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Photonics

Prisoners’ Dilemma in a Spatially Separated System Based on Spin–Photon Interactions

Azmi Ali Altıntaṣ, Fatih Özaydin, Cihan Bayındır, Veysel Bayrakci

Published Aug 30, 2022• 18 citations
Publisher

Abstract

Having access to ideal quantum mechanical resources, the prisoners’ dilemma can be ceased. Here, we propose a distributed quantum circuit to allow spatially separated prisoners to play the prisoners’ dilemma game. Decomposing the circuit into controlled-Z and single-qubit gates only, we design a corresponding spin–photon-interaction-based physical setup within the reach of current technology. In our setup, spins are considered to be the players’ logical qubits, which can be realized via nitrogen-vacancy centers in diamond or quantum dots coupled to optical cavities, and the game is played via a flying photon realizing logic operations by interacting with the spatially separated optical cavities to which the spin qubits are coupled. We also analyze the effect of the imperfect realization of two-qubit gates on the game, and discuss the revival of the dilemma and the emergence of new Nash equilibria.

DOI
10.3390/photonics9090617

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Having access to ideal quantum mechanical resources, the prisoners’ dilemma can be ceased. Here, we propose a distributed quantum circuit to allow spatially separated prisoners to play the prisoners’ dilemma game. Decomposing the circuit into controlled-Z and single-qubit gates only, we design a corresponding spin–photon-interaction-based physical setup within the reach of current technology. In our setup, spins are considered to be the players’ logical qubits, which can be realized via nitrogen-vacancy centers in diamond or quantum dots coupled to optical cavities, and the game is played via a flying photon realizing logic operations by interacting with the spatially separated optical cavities to which the spin qubits are coupled. We also analyze the effect of the imperfect realization of two-qubit gates on the game, and discuss the revival of the dilemma and the emergence of new Nash equilibria.
Decision-making by the two negotiating parties is simulated by a prisoner's dilemma game.The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options.Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the resulting transformed state.The range of strategies allowed for quantum players is richer than in case of a classical game and therefore the result of the game can be better optimized.On the other hand, the quantum game is save against eavesdropping and the players can be assured that this type of quantum arbitration is fair.We show that quantum prisoner's dilemma has more favorable Nash equilibria than its classical analog and they are close to the Pareto optimal solutions.Some economical examples of utilizing quantum game Nash equilibria are proposed.
A quantized version of the prisoner’s dilemma game based on the Marinatto–Weber quantization scheme is introduced. A four-qubit-basis initial entangled quantum state is assumed. Two domains for Nash equilibria are observed in terms of the degree of entanglement of the game. Each Nash equilibrium point is found to be Pareto optimal in its corresponding domain. Therefore, there is a unique solution to the game dilemma in each domain of the entanglement parameter. The maximally entangled limit is studied, and the players’ payoffs are found to equal the average of the payoff matrix independently of their choices. Consequently, any combination of players’ choices is a solution to the dilemma of the game. The nonentangled limits are also studied and found to coincide with the classical mixed-strategy game. Our results are equivalent to those of the noisy schemes of the quantum prisoner’s dilemma game. In our scheme, entanglement can be interpreted as a noise.
The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.
We report the first demonstration of a quantum game on an all-optical one-way quantum computer. Following a recent theoretical proposal we implement a quantum version of Prisoner's Dilemma, where the quantum circuit is realized by a four-qubit box-cluster configuration and the player's local strategies by measurements performed on the physical qubits of the cluster. This demonstration underlines the strength and versatility of the one-way model and we expect that this will trigger further interest in designing quantum protocols and algorithms to be tested in state-of-the-art cluster resources.
The dynamics of a spatial quantum formulation of the iterated prisoner's dilemma game is studied in this work. The game is played in the cellular automata manner, i.e. with local and synchronous interaction. The evolution is driven by the imitation of the best neighbour. It is shown how spatial structure enables in fully quantum contests a dramatically rapid emergence of mutual cooperation. In unfair contests, such as quantum versus classic players, the velocity in which the quantum players take advantage of its privileged status is also very notable.
The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner's Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies' impact on traditional game-theoretical problems.