A Characterization of Perfect Strategies for Mirror Games
Sizhuo Yan, Jianting Yang, Tianshi Yu, Lihong Zhi
TL;DR
This paper characterizes when mirror games have perfect quantum strategies using algebraic methods and provides an algorithm to certify the absence of such strategies.
Contribution
It introduces an algebraic characterization for perfect strategies in mirror games and develops an algorithm based on noncommutative algebra and semidefinite programming.
Findings
Algebraic criteria for perfect strategies in mirror games
An algorithm to certify the absence of perfect strategies
Application of noncommutative Nullstellensatz and Gr"obner bases
Abstract
We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gr\"obner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.
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Taxonomy
TopicsQuantum Mechanics and Applications · Quantum Information and Cryptography · Advanced Algebra and Logic