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arXiv:2103.11364 · Entropy

Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem

Xin Sun, Feifei He, Mirek Sopek, Meiyun Guo

Published Aug 20, 2021• 1 citation
arXivHTMLPDF

Abstract

We study Arrow's Impossibility Theorem in the quantum setting. Our work is based on the work of Bao and Halpern, in which it is proved that the quantum analogue of Arrow's Impossibility Theorem is not valid. However, we feel unsatisfied about the proof presented in Bao and Halpern's work. Moreover, the definition of Quantum Independence of Irrelevant Alternatives (QIIA) in Bao and Halpern's work seems not appropriate to us. We give a better definition of QIIA, which properly captures the idea of the independence of irrelevant alternatives, and a detailed proof of the violation of Arrow's Impossibility Theorem in the quantum setting with the modified definition.

DOI
10.3390/e23081083

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Quantum voting, inspired by quantum game theory, provides a framework in which the quantum majority rule (QMR) constitution of Bao and Yunger Halpern [Phys. Rev. A 95, 062306 (2017)] violates the quantum analogue of Arrow's impossibility theorem. We evaluate this QMR constitution analytically on classical profile data and implement its final measurement stage as a quantum circuit, running on both noiseless simulators and noisy IBM quantum hardware to map how realistic noise deforms the resulting societal ranking distribution. Moderate-high single-qubit noise does not change the qualitative behavior of QMR, whereas strong noise shifts the distribution toward other dominant winners than the classical one. We quantify this behavior using winner-agreement rates, Condorcet-winner flip rates, and Jensen-Shannon divergence between societal ranking distributions. In a second, exploratory component, we demonstrate an explicitly entanglement-based variant of the QMR constitution that serves as a testbed for multi-voter quantum correlations under noise, which we refer to as the QMR2-inspired variant. There, GHZ-type and separable superpositions over opposite rankings have the same expectation values but respond very differently to noise. Taken together, these two components connect the abstract QMR constitution to concrete implementations on noisy intermediate-scale quantum (NISQ) devices and highlight design considerations for future quantum voting protocols.

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Introduction

Jonathan A. Jones, Dieter Jaksch

Why yet another book on quantum information theory? Like many lecturers we began writing this text because none of the alternatives seemed quite right. This book is aimed squarely at undergraduate physics students who want a brief but reasonably thorough introduction to the exciting ideas of quantum information, including its applications in computation and communication. It is based on a short course we have taught to fourth-year students at Oxford University since 2004; for the most part it only assumes knowledge of elementary quantum mechanics and linear algebra, and so could even be taught to third-year undergraduates. A brief revision guide to quantum mechanics is provided as an appendix, which should cover any minor points that have been missed. As the title implies the book is structured in three parts, starting with the basics of quantum information and then applying this to quantum computation and quantum communication. Part I is self-contained, but contains only the barest hints of the exciting applications which attract many people to this field and so might prove unsatisfying on its own. Parts II and III draw heavily on Part I, but are largely independent of each other, and it would be perfectly possible to study only one of these two without the other. As this text is aimed at physics undergraduates, we believe that it is vital to cover experimental techniques, rather than merely presenting quantum information as a series of abstract quantum operations. We have, however, concentrated on the basic ideas underlying each approach, rather than worrying about particular experimental details.

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Comment on "A short impossibility proof of quantum bit commitment"

Guang Ping He

In a recent letter (Phys. Lett. A 377 (2013) 1076, arXiv:0905.3801), the authors presented an impossibility proof of quantum bit commitment, which attempted to cover all possible protocols that involve both quantum and classical information. Here we show that there are many errors in the proof, thus it fails to exhaust all conceivable protocols.

arXiv:1306.5357•Jun 22, 2013•0 citations
We study Arrow's Impossibility Theorem in the quantum setting. Our work is based on the work of Bao and Halpern, in which it is proved that the quantum analogue of Arrow's Impossibility Theorem is not valid. However, we feel unsatisfied about the proof presented in Bao and Halpern's work. Moreover, the definition of Quantum Independence of Irrelevant Alternatives (QIIA) in Bao and Halpern's work seems not appropriate to us. We give a better definition of QIIA, which properly captures the idea of the independence of irrelevant alternatives, and a detailed proof of the violation of Arrow's Impossibility Theorem in the quantum setting with the modified definition.
Quantum Information is the science that aims to use the unusual behavior of the microscopic world, governed by the laws of Quantum Mechanics, in order to improve the way in which we compute or communicate information. Though the first ideas in this direction come from the early 80’s, it is in the last decade when Quantum Information has suffered an spectacular development. It is impossible to resume in a paper like this one the importance and complexity of the field. Therefore, I will limit to briefly explain some of the initial ideas (considered classical by now), and to briefly suggest some of the modern lines of research. By the nature of this exposition, I have decided to avoid rigor and to concentrate more in ideas and intuitions. Anyhow, I have tried to provide with enough references, in such a way that an interested reader could find there proper theorems and proofs.
Quantum voting, inspired by quantum game theory, provides a framework in which the quantum majority rule (QMR) constitution of Bao and Yunger Halpern [Phys. Rev. A 95, 062306 (2017)] violates the quantum analogue of Arrow's impossibility theorem. We evaluate this QMR constitution analytically on classical profile data and implement its final measurement stage as a quantum circuit, running on both noiseless simulators and noisy IBM quantum hardware to map how realistic noise deforms the resulting societal ranking distribution. Moderate-high single-qubit noise does not change the qualitative behavior of QMR, whereas strong noise shifts the distribution toward other dominant winners than the classical one. We quantify this behavior using winner-agreement rates, Condorcet-winner flip rates, and Jensen-Shannon divergence between societal ranking distributions. In a second, exploratory component, we demonstrate an explicitly entanglement-based variant of the QMR constitution that serves as a testbed for multi-voter quantum correlations under noise, which we refer to as the QMR2-inspired variant. There, GHZ-type and separable superpositions over opposite rankings have the same expectation values but respond very differently to noise. Taken together, these two components connect the abstract QMR constitution to concrete implementations on noisy intermediate-scale quantum (NISQ) devices and highlight design considerations for future quantum voting protocols.
Why yet another book on quantum information theory? Like many lecturers we began writing this text because none of the alternatives seemed quite right. This book is aimed squarely at undergraduate physics students who want a brief but reasonably thorough introduction to the exciting ideas of quantum information, including its applications in computation and communication. It is based on a short course we have taught to fourth-year students at Oxford University since 2004; for the most part it only assumes knowledge of elementary quantum mechanics and linear algebra, and so could even be taught to third-year undergraduates. A brief revision guide to quantum mechanics is provided as an appendix, which should cover any minor points that have been missed. As the title implies the book is structured in three parts, starting with the basics of quantum information and then applying this to quantum computation and quantum communication. Part I is self-contained, but contains only the barest hints of the exciting applications which attract many people to this field and so might prove unsatisfying on its own. Parts II and III draw heavily on Part I, but are largely independent of each other, and it would be perfectly possible to study only one of these two without the other. As this text is aimed at physics undergraduates, we believe that it is vital to cover experimental techniques, rather than merely presenting quantum information as a series of abstract quantum operations. We have, however, concentrated on the basic ideas underlying each approach, rather than worrying about particular experimental details.