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Quantum Games with Correlated Noise

Nawaz, A, A. H. Toor

Published Mar 7, 2006• 23 citations
OpenAlex

Abstract

We analyze quantum game with correlated noise through generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analyzed. It is shown that the advantage that a quantum player can get by exploiting quantum strategies is only valid when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.

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Quantum games with correlated noise

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arXiv:1209.5332•Sep 24, 2012•0 citations
We analyze quantum game with correlated noise through generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analyzed. It is shown that the advantage that a quantum player can get by exploiting quantum strategies is only valid when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.
We analyse quantum games with correlated noise through a generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analysed. It is shown that the quantum player only enjoys an advantage over the classical player when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.
This work is mainly based on quantum game-theoretic techniques and their application to quantum information theory.Quantum game theory is an important development in quantum computation and may have implications for quantum information and quantum communication.The select of quantum memory on quantum games have been studied.Three di􀀞erent games such as, Prisoner Dilemma, Battle of the Sexes and Chicken have been analyzed in the quantum domain. By considering the restricted game \nsituation, it is shown that the ects of quantum memory and decoherence become e􀀞ective in a maximally entangled case.For which the quantum player can out perform the classical player in all the three games. It is also shown that the quantum player enjoys an advantage over the classical player in Battle of the Sexes game for the amplitude-damping and the depolarizing channels.The quantum memory compensates the reduction in player pay is due to decoherence. It has no ect on the Nash equilibria of the three games.A generalization of two-player quantization scheme to a three-player Prisoners Dilemma game under the ect of correlated noise is also presented.In a restricted game scenario, it is shown the quantumplayer is always better; for all values of the decoherence parameter for the entire range of memory parameter.It is seen that for maximum degree of correlation, the game does not become \nnoiseless and quantum player can still outscore the classical players for the entire range of the decoherence parameter producing an interesting result in comparison with a two-player quantum game.It is also shown that correlated noise has not on the Nash equilibrium of the game.By exploiting the three-player quantization scheme, we have studiedthe communication aspects of a three-player Prisoner Dilemma quantum game. It is shown that entanglement plays a dominant role in a three-player quantum game.On the baois of initial state and the measurement basis entanglement parameters, a relation among different pay is established.It is investigated that the strategies of the players act as information carriers in quantum games. A relationship for the information shared among the parties is also established.A scheme for quantum key distribution is proposed in which a secret key can be generated from the data coming through a partially entanglement breaking channel.This scheme is rather deterministic andcient in the sense that two classical bits can be transferred \nper entangled pair of qubits.Furthermore, it is important to point out that, in this scheme, same symbol can be used for Eve detection and key generation. It is also worth noting that the eavesdropper, Eve, can be detected very easily from the disturbance of the elements of the decoding bimatrix. We have checked for the \nsecurity of the scheme and found it secure against individual attacks.It is further investigated that quantum games are useful in developing quantum cryptographic protocols.A multiparty quantum cryptographic protocol using tripartite entangled GHZ states is devised using game-theoretic techniques.In this protocol, two classical bits can be transferred per entangled pair of qubits to the communicating parties.Unitary operators applied by the sender on a tripartite entangled state encodes a classical symbol that can be decoded at receiver end with the help of a decoding matrix. In this protocol, if Eve interferes during the transmission, she can be detected by the disturbance of the decoding matrix. Our security analysis shows that this protocol is also secure against intercept/re-send attacks.
A protocol for considering decoherence in quantum games is presented. Results for general two-player, two-strategy quantum games subject to decoherence are derived and some specific examples are given. Decoherence in other types of quantum games is also considered. As expected, the advantage that a quantum player achieves over a player restricted to classical strategies is diminished for increasing decoherence but only vanishes in the limit of maximum decoherence when the results of the game are randomized.
We study quantum games with correlated noise through a generalized quantization scheme. We investigate the effects of memory on quantum games, such as Prisoner's Dilemma, Battle of the Sexes and Chicken, through three prototype quantum-correlated channels. It is shown that the quantum player enjoys an advantage over the classical player for all nine cases considered in this paper for the maximally entangled case. However, the quantum player can also outperform the classical player for subsequent cases that can be noted in the case of the Battle of the Sexes game. It can be seen that the Nash equilibria do not change for all the three games under the effect of memory.
It has recently been established that quantum strategies have great advantage over classical ones in quantum games. However, quantum states are easily affected by the quantum noise resulting in decoherence. In this paper, we investigate the effect of quantum noise on the restricted quantum game in which one player is restricted in classical strategic space, another in quantum strategic space and only the quantum player is affected by the quantum noise. Our results show that in the maximally entangled state, no Nash equilibria exist in the range of 0< p ⩽0.422 ( p is the quantum noise parameter), while two special Nash equilibria appear in the range of 0.422< p <1. The advantage that the quantum player diminished only in the limit of maximum quantum noise. Increasing the amount of quantum noise leads to the increase of the classical player's payoff and the reduction of the quantum player's payoff, but is helpful in forming two Nash equilibria.
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is classical correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner's Dilemma into the game of Chicken. We show that this behaviour, and the associated enhanced equilibrium payoff over playing the game with quantum objects in non-entangled states, is entirely due to the classical part of the correlation. Generalizing these games to the pure strategy 2-player quantum game where the players have finite strategy sets and a projective joint measurement is made on the output state produced by the players, we show that a given quantum game of this form can always be reproduced by a classical model, such as a communication channel. Where entanglement is a feature of the these 2-player quantum games the matrix of expected outcomes for the players can be reproduced by a classical channel with correlated noise.