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arXiv:1012.1428

Quantum Games and Quantum Discord

Ahmad Ahsan Nawaz, A. H. Toor

Published Dec 7, 2010• 2 citations
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Abstract

We quantize prisoners dilemma and chicken game by our generalized quantization scheme to explore the role of quantum discord in quantum games. In order to establish this connection we use Werner-like state as an initial state of the game. In this quantization scheme measurement can be performed in entangled as well as in product basis. For the measurement in entangled basis the dilemma in both the games can be resolved by separable states with non-zero quantum discord. Similarly for product basis measurement the payoffs are quantum mechanical only for nonzero values of quantum discord.

DOI
10.48550/arxiv.1012.1428

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We quantize prisoners dilemma and chicken game by our generalized quantization scheme to explore the role of quantum discord in quantum games. In order to establish this connection we use Werner-like state as an initial state of the game. In this quantization scheme measurement can be performed in entangled as well as in product basis. For the measurement in entangled basis the dilemma in both the games can be resolved by separable states with non-zero quantum discord. Similarly for product basis measurement the payoffs are quantum mechanical only for nonzero values of quantum discord.
In a normal form, prisoners' dilemma (PD) is represented by a payoff matrix showing players' strategies and payoffs. To obtain the distinguishing trait and strategic form of PD, certain constraints are imposed on the elements of its payoff matrix. We quantize PD by a generalized quantization scheme to analyze its strategic behavior in the quantum domain. The game starts with a general entangled state of the form and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements, there exist respective cutoff values of entanglement of the initial quantum state up to which the strategic form of the game remains intact. Beyond these cutoffs the quantized PD behaves like the chicken game (CG) up to another cutoff value. For the measurement in the entangled basis the dilemma is resolved for sinξ > 1/7 with Q ⊗ Q as a Nash Equilibrium (NE). However, the quantized game behaves like PD when sinξ > 1/3; whereas in the range 1/7 < sinξ < 1/3 it behaves like CG with Q ⊗ Q as an NE. For the measurement in the product basis the quantized PD behaves like classical PD for with D ⊗ D as an NE. In region , the quantized PD behaves like classical CG with C ⊗ D and D ⊗ C as NEs.
In its normal form prisoners' dilemma (PD) is represented by a payoff matrix showing players strategies and payoffs. To obtain distinguishing trait and strategic form of PD certain constraints are imposed on the elements of its payoff matrix. We quantize PD by generalized quantization scheme to analyze its strategic behavior in quantum domain. The game starts with general entangled state of the form $\left}ψ\right\rangle =\cos\fracξ% {2}\left|00\right\rangle +i\sin\fracξ{2}\left|11\right\rangle $ and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements there exist respective cutoff values of entanglement of initial quantum state up to which strategic form of game remains intact. Beyond these cutoffs the quantized PD behaves like chicken game up to another cutoff value. For the measurement in entangled basis the dilemma is resolved for\ sin⁡ξ>17\sinξ>\frac{1}{7}sinξ>71​ with Q⊗QQ\otimes QQ⊗Q as a NE but the quantized game behaves like PD when sin⁡ξ>13\sinξ>\frac{1}{3}sinξ>31​; whereas in the range $\frac{1}{7}
We present a quantization scheme for a three-player Prisoner's Dilemma game. It is shown that entanglement plays a dominant role in the three-player quantum game. Four different types of payoffs are identified on the basis of different combinations of initial state and measurement basis entanglement parameters. A relation among these different payoffs is also established. We also study the communication aspects of the three-player game. By exploiting different combinations of initial state and measurement basis entanglement parameters, we establish a relationship for the information shared among the parties. It is seen that the strategies of the players act as carriers of information in quantum games.
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.