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European Quantum Electronics Conference (EQEC)

Experimental implementation of a quantum game

Carsten Schuck, O. Schulz, Christian Kurtsiefer, Harald Weinfurter

Published Nov 15, 2004• 2 citations
Publisher

Abstract

Experimental realization of the game is presented using polarization-time-entangled photon pairs on which arbitrary (quantum-) strategies can be performed with adjustable birefringent elements. For the demonstration of the efficiency of quantum strategies, the strategy of each of the two players can be realized as a unitary transformation performed on one spin-1/2 particle. The polarization-entangled photon pairs are generated with high temporal definition in a parametric down-conversion process. Initially they are prepared in a Kraus state on which the two players perform their single-particle operations using the corresponding angular settings of a combination of /spl lambda//2- and /spl lambda//4-waveplates.

DOI
10.1109/eqec.2003.1314280

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arXiv:quant-ph/0508105•Aug 15, 2005•0 citations
Experimental realization of the game is presented using polarization-time-entangled photon pairs on which arbitrary (quantum-) strategies can be performed with adjustable birefringent elements. For the demonstration of the efficiency of quantum strategies, the strategy of each of the two players can be realized as a unitary transformation performed on one spin-1/2 particle. The polarization-entangled photon pairs are generated with high temporal definition in a parametric down-conversion process. Initially they are prepared in a Kraus state on which the two players perform their single-particle operations using the corresponding angular settings of a combination of /spl lambda//2- and /spl lambda//4-waveplates.
Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now, only two- and three-player quantum games have been demonstrated with restricted strategy sets. Here, we report the first experiment that implements a four-player quantum minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate player strategies gives equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our results demonstrate for the first time how nontrivial equilibria can arise in a competitive situation involving quantum agents.
We quantize the classical game of simultaneous two-person duel. In this game, the players Alice and Bob, have a two level system, for example, a spin 1/2 particle, which models alive and dead states. Through physical considerations we write up the unitary operator which represents the classical action of firing, which quantum mechanically is represented by flipping the opponent’s spin and test its classical limits. We study three representative special cases of such duel where players start the duel: both alive; in a Bell state; and when one of them has her skills decreasing along the duel. We also present a proposal of experimental realization of this quantum game using twin photons produced by spontaneous parametric down conversion.
Abstract We demonstrate a Bayesian quantum game on an ion-trap quantum computer with five qubits. The players share an entangled pair of qubits and perform rotations on their qubit as the strategy choice. Two five-qubit circuits are sufficient to run all 16 possible strategy choice sets in a game with four possible strategies. The data are then parsed into player types randomly in order to combine them classically into a Bayesian framework. We exhaustively compute the possible strategies of the game so that the experimental data can be used to solve for the Nash equilibria of the game directly. Then we compare the payoff at the Nash equilibria and location of discontinuous changes in the payoff obtained from the experimental data to the theory, and study how it changes as a function of the amount of entanglement.
Invariant entangled states remain unchanged under simultaneous identical unitary transformations of all their subsystems. We experimentally generate and characterize such invariant two-, four-, and six-photon polarization entangled states. This is done only with a suitable filtering procedure of multiple emissions of entangled photon pairs from a single source without any interferometric overlaps. We get the desired states utilizing bosonic emission enhancement due to indistinguishability. The setup is very stable and gives high interference contrasts. Thus, the process is a very likely candidate for various photonic demonstrations of quantum information protocols.
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.
A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. Moreover, it became clear that the players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2x2 games, it became generally accepted that quantum versions can be easily extended to N-player situations by simply allowing N-partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.