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International Conference on Data Analytics for Business and Industry (ICDABI)

Initial State Time-Dependent Quantum Game

A. T. M. Makram-Allah, Mahmoud Youssef Abd‐Rabbou, NASSER METWALLY

Published Oct 23, 2024• 1 citation
Publisher

Abstract

The possibility of implementing a time dependent version of Eisert, Wilkens, and Lewenstein (EWL), quantum game is discussed. It is assumed that the players initial states are time dependent, while the entangling quantum gate either time dependent or independent during the game. The obtained results shows that the players payoff depend on their initial state, while The time-independent entangling gate has a minor impact on the game compared to its time-dependent counterpart. However, one can improve the lower bounds of the payoffs by using time dependent gate. Moreover, the exponential form of the entangling power is more powerful than the linear one, where the payoffs are much better. However, the maximum potential rewards in this game may be lower than those attainable in the classical game, the minimum guaranteed outcomes are much better.

DOI
10.1109/icdabi63787.2024.10800664

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The possibility of implementing a time dependent version of Eisert, Wilkens, and Lewenstein (EWL), quantum game is discussed. It is assumed that the players initial states are time dependent, while the entangling quantum gate either time dependent or independent during the game. The obtained results shows that the players payoff depend on their initial state, while The time-independent entangling gate has a minor impact on the game compared to its time-dependent counterpart. However, one can improve the lower bounds of the payoffs by using time dependent gate. Moreover, the exponential form of the entangling power is more powerful than the linear one, where the payoffs are much better. However, the maximum potential rewards in this game may be lower than those attainable in the classical game, the minimum guaranteed outcomes are much better.
Abstract The possibility of implementing the Prisoner's Dilemma game using different classes of initial states between the players is discussed. Different behaviors of the payoffs have been exhibited, may it increases, decreases or fixed. These behaviors depend on the used initial state either maximum/partial entangled, the chosen entangling gate, and the strategies employed by each player. The increased payoff is displayed when the players adopt different defect strategies, while the fixed payoff is shown when both players apply the same defect strategy. The maximum bounds of the payoff can be attained if the players initially share a maximum entangled state. The maximum /minimum fixed payoff depends on the used entangling gate, where the maximum one is shown if the players send their final states to entangling gate. It is shown that, the nature of the payoff, whether stationary, increasing or decreasing is contingent upon the polarization of the initial state and the used defect strategy. However, if both states are polarized on the same axis direction, the predicted maximum payoff is smaller than that obtained if the initial state are polarized in different polarization of the same axis. Therefore, the initial state settings, entangling parameter and the choice of the defect strategy can be used as control tools to maximize/minimize the obtained payoff.
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.
Abstract The maximally entangled (ME) Eisert–Wilkens–Lewenstein (EWL) quantization scheme efficiently solves the classical prisoner’s dilemma (PD). Despite the intensive research on the EWL scheme, little is known about the role of the arbiter in controlling the scheme properties. We redefine the EWL scheme by introducing an arbitrary phase shift into the ME initial state. Four basic initial states were analyzed in this study. It is found that the arbiter completely controls the characteristics of the arbitrary ME quantum PD. The arbiter can adjust the ME initial state to obtain an efficient symmetric Nash equilibrium. In contrast, the arbiter can nullify the quantization advantages. Moreover, it can preserve/violate the symmetry of the game or even turn it into a potential game.
We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to Quantumize\textit{Quantumize}Quantumize any finite classical nnn-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.
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.