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arXiv:1102.1351 · Journal of Physics A Mathematical and Theoretical

Relativistic quantum games in noninertial frames

Salman Khan, Majid Khan

Published Aug 10, 2011• 23 citations
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Abstract

We study the influence of the Unruh effect on quantum non-zero sum games. In particular, we investigate the quantum Prisoners’ Dilemma both for entangled and unentangled initial states and show that the acceleration of the noninertial frames disturbs the symmetry of the game. It is shown that for the maximally entangled initial state, the classical strategy (cooperation) becomes the dominant strategy. Our investigation shows that any quantum strategy does no better for any player against the classical strategies. The miracle move of Eisert et al (1999 Phys. Rev. Lett. 83 3077) is no more a superior move. We show that the dilemma-like situation is resolved in favor of one player or the other.

DOI
10.1088/1751-8113/44/35/355302

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We study the influence of the Unruh effect on quantum non-zero sum games. In particular, we investigate the quantum Prisoners’ Dilemma both for entangled and unentangled initial states and show that the acceleration of the noninertial frames disturbs the symmetry of the game. It is shown that for the maximally entangled initial state, the classical strategy (cooperation) becomes the dominant strategy. Our investigation shows that any quantum strategy does no better for any player against the classical strategies. The miracle move of Eisert et al (1999 Phys. Rev. Lett. 83 3077) is no more a superior move. We show that the dilemma-like situation is resolved in favor of one player or the other.
The influence of noise and of Unruh effect on quantum Prisoners’ dilemma is investigated both for entangled and unentangled initial states. The noise is incorporated through amplitude damping channel. For unentangled initial state, the decoherence compensates for the adverse effect of acceleration of the frame and the effect of acceleration becomes irrelevant provided the game is fully decohered. It is shown that the inertial player always out scores the noninertial player by choosing defection. For maximally entangled initially state, we show that for fully decohered case every strategy profile results in either of the two possible equilibrium outcomes. Two of the four possible strategy profiles become Pareto optimal and Nash equilibrium and no dilemma is leftover. It is shown that other equilibrium points emerge for different region of values of decoherence parameter that are either Pareto optimal or Pareto inefficient in the quantum strategic spaces. It is shown that the Eisert et al. (Phys Rev Lett 83:3077, 1999) miracle move is a special move that leads always to distinguishable results compare to other moves. We show that the dilemma like situation is resolved in favor of one player or the other.
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The effect of increasing quantum bits and Unruh effect on quantum Prisoners’ dilemma has been investigated for both entangled and unentangled initial states. The Nash equilibrium, as an important result of quantum game theory, was obtained through the different payoffs resulted from choosing various strategies. It has been shown that the non-inertial frame disturbs the symmetry of the game. Actually, selection of the basic quantum strategy by players and calculating the payoff of the game via the density matrix of 4-qubit quantum states can represent a scale of influence of entanglement in a quantum mechanical system
We investigate the influence of the Unruh effect on the quantum Battle of the Sexes (BOF). We show that the acceleration of a noninertial frame greatly affects the payoffs of the players. In the case of Battle of the Sexes, the Nash Equilibra in the inertial frame may be equilibrium but in other games this conclusion may not be true anymore. Furthermore, we also show that the new Nash Equilibria of the game played in the noninertial frame and the probability distributions of the players in the new Nash Equilibria vary with the acceleration of moving player and the degree of entanglement in the initial state shared by the players.
In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.
Effects of quantum and classical correlations on game theory are studied to clarify the new aspects brought into game theory by the quantum mechanical toolbox. In this study, we compare quantum correlation represented by a maximally entangled state and classical correlation that is generated through phase damping processes on the maximally entangled state. Thus, this also sheds light on the behavior of games under the influence of noisy sources. It is observed that the quantum correlation can always resolve the dilemmas in non-zero sum games and attain the maximum sum of both players' payoffs, while the classical correlation cannot necessarily resolve the dilemmas.