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arXiv:2211.17034

Probabilistic cellular automaton for quantum particle in a potential

CHRISTOF WETTERICH

Published Nov 30, 2022• 0 citations
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Abstract

We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced by a probability distribution over initial conditions. The proposed automaton involves right- and left-movers, jumping from one cell to a neighboring one. They change their direction of motion at each randomly distributed disorder or scattering point. The continuum limit of an infinite number of cells yields a Dirac equation, and in the non-relativistic limit the familiar Schrödinger equation, with potential determined by the spacetime-distribution of scattering points. These equations describe the time evolution of the probabilistic information for the position of the particle. All quantum rules for observables, both for discrete possible measurement values and continuous expectation values, follow from the classical statistical laws.

DOI
10.48550/arxiv.2211.17034

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We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced by a probability distribution over initial conditions. The proposed automaton involves right- and left-movers, jumping from one cell to a neighboring one. They change their direction of motion at each randomly distributed disorder or scattering point. The continuum limit of an infinite number of cells yields a Dirac equation, and in the non-relativistic limit the familiar Schrödinger equation, with potential determined by the spacetime-distribution of scattering points. These equations describe the time evolution of the probabilistic information for the position of the particle. All quantum rules for observables, both for discrete possible measurement values and continuous expectation values, follow from the classical statistical laws.
Probabilistic cellular automata with deterministic updating are discrete quantum systems. They are characterized by a probability distribution over initial configurations. We employ the quantum formalism for an investigation of random probabilistic cellular automata for which the properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells or space points. We consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows us to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of the stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a generalized random potential.
A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.
A simple probabilistic cellular automaton is shown to be equivalent to a relativistic fermionic quantum field theory with interactions. Occupation numbers for fermions are classical bits or Ising spins. The automaton acts deterministically on bit configurations. The genuinely probabilistic character of quantum physics is realized by probabilistic initial conditions. In turn, the probabilistic automaton is equivalent to the classical statistical system of a generalized Ising model. For a description of the probabilistic information at any given time quantum concepts as wave functions and non-commuting operators for observables emerge naturally. Quantum mechanics can be understood as a particular case of classical statistics. This offers prospects to realize aspects of quantum computing in the form of probabilistic classical computing. This article is part of the theme issue 'Quantum technologies in particle physics'.
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.
Simple algebraic cellular automata and a diffusion confinement are used to model quantum behavior of a relativistic particle in a 1-D box. Comparisons and contrasts are drawn between such a quantum cellular automaton (QCA) and both orthodox and unorthodox interpretations of quantum mechanics. The persistent tension between deterministic pictures and probabilistic descriptions of quantum reality are clarified for the case of a 1-D quantum particle viewed as a cellular automaton surging for survival in space and time.
We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.