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arXiv:0907.2293 · Communications in Theoretical Physics

Quantum Monty Hall Problem under Decoherence

Salman Khan, M Ramzan, M.K Khan

Published Jul 1, 2010• 13 citations
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Abstract

We study the effect of decoherence on quantum Monty Hall problem under the influence of amplitude damping, depolarizing, and dephasing channels. It is shown that under the effect of decoherence, there is a Nash equilibrium of the game in case of depolarizing channel for Alice's quantum strategy. Whereas in case of dephasing noise, the game is not influenced by the quantum channel. For amplitude damping channel, Bob's payoffs are found symmetrical about a decoherence of 50% and the maximum occurs at this value of decoherence for his classical strategy. However, it is worth-mentioning that in case of depolarizing channel, Bob's classical strategy remains always dominant against any choice of Alice's strategy.

DOI
10.1088/0253-6102/54/1/10

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We study the effect of decoherence on quantum Monty Hall problem under the influence of amplitude damping, depolarizing, and dephasing channels. It is shown that under the effect of decoherence, there is a Nash equilibrium of the game in case of depolarizing channel for Alice's quantum strategy. Whereas in case of dephasing noise, the game is not influenced by the quantum channel. For amplitude damping channel, Bob's payoffs are found symmetrical about a decoherence of 50% and the maximum occurs at this value of decoherence for his classical strategy. However, it is worth-mentioning that in case of depolarizing channel, Bob's classical strategy remains always dominant against any choice of Alice's strategy.
We reformulate the quantum Monty Hall problem in the presence of decoherence. The decoherence destroys the fairness of the game. A new Nash equilibrium for a particular strategy profile in the presence of decoherence emerges. It is shown that in the presence of decoherence under the action of amplitude damping channel, Bob's winning probability is always higher than three-forth, irrespective of Alice's strategy, if he does not switch to the other door and always wins for a fully decohered case of the channel. Depolarizing channel damps up Bob's winning probability and gets better off if he sticks to his current selection. Phase damping channel leaves the winning probability unaffected. Unlike the classical and the quantum forms of the game, Bob's dominant strategy in the presence of decoherence is not switching.
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 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.
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.
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 influence of entanglement and correlated noise using correlated amplitude damping, depolarizing and phase damping channels on the quantum Stackelberg duopoly. Our investigations show that under the influence of an amplitude damping channel a critical point exists for an unentangled initial state at which firms get equal payoffs. The game becomes a follower advantage game when the channel is highly decohered. Two critical points corresponding to two values of the entanglement angle are found in the presence of correlated noise. Within the range of these limits of the entanglement angle, the game is a follower advantage game. In the case of a depolarizing channel, the payoffs of the two firms are strongly influenced by the memory parameter. The presence of quantum memory ensures the existence of the Nash equilibrium for the entire range of decoherence and entanglement parameters for both the channels. A local maximum in the payoffs is observed which vanishes as the channel correlation increases. Moreover, under the influence of the depolarizing channel, the game is always a leader advantage game. Furthermore, it is seen that the phase damping channel does not affect the outcome of the game.