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arXiv:1306.5957 · Quantum Information Processing

Decoherence effects in the quantum qubit flip game using Markovian approximation

Piotr Gawron, Dariusz Kurzyk, Łukasz Pawela

Published Nov 16, 2013• 23 citations
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Abstract

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.

DOI
10.1007/s11128-013-0681-y

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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 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.
In this paper, we investigate the influence of entanglement and decoherence on the quantum Stackelberg duopoly (QSD) game. It is shown that the first-mover advantage can be weakened or enhanced due to the existence of entanglement for the QSD game without decoherence. The influence of decoherence induced by the amplitude damping and the phase damping are explicitly studied in the formalism of Kraus operator representations. We show that the amplitude damping drastically changes the Nash equilibrium of the QSD game and the profits of the two players while the phase damping does not affect the Nash equilibrium and the profits of the two players. It is found that under certain conditions there exists a 'critical point' of the damping parameter for the amplitude damping environment. At the 'critical point' the two players have the same moves and payoffs. The QSD game can change from the first-mover advantage game into the follower-mover advantage game when the damping parameter varies from the left-hand-side regime of the 'critical point' to the right-hand-side regime.
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.
A quantum version of the Minority game for an arbitrary number of agents is studied. When the number of agents is odd, quantizing the game produces no advantage to the players, however, for an even number of agents new Nash equilibria appear that have no classical analogue. The new Nash equilibria provide far preferable expected payoffs to the players compared to the equivalent classical game. The effect on the Nash equilibrium payoff of reducing the degree of initial entanglement between the players' qubits or of introducing various forms of decoherence is discussed. As the number of players increases the multipartite GHZ state becomes increasingly fragile, as indicated by the smaller error probability required to reduce the Nash equilibrium payoff to the classical level.
A quantum version of the Minority game for an arbitrary number of agents is considered. It is known that when the number of agents is odd, quantizing the game produces no advantage to the players, but for an even number of agents new Nash equilibria appear that have no classical analogue and have improved payoffs. We study the effect on the Nash equilibrium payoff of various forms of decoherence. As the number of players increases the multipartite GHZ state becomes increasingly fragile, as indicated by the smaller error probability required to reduce the Nash equilibrium payoff to the classical level.