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arXiv:1102.5056 · Fluctuation and Noise Letters

ENVIRONMENT-ASSISTED QUANTUM MINORITY GAMES

Muhammad Asif Ramzan, Majid Khan

Published Nov 18, 2013• 13 citations
arXivHTMLPDF

Abstract

The effect of entanglement and correlated noise in a four-player quantum Minority game (QMG) is investigated. Different time correlated quantum memory channels are considered to analyze the Nash equilibrium (NE) payoff of the first player. It is seen that the NE payoff is substantially enhanced due to the presence of correlated noise. The behavior of damping channels (amplitude damping (AD) and phase damping) is approximately similar. However, bit-phase flip channel heavily influences the Minority game as compared to other channels in the presence of correlated noise. On the other hand, phase flip channel has a symmetrical behavior around 50% noise threshold. The significant reduction in payoffs due to decoherence is well compensated due to the presence of correlated noise. However, the NE of the game does not change in the presence of noise. It is seen that in case of generalized AD channel, entanglement plays a significant role at lower level of decoherence. The channel has less dominant effects on the payoff at higher values of decoherence. Furthermore, AD and generalized amplitude damping channels have almost comparable effects at lower level of decoherence (p < 0.5). Therefore, the game deserves careful study during its implementation due to prominent role of noise for different channels.

DOI
10.1142/s0219477513500259

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The effect of entanglement and correlated noise in a four-player quantum Minority game (QMG) is investigated. Different time correlated quantum memory channels are considered to analyze the Nash equilibrium (NE) payoff of the first player. It is seen that the NE payoff is substantially enhanced due to the presence of correlated noise. The behavior of damping channels (amplitude damping (AD) and phase damping) is approximately similar. However, bit-phase flip channel heavily influences the Minority game as compared to other channels in the presence of correlated noise. On the other hand, phase flip channel has a symmetrical behavior around 50% noise threshold. The significant reduction in payoffs due to decoherence is well compensated due to the presence of correlated noise. However, the NE of the game does not change in the presence of noise. It is seen that in case of generalized AD channel, entanglement plays a significant role at lower level of decoherence. The channel has less dominant effects on the payoff at higher values of decoherence. Furthermore, AD and generalized amplitude damping channels have almost comparable effects at lower level of decoherence (p < 0.5). Therefore, the game deserves careful study during its implementation due to prominent role of noise for different channels.
We study the effect of quantum noise in 3 × 3 entangled quantum games. By taking into account different noisy quantum channels, we analyze how a two-player, three-strategy Rock–Scissor–Paper game is influenced by the quantum noise. We consider the winning non-transitive strategies R , S and P such that R beats S , S beats P and P beats R . The game behaves as a noiseless game for the maximum value of the quantum noise. It is seen that Alice's payoff is heavily influenced by the depolarizing noise as compared to the amplitude damping noise. A depolarizing channel causes a monotonic decrease in players' payoffs as we increase the amount of quantum noise. In the case of the amplitude damping channel, Alice's payoff function reaches its minimum for α = 0.5 and is symmetrical. This means that larger values of quantum noise influence the game weakly. On the other hand, the phase damping channel does not influence the game. Furthermore, the Nash equilibrium and non-transitive character of the game are not affected under the influence of quantum noise.
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Abstract We study the influence of memory effects on the quantum Stackelberg duopoly game through an amplitude-damping channel where successive uses of the channels are correlated. It is shown that the memory effects can drastically change the Nash equilibrium and the payoffs of the two firms. As the degree of channel memory increases, there exists a Nash equilibrium for the entire range of the entanglement parameter. Similarly, the presence of memory ensures the existence of the Nash equilibrium for the whole range of the decoherence parameter both for entangled and unentangled initial states. Moreover, quantum memory leads to a ‘critical point’ of the damping parameter, at which both firms are equally benefited. With the maximum-memory effect, the position of the ‘critical point’ moves to the place of fully decohered.
We study the behavior of cooperative multiplayer quantum games [35,36] in the presence of decoherence using different quantum channels such as amplitude damping, depolarizing and phase damping. It is seen that the outcomes of the games for the two damping channels with maximum values of decoherence reduce to same value. However, in comparison to phase damping channel, the payoffs of cooperators are strongly damped under the influence\ amplitude damping channel for\ the lower values of decoherence parameter. In the case of depolarizing channel, the game is a no-payoff game irrespective of the degree of entanglement in the initial state for the larger values of decoherence parameter. The decoherence gets the cooperators worse off.
We are considering a quantum version of the penny flip game, whose implementation is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. We focus our attention on the phase damping, amplitude damping and amplitude raising channels. Our results show that the Pauli strategy is no longer a Nash equilibrium under decoherence. We attempt to optimize the players’ control pulses in the aforementioned setup to allow them to achieve higher probability of winning the game compared with the Pauli strategy.
We study the behavior of cooperative multiplayer quantum games [Q. Chen, Y. Wang, J.T. Liu, and K.L. Wang, Phys. Lett. A 327 (2004) 98; A.P. Flitney and L.C.L. Hollenberg, Quantum Inf. Comput. 7 (2007) 111] in the presence of decoherence using different quantum channels such as amplitude damping, depolarizing and phase damping. It is seen that the outcomes of the games for the two damping channels with maximum values of decoherence reduce to same value. However, in comparison to phase damping channel, the payoffs of cooperators are strongly damped under the influence amplitude damping channel for the lower values of decoherence parameter. In the case of depolarizing channel, the game is a no-payoff game irrespective of the degree of entanglement in the initial state for the larger values of decoherence parameter. The decoherence gets the cooperators worse off.