Schr\"odinger's Ballot: Quantum Information and the Violation of Arrow's Impossibility Theorem

TL;DR

This paper revisits Arrow's Impossibility Theorem in quantum settings, proposing a refined definition of Quantum Independence of Irrelevant Alternatives and providing a detailed proof of the theorem's violation.

Contribution

It introduces an improved definition of QIIA and offers a rigorous proof demonstrating the violation of Arrow's theorem in quantum contexts.

Findings

Quantum analogue of Arrow's theorem is violated under the new QIIA definition.

The revised QIIA better captures independence of irrelevant alternatives in quantum settings.

The paper clarifies previous proofs and strengthens the understanding of social choice in quantum theory.

Abstract

In this paper 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 there. Moreover, the definition of Quantum Independence of Irrelevant Alternatives (QIIA) seems not appropriate to us. In this paper, 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.