Quantum Games under Decoherence

Abstract

Quantum systems are easily influenced by ambient environments. Decoherence is generated by system interaction with external environment. In this paper, we analyse the effects of decoherence on quantum games with Eisert-Wilkens-Lewenstein (EWL) (Eisert et al., Phys. Rev. Lett. 83(15), 3077 1999) and Marinatto-Weber (MW) (Marinatto and Weber, Phys. Lett. A 272, 291 2000) schemes. Firstly, referring to the analytical approach that was introduced by Eisert et al. (Phys. Rev. Lett. 83(15), 3077 1999), we analyse the effects of decoherence on quantum Chicken game by considering different traditional noisy channels. We investigate the Nash equilibria and changes of payoff in specific two-parameter strategy set for maximally entangled initial states. We find that the Nash equilibria are different in different noisy channels. Since Unruh effect produces a decoherence-like effect and can be perceived as a quantum noise channel (Omkar et al., arXiv:1408.1477v1), with the same two parameter strategy set, we investigate the influences of decoherence generated by the Unruh effect on three-player quantum Prisoners' Dilemma, the non-zero sum symmetric multiplayer quantum game both for unentangled and entangled initial states. We discuss the effect of the acceleration of noninertial frames on the the game's properties such as payoffs, symmetry, Nash equilibrium, Pareto optimal, dominant strategy, etc. Finally, we study the decoherent influences of correlated noise and Unruh effect on quantum Stackelberg duopoly for entangled and unentangled initial states with the depolarizing channel. Our investigations show that under the influence of correlated depolarizing channel and acceleration in noninertial frame, some critical points exist for an unentangled initial state at which firms get equal payoffs and the game becomes a follower advantage game. Int J Theor PhysIt is shown that the game is always a leader advantage game for a maximally entangled initial state and there appear some points at which the payoffs become zero.

Quantum version of Prisoners' Dilemma under Interacting Environment

“…has been represented pictorially by Figure 1. Figure 1(a) shows the distribution of Alice's payoff at t = 0, which is similar to Eisert et al [1] and that after t = 1000 is shown by Figure 1(b), which is consistent with Huang and Qiu [6]. The Nash equilibrium ( Q, Q) disappears as the time step runs from 0 to 1000.…”

A quantum Samaritan’s dilemma cellular automaton

“…The effect of other types of quantum noise [16] will be taken into account in future work. In particular that of phase-damping, structurally close to amplitude-damping as both damping noises have the same K 1 Kraus operator, deferring in $K_2=false(\begin{array}{cc}.2em0.333em0& 0\\ 0& \sqrt{\mu }\end{array}false)$ with phase damping.…”

“…Let us remark that only amplitude-damping quantum noise will be under scrutiny in this study. The effect of other types of quantum noise [ 16 ] will be taken into account in future work. In particular that of phase-damping, structurally close to amplitude-damping as both damping noises have the same K 1 Kraus operator, deferring in with phase damping.…”

“…General pay-off matrices [7] and other noise types [16,[29][30][31] are to come under scrutiny in subsequent studies on iterated quantum games in the spatial context. Other quantization schemes [32][33][34][35], multiparty games [36,37] as well as games with imperfect information [23,[38][39][40][41] deserve particular studies.…”

Exploring the impact of spontaneous emission in two quantum game-based key-distribution protocols

“…When dealing with real-world implementations of quantum systems, noise and decoherence must be taken into account to some extent in the theoretical model, quantum games are no exception [32][33][34]. In particular, a noisy version of Flitney and Abbot's Monty Hall quantum game was analyzed back in 2010 [35].…”