conditions and structural insights into solution states.

Findings

Perfect quantum solution states have a specific algebraic-combinatorial structure.

No perfect strategies exist for 2×n games with odd n.

Introduces a quantum magic rectangle inequality for non-perfect strategies.

Abstract

We provide a complete structural characterization of perfect quantum strategies for arbitrary quantum magic rectangle games. We derive necessary and sufficient conditions that jointly constrain the shared state and measurement operators, establishing a unified analytical framework for perfect nonlocal strategies in this setting. Our results show that all perfect quantum solution states (PQSS) must exhibit a specific algebraic--combinatorial structure, ruling out a priori assumptions about particular entangled resources and clarifying the full class of states compatible with perfect correlations. We further show that perfect quantum strategies do not exist for $2 \times n$ quantum magic rectangle games with odd $n$, and introduce a corresponding quantum magic rectangle inequality to characterize optimal non-perfect strategies. While our results are structural, they may provide a foundation for future developments in quantum information and quantum cryptography based on perfect nonlocal correlations.