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Advanced Quantum Technologies

Experimental Realization of Parrondo's Paradox in 1D Quantum Walks

Munsif Jan, Qin‐Qin Wang, Xiao‐Ye Xu, Wei‐Wei Pan, Zhe Chen, Yong‐Jian Han, Chuan‐Feng Li, Guang‐Can Guo, Derek Abbott

Published May 10, 2020• 23 citations

Abstract

Abstract The Parrondo effect is a well‐known apparent paradox where a combination of biased random walks displays a counterintuitive reversal in direction. These random walks can be expressed in terms of classical coin tossing games, leading to the surprising result that a combination of losing games can result in a winning game. There is now a large body of literature on quantum walks theoretically analyzing the quantum version of this effect, but to date, there have been no experimental observations of quantum Parrondo walks. Here, the first experimental verification of a quantum Parrondo walk within a quantum optics scenario is demonstrated. Based on the compact large‐scale experimental quantum‐walk platform, two rotation operators are implemented to realize the quantum Parrondo effect. The effect of quantum coherence in a quantum Parrondo walk is also investigated based on a delayed‐choice scheme that cannot be realized with classical light. It is demonstrated that the Parrondo effect vanishes when the quantum walk has a completely decoherent initial state in a delayed‐choice setting. Quantum walks are fundamental to multiple quantum algorithms, and this research provides motivation to expand the results to further explore quantum Parrondo walks.

DOI: 10.1002/qute.201900127

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Abstract The Parrondo effect is a well‐known apparent paradox where a combination of biased random walks displays a counterintuitive reversal in direction. These random walks can be expressed in terms of classical coin tossing games, leading to the surprising result that a combination of losing games can result in a winning game. There is now a large body of literature on quantum walks theoretically analyzing the quantum version of this effect, but to date, there have been no experimental observations of quantum Parrondo walks. Here, the first experimental verification of a quantum Parrondo walk within a quantum optics scenario is demonstrated. Based on the compact large‐scale experimental quantum‐walk platform, two rotation operators are implemented to realize the quantum Parrondo effect. The effect of quantum coherence in a quantum Parrondo walk is also investigated based on a delayed‐choice scheme that cannot be realized with classical light. It is demonstrated that the Parrondo effect vanishes when the quantum walk has a completely decoherent initial state in a delayed‐choice setting. Quantum walks are fundamental to multiple quantum algorithms, and this research provides motivation to expand the results to further explore quantum Parrondo walks.

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.

Parrondo's effect is a well-known apparent paradox where a combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's effect can occur not only in the case of one-dimensional discrete quantum walks with random or deterministic periodic sequence of two- or multistate quantum coins but also in the case of one-dimensional discrete quantum walks with deterministic aperiodic sequence of two-state quantum coins. Moreover, we show how Parrondo's effect affects the time evolution of the walker-coin quantum entanglement.

Parrondo's paradox refers to the apparently paradoxical effect whereby a certain combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's paradox can occur not only in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of two quantum coins but also in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of three quantum coins. Moreover, we show how Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

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.

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 ubiquitous in games, ratchets and random walks.The apparent paradox, devised by J. M. R. Parrondo, that two losing games

A

and

B

can produce an winning outcome has been adapted in many physical and biological systems to explain their working. However, proposals on demonstrating Parrondo's paradox using quantum walks failed in the asymptotic limits. In this work, we show that instead of a single coin if we consider a two coin initial state which may or may not be entangled, we can observe a genuine Parrondo's paradox with quantum walks. Further we focus on reasons for this and pin down the asymmetry in initial two-coin state or asymmetry in shift operator, either of which are necessary for observing a genuine Parrondo's paradox. We extend our work to a 3-coin initial state too with similar results. The implications of our work for observing quantum ratchet like behavior using quantum walks is also discussed.