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arXiv:2411.10041

A quantum approach for determining a state of the opinion

François Dubois

Published Nov 15, 2024• 0 citations
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Abstract

We propose to define a notion of state of the opinion in order to link politician popularity estimations and voting intentions. We present two ways of modelling: a classical approach and quantum modelling. We test these ideas on data obtained during the French presidential election of April 2012.
DOI
10.48550/arxiv.2411.10041

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In this contribution, we construct a connection between two quantum voting models presented previously. We propose to try to determine the result of a vote from associated given opinion polls. We introduce a density operator relative to the family of all candidates to a particular election. From an hypothesis of proportionality between a family of coefficients which characterize the density matrix and the probabilities of vote for all the candidates, we propose a numerical method for the entire determination of the density operator. This approach is a direct consequence of the Perron-Frobenius theorem for irreductible positive matrices. We applyour algorithm to synthetic data and to operational results issued from the French presidential election of April 2012.
This paper is devoted to the application of the mathematical formalism of quantum mechanics to social (political) science. By using the quantum dynamical equations we model the process of decision making in US elections. The crucial point we attempt to make is that the voter's mental state can be represented as a superposition of two possible choices for either republicans or democrats. However, reality dictates a more complicated situation: typically a voter participates in two elections, i.e. the congress and the presidential elections. In both elections he/she has to decide between two choices. This very feature of the US election system requires that the mental state is represented by a 2-qubit state corresponding to the superposition of 4 different choices (e.g. for republicans in the congress; for the president as a democrat). The main issue of this paper is to describe the dynamics of the voters' mental states taking in account the mental and socio- political environment. What is truly novel in this paper is that instead of using Schrödinger's equation to describe the dynamics in an absence of interactions, we here apply the quantum master equation. This equation describes quantum decoherence, i.e., resolution from superposition to a definite choice.
Classical models of opinion dynamics represent individual opinions as scalar or vector values governed by the classical probability theory, either as deterministic quantities or random variables. This framework does not account for empirically observed phenomena such as cognitive ambivalence (where an individual simultaneously holds conflicting views) and order effects (where survey responses depend on the order in which questions are asked). We propose a quantum model of opinion dynamics in which each agent's cognitive state is represented by a density matrix that encodes both the expressed opinion and cognitive ambivalence. Survey questions become non-commuting self-adjoint operators, which provides a principled explanation for order effects. Our model also identifies quantities without classical counterparts, including quantum coherence and pairwise opinion covariances. Under a product state approximation, the quantum model reduces to the classical Friedkin--Johnsen opinion model. We test the framework on synthetic and real-world networks and observe that pairwise correlations follow network-dependent transient dynamics but converge to the same steady state regardless of the network, and that quantum coherence decays exponentially at a rate independent of the network.
Quantum communication protocols seek to leverage the unique properties of quantum systems for coordination or communication tasks, usually with guarantees of security or anonymity that exceed what is possible classically. One promising domain of application is elections, where strong such guarantees are essential to ensure legitimacy. We experimentally implement a recently proposed election protocol from Centrone et al. such that no one, including a potential central authority, can know the preferred candidate of any voter other than themself. We conduct a four-party election, generating and distributing four-partite GHZ states with ≈89%\approx 89\%≈89% fidelity and successfully recording voters' intentions ≈87%\approx 87\%≈87% of the time.