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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum models of Parrondo's games

Adrian P. Flitney, Derek Abbott

Published Nov 1, 2002• 0 citations
Publisher

Abstract

It is possible to have two games that are losing when played in isolation but that, because of some form of feedback, produce a winning game when played alternately or even in a random mixture. This effect is known as Parrondo's paradox. Quantum mechanics provides novel methods of combining two games through interference and entanglement. Two models of quantum Parrondo's games have been published and these are reviewed here. We speculate on a model of a quantum Parrondo's game using entanglement. Such games could find a use in the development of algorithms for quantum computers.

DOI
10.1117/12.469069

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It is possible to have two games that are losing when played in isolation but that, because of some form of feedback, produce a winning game when played alternately or even in a random mixture. This effect is known as Parrondo's paradox. Quantum mechanics provides novel methods of combining two games through interference and entanglement. Two models of quantum Parrondo's games have been published and these are reviewed here. We speculate on a model of a quantum Parrondo's game using entanglement. Such games could find a use in the development of algorithms for quantum computers.
The well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones, is called Parrondo's paradox. Here, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of the different periods. Moreover, we also analyze the relationships between Parrondo's games and quantum entanglement in our scheme. Along with the applications of different kinds of quantum walks, our outcomes potentially encourage the development of new quantum algorithms.
Parrondo's paradox is a well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones. In this paper, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of different periods for distinct choices of the relative phase. Furthermore, we investigate the regimes of the relative phase of the initial state of coins where Parrondo games exist. Moreover, we also analyze the relationships between Parrondo's game and quantum entanglement and show regimes where the Parrondo sequence may generate a maximal entangler state in our scheme. Along with the applications of different kinds of quantum walks, our outcomes potentially encourage the development of new quantum algorithms.
Classical Parrondo game shows that the combination of two losing games can achieve a winning result. We investigate such an effect under quantum models, and make a complete research about the initial states that can be adapted. By the comparison of the payoffs which come from the GHZ class, W class and initial states without entanglement, we find out that entanglement plays an important role here. Moreover, we reveal that, particular initial state without entanglement can even provide us optimal payoff. Thorough discussion and exact results in this article will help us not only to make appropriate choices in quantum Parrondo game, but also have a deeper understanding about entanglement and quantum game.
Game A + Game B = Game C. Parrondo's games follow this basic structure where A and B are losing games and C is a winning game-a phenomenon called Parrondo's paradox. These games can take on a wider class of definitions and exhibit these paradoxical results. In this paper, we show three paradoxical cases. (i) The successive "tossing" of a single fair quantum coin gives a biased result, a previously known result. (ii) The random tossing of two quantum coins, each with successive biased expectations, gives an average random walk position of approximately zero. (iii) The sequential periodic tossing of two quantum coins, each with successive negative biased expectations, gives an average random walk with positive expectation. Using these results, we then propose a protocol for identifying and classifying quantum operations that span the same Hilbert space for a two-level quantum system.
We study a quantum walk in one-dimension using two different "coin" operators. By mixing two operators, both of which give a biased walk with negative expectation value for the walker position, it is possible to reverse the bias through interference effects. This effect is analogous to that in Parrondo's games, where alternating two losing (gambling) games can produce a winning game. The walker bias is produced by introducing a phase factor into the coin operator, with two different phase factors giving games AAA and BBB. We give the range of phases for which the Parrondo effect can be obtained with AAA and BBB played alternately or in other (repeated) deterministic sequences. The effect is transitory. For sufficiently large times the original bias resumes.