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.

Highlights

  • Many voting protocols based on classical cryptography have been developed and successfully applied in the last two decades [1,2]

  • To react to the risk posed by forthcoming quantum computers, a number of quantum voting protocols have been developed in the last decade [3,4,5,6,7,8,9,10,11,12,13]

  • While all these works have focused on the security problems of voting from a cryptographic perspective, Bao and Halpern [14] studied quantum voting from a social choice theoretic perspective by showing that the quantum analog of Arrow’s Impossibility Theorem is violated in the quantum setting

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Summary

Introduction

Many voting protocols based on classical cryptography have been developed and successfully applied in the last two decades [1,2]. To react to the risk posed by forthcoming quantum computers, a number of quantum voting protocols have been developed in the last decade [3,4,5,6,7,8,9,10,11,12,13] While all these works have focused on the security problems of voting from a cryptographic perspective, Bao and Halpern [14] studied quantum voting from a social choice theoretic perspective by showing that the quantum analog of Arrow’s Impossibility Theorem is violated in the quantum setting. Facing an inappropriate definition of QIIA and an unsatisfying proof in [14], we can still question whether Arrow’s Impossibility Theorem is violated in the quantum setting. Some primitives of the quantum information theory which are used in this paper are collected in Appendix A

Classical Voting System
Quantum Voting System
A QSWF E satisfies unsharp IIA if the following condition is satisfied
A QSWF E satisfies unsharp dictatorship if there is a voter vi such that
Quantum Condorcet Voting and Arrow’s Impossibility Theorem
Security of Quantum Voting
Probabilistic Social Choice
Conclusions and Future Work
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