Permissible four-strategy quantum extensions of classical games

TL;DR

This paper investigates quantum extensions of classical games using two-unitary operations, establishing conditions for isomorphism invariance and classifying five main types of such quantum games.

Contribution

It introduces new conditions for isomorphism invariance in quantum game extensions and classifies five main classes of these extended games.

Findings

Identified five main classes of quantum game extensions.

Derived a practical criterion for isomorphism of quantum games.

Explored interdependencies and limit cases among different classes.

Abstract

The study focuses on strategic-form games extended in the Eisert-Wilkens-Lewenstein scheme by two unitary operations. Conditions are determined under which the pair of unitary operators, along with classical strategies, form a game invariant under isomorphic transformations of the input classical game. These conditions are then applied to determine these operators, resulting in five main classes of games satisfying the isomorphism criterion, and a theorem is proved providing a practical criterion for this isomorphism. The interdependencies between different classes of extensions are identified, including limit cases in which one class transforms into another.