Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2006.11027 · The Annals of Applied Probability

Optimal bounds in normal approximation for many interacting worlds

Louis H. Y. Chen, Lê Vǎn Thành

Published Mar 22, 2023• 4 citations
arXivHTMLPDF

Abstract

In this paper, we use Stein’s method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540–2555).

DOI
10.1214/21-aap1747

Related papers

Stein's method, many interacting worlds and quantum mechanics

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds."A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall, Deckert and Wiseman (2014) and proven by McKeague and Levin (2016) using Stein's method.In this article we propose new connections between Stein's method and Many Interacting Worlds (MIW) theory.In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein's method.These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.

Bernoulli•Jan 1, 2017•2 citations

Higher energy state approximations in the “Many interacting worlds” model

Alex Loomis, Sunder Sethuraman

In the “Many Interacting Worlds” (MIW) discrete Hamiltonian system approximation of Schrödinger’s wave equation, introduced in Ref. 11, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein’s method, in Wasserstein-[Formula: see text] distance with rate [Formula: see text] has been shown Refs. 5, 13, and 15. In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-[Formula: see text] distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the “differential equation” approach to Stein’s method, which allows to handle behavior near zeros of the higher energy states.

arXiv:2401.15512•Infinite Dimensional Analysis Quantum Probability and Related Topics•May 17, 2024•0 citations

Stein’s method and approximating the quantum harmonic oscillator

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

Hall et al. (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein's method These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.

Bernoulli•Dec 12, 2018•8 citations

Stein’s method and approximating the multidimensional quantum harmonic oscillator

Ian W. McKeague, Yvik C. Swan

Abstract Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.

arXiv:2105.13499•Journal of Applied Probability•Apr 11, 2023•2 citations

Stein's method, many interacting worlds and quantum mechanics

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall, Deckert and Wiseman (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we propose new connections between Stein's method and Many Interacting Worlds (MIW) theory. In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein's method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.

arXiv:1606.06618•Jun 21, 2016•2 citations

Convergence of empirical distributions in an interpretation of quantum mechanics

Ian W. McKeague, Bruce Levin

From its beginning, there have been attempts by physicists to formulate quantum mechanics without requiring the use of wave functions. An interesting recent approach takes the point of view that quantum effects arise solely from the interaction of finitely many classical “worlds.” The wave function is then recovered (as a secondary object) from observations of particles in these worlds, without knowing the world from which any particular observation originates. Hall, Deckert and Wiseman [Phys. Rev. X 4 (2014) 041013] have introduced an explicit many-interacting-worlds harmonic oscillator model to provide support for this approach. In this note, we provide a proof of their claim that the particle configuration is asymptotically Gaussian, thus matching the stationary ground-state solution of Schrödinger’s equation when the number of worlds goes to infinity. We also construct a Markov chain based on resampling from the particle configuration and show that it converges to an Ornstein–Uhlenbeck process, matching the time-dependent solution as well.

arXiv:1412.1563•The Annals of Applied Probability•Aug 1, 2016•11 citations
In this paper, we use Stein’s method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540–2555).
Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds."A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall, Deckert and Wiseman (2014) and proven by McKeague and Levin (2016) using Stein's method.In this article we propose new connections between Stein's method and Many Interacting Worlds (MIW) theory.In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein's method.These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.
In the “Many Interacting Worlds” (MIW) discrete Hamiltonian system approximation of Schrödinger’s wave equation, introduced in Ref. 11, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein’s method, in Wasserstein-[Formula: see text] distance with rate [Formula: see text] has been shown Refs. 5, 13, and 15. In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-[Formula: see text] distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the “differential equation” approach to Stein’s method, which allows to handle behavior near zeros of the higher energy states.
Hall et al. (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein's method These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.
Abstract Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.
Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall, Deckert and Wiseman (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we propose new connections between Stein's method and Many Interacting Worlds (MIW) theory. In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein's method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.
From its beginning, there have been attempts by physicists to formulate quantum mechanics without requiring the use of wave functions. An interesting recent approach takes the point of view that quantum effects arise solely from the interaction of finitely many classical “worlds.” The wave function is then recovered (as a secondary object) from observations of particles in these worlds, without knowing the world from which any particular observation originates. Hall, Deckert and Wiseman [Phys. Rev. X 4 (2014) 041013] have introduced an explicit many-interacting-worlds harmonic oscillator model to provide support for this approach. In this note, we provide a proof of their claim that the particle configuration is asymptotically Gaussian, thus matching the stationary ground-state solution of Schrödinger’s equation when the number of worlds goes to infinity. We also construct a Markov chain based on resampling from the particle configuration and show that it converges to an Ornstein–Uhlenbeck process, matching the time-dependent solution as well.