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International Journal of Theoretical Physics

Quantum Games with Strategies Induced by Basis Change Rules

Piotr Frąckiewicz, Jarosław Pykacz

Published May 26, 2017• 6 citations
Publisher

Abstract

The aim of the paper is to draw attention to a special class of two parameter unitary strategies in the Eisert-Wilkens-Lewenstein scheme for quantum games. We identify the players’ strategies with basis change matrices. Then we prove that the resulting quantum game is invariant with respect to isomorphic transformations of the input game. Moreover, it is shown that the game so obtained may not be trivial with respect to pure Nash equilibria, compared with the model with strategies being the special unitary group SU (2).

DOI
10.1007/s10773-017-3423-6

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The aim of the paper is to draw attention to a special class of two parameter unitary strategies in the Eisert-Wilkens-Lewenstein scheme for quantum games. We identify the players’ strategies with basis change matrices. Then we prove that the resulting quantum game is invariant with respect to isomorphic transformations of the input game. Moreover, it is shown that the game so obtained may not be trivial with respect to pure Nash equilibria, compared with the model with strategies being the special unitary group SU (2).
The aim of the paper is to examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2. The strategies of quantum players are specific types of two-parameter unitary operations such that the resulting quantum game is invariant under isomorphic transformations of the input classical game. We formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency. We prove that, depending on the payoffs of a classical game, the corresponding quantum game may or may not have Nash equilibria in the set of unitary strategies under study. Some of the equilibria cease to be equilibria if the players’ strategy set is the three-parameter special unitary group.
We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
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.
Two-player |NNN|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.
Over the last twenty years, quantum game theory has given us many ideas of how quantum games could be played. One of the most prominent ideas in the field is a model of quantum playing bimatrix games introduced by J. Eisert, M. Wilkens and M. Lewenstein. The scheme assumes that players' strategies are unitary operations and the players act on the maximally entangled two-qubit state. The quantum nature of the scheme has been under discussion since the article by Eisert et al. came out. The aim of our paper was to identify some of non-classical features of the quantum scheme.
The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.