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A New Quantum Game Based on CHSH Game

Alan Bojić

Published Jun 1, 2013• 0 citations
OpenAlex

Abstract

The Quantum game theory is an extension of the classical game theory which describes games in which players share quantum information during the game. The use of the quantum mechanical effects enables players’ better results than they would achieve in the classical game without sharing any information at all. In this paper I would like to describe a new two player cooperative game which is based on the well known CHSH game. A quantum strategy for a new game will be introduced and it will be compared to the classical strategy and CHSH game strategy.

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Abstract In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a $$\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )$$ | G H Z 4 ⟩ = 1 2 ( | 0000 ⟩ + | 1111 ⟩ ) state, breaking the traditional 10% gain observed in 2- and 3-qubit CHSH games and achieving a 22.5% gap. In the other game, players gain a greater advantage using a $$\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )$$ | W 4 ⟩ = 1 2 ( | 0001 ⟩ + | 0010 ⟩ + | 0100 ⟩ + | 1000 ⟩ ) state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.

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Matej Pivoluska, Martin Plesch

A CHSH q game is a generalization of the standard two player CHSH game, with q different input and output options. In contrast to the binary game, the best classical and quantum winning strategies are not known exactly. In this paper we provide a constructive classical strategy for winning a CHSH q game, with q being a prime. Our construction achieves a winning probability better than , which is in contrast with the previously known constructive strategies achieving only the winning probability of .

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The Quantum game theory is an extension of the classical game theory which describes games in which players share quantum information during the game. The use of the quantum mechanical effects enables players’ better results than they would achieve in the classical game without sharing any information at all. In this paper I would like to describe a new two player cooperative game which is based on the well known CHSH game. A quantum strategy for a new game will be introduced and it will be compared to the classical strategy and CHSH game strategy.
Abstract In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a ∣GHZ4⟩=12(∣0000⟩+∣1111⟩)\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )∣GHZ4​⟩=2​1​(∣0000⟩+∣1111⟩) | G H Z 4 ⟩ = 1 2 ( | 0000 ⟩ + | 1111 ⟩ ) state, breaking the traditional 10% gain observed in 2- and 3-qubit CHSH games and achieving a 22.5% gap. In the other game, players gain a greater advantage using a ∣W4⟩=12(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩)\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )∣W4​⟩=21​(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩) | W 4 ⟩ = 1 2 ( | 0001 ⟩ + | 0010 ⟩ + | 0100 ⟩ + | 1000 ⟩ ) state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.
In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a GHZGHZGHZ-state, breaking the traditional 10\% gain observed in 2 and 3 qubit CHSH games and achieving a 22.5\% gap. In the other game, players gain a greater advantage using a WWW-state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.
In this paper, we generalize to three players the well-known CHSH quantum game. To do so, we consider all possible 3-variables Boolean functions and search among them which ones correspond to a game scenario with a quantum advantage (for a given entangled state). In particular, we provide two new three-player quantum games where, in one case, the best quantum strategy is obtained when the players share a GHZGHZGHZ state, while in the other one, the players have a better advantage when they use a WWW state as their quantum resource. To illustrate our findings we implement our game scenarios on an online quantum computer and prove experimentally the advantage of the corresponding quantum resource for each game.
Recently the concept of quantum information has been introduced into game theory. Here we present the first study of quantum games with more than two players. We discover that such games can possess an alternative form of equilibrium strategy, one which has no analog either in traditional games or even in two-player quantum games. In these ``coherent'' equilibria, entanglement shared among multiple players enables different kinds of cooperative behavior: indeed it can act as a contract, in the sense that it prevents players from successfully betraying one another.
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.
An explicit classical strategy for winning a CHSHq{\mathrm{CHSH}}_{q}CHSHq​ game
A CHSH q game is a generalization of the standard two player CHSH game, with q different input and output options. In contrast to the binary game, the best classical and quantum winning strategies are not known exactly. In this paper we provide a constructive classical strategy for winning a CHSH q game, with q being a prime. Our construction achieves a winning probability better than , which is in contrast with the previously known constructive strategies achieving only the winning probability of .