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

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Classical Voting System

**Definition**

**1**

**.**An SWF F satisfies the unanimity condition if, whenever all voters rank x above y, then so does society:

**Definition**

**2**

**.**An SWF F satisfies IIA if the relative social ranking of two candidates only depends on their relative voter rankings:

**Definition**

**3**

**.**An SWF F satisfies dictatorship if there is a voter ${v}_{i}\in \mathcal{V}$ such that $F\left(\mathbf{R}\right)={R}_{i}$ for every profile $\mathbf{R}=({R}_{1},\dots ,{R}_{n})$.

**Theorem**

**1**

**.**Any SWF for three or more candidates that satisfies unanimity and the IIA must also satisfy dictatorship.

#### 2.2. Quantum Voting System

**Definition**

**4**

**.**A QSWF $\mathcal{E}$ satisfies the sharp unanimity condition if it satisfies the following:

- For all quantum ballot profiles ρ and all pairs of candidates $(x,y)$, if $Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left(\rho \right)\right)\right)=1$ for each voter ${v}_{i}$, then $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)=1$.

- For all quantum ballot profiles ρ and all pairs of candidates $(x,y)$, if $Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left(\rho \right)\right)\right)>0$ for each voter ${v}_{i}$, then $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)>0$.

**Definition**

**5**

**.**A QSWF $\mathcal{E}$ satisfies the sharp IIA condition if it satisfies the following:

- For all quantum ballot profiles ρ and ${\rho}^{\prime}$ and all pairs of candidates $(x,y)$, if $Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left(\rho \right)\right)\right)=Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left({\rho}^{\prime}\right)\right)\right)$ for each voter ${v}_{i}$, then $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)=1$ implies that $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left({\rho}^{\prime}\right)\right)\right)=1$.

- For all quantum ballot profiles $\rho ,{\rho}^{\prime}$ and all pairs of candidates $(x,y)$, if $Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left(\rho \right)\right)\right)=Tr\left({\Pi}^{x\succ y}\left({Tr}_{\ne i}\left({\rho}^{\prime}\right)\right)\right)$ for each voter ${v}_{i}$, then $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)>0$ implies that $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left({\rho}^{\prime}\right)\right)\right)>0$.

**Definition**

**6**

- For all quantum ballot profiles $\rho =({\rho}_{1},\dots ,{\rho}_{n})$ and all pairs of candidates $(x,y)$, $Tr\left({\Pi}^{x\succ y}{\rho}_{i}\right)=1$ iff $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)=1$.

- For all quantum ballot profiles $\rho =({\rho}_{1},\dots ,{\rho}_{n})$ and all pairs of candidates $(x,y)$, $Tr\left({\Pi}^{x\succ y}{\rho}_{i}\right)>0$ iff $Tr\left({\Pi}^{x\succ y}\left(\mathcal{E}\left(\rho \right)\right)\right)>0$.

## 3. Quantum Condorcet Voting and Arrow’s Impossibility Theorem

**Definition**

**7**

**.**Let ${\rho}_{1}\otimes \dots \otimes {\rho}_{n}$ be a basis quantum ballot profile. The quantum Condorcet voting ${\mathcal{E}}_{qcv}$ operates in the following steps:

- 1.
- Calculates the Condorcet score of each candidate according to ${\rho}_{1}\otimes \dots \otimes {\rho}_{n}$. 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 ${S}_{c}\left(x\right)$ is $\left|\right\{y\in \mathcal{C}:\left|{\mathcal{V}}_{x\succ y}^{\mathbf{R}}\right|\ge \left|{\mathcal{V}}_{y\succ x}^{\mathbf{R}}\right|\left\}\right|$ where R is the classical ballot profile corresponding to ${\rho}_{1}\otimes \dots \otimes {\rho}_{n}$.
- 2.
- Generate a weak order ⪰ over all candidates according to their Condorcet score. That is, $x\u2ab0y$ iff ${S}_{c}\left(x\right)\ge {S}_{c}\left(y\right)$.
- 3.
- Complete the weak order. That is, generate the set $\{{\succ}^{1},\dots ,{\succ}^{m}\}$ in which each ${\succ}^{i}$ is a linear order that extends ⪰ and $\{{\succ}^{1},\dots ,{\succ}^{m}\}$ contains all extensions of ⪰.
- 4.
- Transform the linear order into a quantum state. That is, for $\{{\succ}^{1},\dots ,{\succ}^{m}\}$ we create a quantum state ${\sigma}^{1}=\frac{1}{m}{\displaystyle \sum _{i}}{\sigma}_{i}$, where each ${\sigma}_{i}$ is the basis ballot that corresponds to ${\succ}^{i}$.
- 5.
- Give the minority a shot. For any candidate pair $(x,y)$ which is encoded by at least one ${\rho}_{i}$, We spread an amount $\delta \in (0,1)$ of weight across the $x\succ y$ subspace. That is, ${\sigma}^{1}$ is changed to ${\sigma}^{2}=(1-k\delta ){\sigma}^{1}+\delta {\Omega}^{{x}_{1}\succ {y}_{1}}+\dots +\delta {\Omega}^{{x}_{k}\succ {y}_{k}}$, where $({x}_{1},{y}_{1}),\dots ,({x}_{k},{y}_{k})$ ranges over all candidate pairs that are encoded by at least one ${\rho}_{i}$. The parameter δ is required to satisfy that $\delta <\frac{1}{{\left|\mathcal{C}\right|}^{2}}$.
- 6.
- Enforce unanimity. For any candidate pair $(x,y)$ which is encoded by all the ${\rho}_{i}$, we project ${\sigma}^{2}$ onto the $x\succ y$ subspace. That is, ${\sigma}^{2}$ is changed to ${\sigma}^{3}=\frac{{\Pi}^{{x}_{k}\succ {y}_{k}}\dots {\Pi}^{{x}_{1}\succ {y}_{1}}{\sigma}^{2}{\Pi}^{{x}_{1}\succ {y}_{1}}\dots {\Pi}^{{x}_{k}\succ {y}_{k}}}{Tr({\Pi}^{{x}_{k}\succ {y}_{k}}\dots {\Pi}^{{x}_{1}\succ {y}_{1}}{\sigma}^{2})}$, where $({x}_{1},{y}_{1}),\dots ,({x}_{k},{y}_{k})$ ranges over all candidate pairs that are encoded by all the ${\rho}_{i}$.

**Theorem**

**2.**

**Proof:**

**Theorem**

**3.**

**Proof:**

**Theorem**

**4.**

**Proof:**

**Theorem**

**5.**

**Proof**:

**Theorem**

**6.**

**Proof:**

**Theorem**

**7.**

**Proof:**

**Corollary**

**1.**

## 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

**Definition**

**A1**

**.**A (finite-dimensional) Hilbert space $\mathsf{H}$ is a

- 1.
- complex vector space, that is,$$\varphi ,\psi \in \mathsf{H}anda,b\in \mathbb{C}\Rightarrow a\varphi +b\psi \in \mathsf{H},$$
- 2.
- with a (positive-definite) scalar product $\langle \xb7|\xb7\rangle :\mathsf{H}\times \mathsf{H}\mapsto \mathbb{C}$ such that for all $\varphi ,\psi ,{\varphi}_{1},{\varphi}_{2}\in \mathsf{H}$ and $a,b\in \mathbb{C}$
- (a)
- $\langle \varphi |\psi \rangle =\overline{\langle \psi |\varphi \rangle}$
- (b)
- $\langle \varphi |\varphi \rangle \ge 0$
- (c)
- $\langle \varphi |\varphi \rangle =0$ iff $\varphi =0$
- (d)
- $\langle \psi |a{\varphi}_{1}+b{\varphi}_{2}\rangle =a\langle \psi |{\varphi}_{1}\rangle +b\langle \psi |{\varphi}_{2}\rangle $

**Definition**

**A2**

**.**An orthonormal basis $\left\{\right|{\varphi}_{i}\rangle \}$ for a Hilbert space $\mathsf{H}$ is a basis of $\mathsf{H}$ whose vectors are unit vectors and are orthogonal to each other, that is, for any $|{\varphi}_{i}\rangle ,|{\varphi}_{j}\rangle $, $\parallel {\varphi}_{i}\parallel =1$ and $\langle {\varphi}_{i}|{\varphi}_{j}\rangle =0$. The dimension of a Hilbert space is the number of vectors of an orthonormal basis.

**Definition**

**A3**

**.**Given Hilbert spaces V and W of dimension m and n respectively, their tensor product, denoted $V\otimes W$, is a $mn$-dimensional space consisting of linear combinations of outer products $|v\rangle \otimes |w\rangle $ of vectors $|v\rangle ={({v}_{1},{v}_{2},\cdots ,{v}_{m})}^{T}\in V$ and $|w\rangle ={({w}_{1},{w}_{2},\cdots ,{w}_{n})}^{T}\in W$, where

**Definition**

**A4**

**.**A subspace of a Hilbert space V is a subset W of V such that W is also a Hilbert space.

**Definition**

**A5**

**.**A linear map $A:\mathsf{H}\mapsto \mathsf{H}$ is called an operator on $\mathsf{H}$.

**Definition**

**A6**

**.**The operator ${A}^{*}:\mathsf{H}\mapsto \mathsf{H}$ that satisfies $\langle {A}^{*}\varphi |\psi \rangle =\langle \varphi |A\psi \rangle $ for all $\varphi ,\psi \in \mathsf{H}$ is called the adjoint operator to A.

**Definition**

**A7**

**.**A projector of a Hilbert space $\mathsf{H}$ is a linear map $P:\mathsf{H}\mapsto \mathsf{H}$ such that ${P}^{2}=P$ and ${P}^{*}=P$.

**Definition**

**A8**

**.**Let $\mathsf{H}$ be a Hilbert space and ρ be an operator on $\mathsf{H}$. The trace of ρ is defined by

**Definition**

**A9**

**.**An operator $A:\mathsf{H}\mapsto \mathsf{H}$ is positive semidefinite if it holds that $A={B}^{*}B$ for some operator $B\in L\left(\mathsf{H}\right)$.

**Definition**

**A10**

**.**A positive semidefinite operator ρ on $\mathsf{H}$ is a density operator if it holds that $\rho ={\rho}^{*}$ and $Tr\left(\rho \right)=1$.

**Definition**

**A11**

**.**Suppose the composite system of two subsystems A and B is described by the density operator ${\rho}_{AB}$. The partial trace over B is defined by

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Sun, 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