Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem
Abstract
:1. Introduction
2. Background
2.1. Classical Voting System
2.2. Quantum Voting System
- For all quantum ballot profiles ρ and all pairs of candidates , if for each voter , then .
- For all quantum ballot profiles ρ and all pairs of candidates , if for each voter , then .
- For all quantum ballot profiles ρ and and all pairs of candidates , if for each voter , then implies that .
- For all quantum ballot profiles and all pairs of candidates , if for each voter , then implies that .
- For all quantum ballot profiles and all pairs of candidates , iff .
- For all quantum ballot profiles and all pairs of candidates , iff .
3. Quantum Condorcet Voting and Arrow’s Impossibility Theorem
- 1.
- Calculates the Condorcet score of each candidate according to . The Condorcet score of a candidate is the number of winning in pairwise comparison with other candidates. That is, for a candidate x, his Condorcet score is where R is the classical ballot profile corresponding to .
- 2.
- Generate a weak order ⪰ over all candidates according to their Condorcet score. That is, iff .
- 3.
- Complete the weak order. That is, generate the set in which each is a linear order that extends ⪰ and contains all extensions of ⪰.
- 4.
- Transform the linear order into a quantum state. That is, for we create a quantum state , where each is the basis ballot that corresponds to .
- 5.
- Give the minority a shot. For any candidate pair which is encoded by at least one , We spread an amount of weight across the subspace. That is, is changed to , where ranges over all candidate pairs that are encoded by at least one . The parameter δ is required to satisfy that .
- 6.
- Enforce unanimity. For any candidate pair which is encoded by all the , we project onto the subspace. That is, is changed to , where ranges over all candidate pairs that are encoded by all the .
4. Related Work
4.1. Security of Quantum Voting
4.2. Probabilistic Social Choice
5. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Basics of Quantum Information
- 1.
- complex vector space, that is,
- 2.
- with a (positive-definite) scalar product such that for all and
- (a)
- (b)
- (c)
- iff
- (d)
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Sun, X.; He, F.; Sopek, M.; Guo, M. Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem. Entropy 2021, 23, 1083. https://doi.org/10.3390/e23081083
Sun X, He F, Sopek M, Guo M. Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem. Entropy. 2021; 23(8):1083. https://doi.org/10.3390/e23081083
Chicago/Turabian StyleSun, Xin, Feifei He, Mirek Sopek, and Meiyun Guo. 2021. "Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem" Entropy 23, no. 8: 1083. https://doi.org/10.3390/e23081083