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arXiv:quant-ph/0311018

Let us play with qubits

Sylvain Gravier, Philippe Jorrand, Mehdi Mhalla, Charles Payan

Published Nov 5, 2003• 0 citations
arXivHTMLPDF

Abstract

Quantum game theory offers a lot of interesting questions, and it is relevant to use the quantum information theory to resolve or improve games with lack of information : how to use the power of quantum entanglement to show the superiority of a quantum player that is allowed to use quantum mechanics versus a classical player, how to use quantum communication properties in cooperative games ... But games are also useful to make notions easier to understand, and permit to apprehend easier new ways of reasoning. The objective of this work is to formalize and to study a simple game with qubits using quantum notions of measurement and superposition but keeping a simple formalism so that knowing quantum mechanics is not necessary to play the game. We solve a quantum combinatorial game by giving a winning strategy for it. We also propose a quantisation of a family of combinatorial games.

DOI
10.48550/arxiv.quant-ph/0311018

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Quantum game theory offers a lot of interesting questions, and it is relevant to use the quantum information theory to resolve or improve games with lack of information : how to use the power of quantum entanglement to show the superiority of a quantum player that is allowed to use quantum mechanics versus a classical player, how to use quantum communication properties in cooperative games ... But games are also useful to make notions easier to understand, and permit to apprehend easier new ways of reasoning. The objective of this work is to formalize and to study a simple game with qubits using quantum notions of measurement and superposition but keeping a simple formalism so that knowing quantum mechanics is not necessary to play the game. We solve a quantum combinatorial game by giving a winning strategy for it. We also propose a quantisation of a family of combinatorial games.
Quantum games offer situations where quantum information theory may help in solving or improving games with lack of information. For example, one may rely upon properties of entanglement for showing the superiority of a quantum player who is allowed to use quantum operations against a classical player (penny flipover [5]). Quantum communication properties enter into play in cooperative games (prisoners game, [3], guessing number [10]). An introduction to this field and an overview of important results are proposed in [7, 8]. But games may also help to make notions easier to understand, and new ways of reasoning to apprehend. The objective of this paper is to formalize and study a simple game with qubits using quantum notions of measurement and superposition, while keeping a simple formalism so that knowing quantum mechanics is not necessary to play the game. We generalize Conway's classical octal games to quantum octal games, and we solve the quantum combinatorial game QO.07 by giving a winning strategy for it. A playable version of the studied qubit game is available at the address .
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.
This paper develops and analyses a novel quantum combinatorial game: quantum checkers (named Cheqqers). The concepts of superposition, entanglement, measurements and interference from quantum mechanics are integrated into the game of checkers by adding new types of legal moves. The addition of these new rules is done gradually by introducing several levels of 'quantumness'. Quantum checkers provides a framework for interpolating between a known and solved classical game and a more complex quantum game, and serves as 1) a benchmark for AI players learning to play quantum games and 2) an interesting game for human players that allows them to build intuition for quantum phenomena. We provide the initial analysis on the complexity of this game using random agents and a Monte Carlo tree search agent.
A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. Moreover, it became clear that the players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2x2 games, it became generally accepted that quantum versions can be easily extended to N-player situations by simply allowing N-partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
Quantum games have proposed a new point of view for the solution of the classical problems and dilemmas in game theory. Certain quantization relationships can be proposed with the objective that a game can be generalized into a quantum domain where the linear superposition of actions is allowed. This quantization let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. This leads us to interesting aspects which only can be observed through the quantization of a game like the possibility of the entanglement between players, the definition of a socioeconomical temperature in a system, the collective welfare principle,the analysis of a game through elements of quantum information theory and the globalization process. The relationships between quantum mechanics and game theory are also studied and its shown how although both systems analyzed are described through two theories apparently different (quantum mechanics and game theory) both are analogous and thus exactly equivalents.