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arXiv:2507.06382 · Physical Review A

Randomly measured quantum particles and thermal noise

Victor Gurarie

Published Jun 12, 2026• 2 citations
arXivHTMLPDF

Abstract

We consider the motion of a quantum particle whose position is measured in random places at random moments in time. We contrast this motion with the motion of a quantum particle in a potential which varies randomly in space and in time, which could also be thought of as (possibly thermal) noise. We calculate expectations of observables both linear and nonlinear in the density matrix. We demonstrate explicitly that while linear observables cannot distinguish between random measurements and random noise, measurable distinctions can be seen in nonlinear observables.

DOI
10.1103/55w8-jy76

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We consider the motion of a quantum particle whose position is measured in random places at random moments in time. We contrast this motion with the motion of a quantum particle in a potential which varies randomly in space and in time, which could also be thought of as (possibly thermal) noise. We calculate expectations of observables both linear and nonlinear in the density matrix. We demonstrate explicitly that while linear observables cannot distinguish between random measurements and random noise, measurable distinctions can be seen in nonlinear observables.
We consider the motion of a quantum particle whose position is measured in random places at random moments in time. We show that a freely moving particle measured in this way undergoes superdiffusion, while a charged particle moving in a magnetic field confined to the lowest Landau level undergoes conventional diffusion. We also look at a particle moving in one dimensional space in a random time-independent potential, so that it is Anderson localized, which is also measured at random points in space and randomly in time. We find that random measurements break localization and this particle also undergoes diffusion. To address these questions, we develop formalism similar to that employed when studying classical and quantum problems with time-dependent noise.
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
In this paper we derive the equations of motion satisfied by the density matrix of an arbitrary physical system in contact with an infinite thermal reservoir of quantum mechanical free particles. We show that for weak but sustained interactions between system and reservoir, the reservoir is represented by a linear time-independent kernel in the system density matrix space. We obtain the kernel for the three possible models of quantum mechanical free particles which are boson and fermion reservoirs in which the system-reservoir interaction conserves the number of particles of the reservoir and a boson reservoir in which reservoir "particles" may be created or destroyed. In order to derive the equations of motion without using the inconsistent "rerandomization of phases" after each interaction, we generalize the Wigner-Weisskopf theory of line broadening of spectral lines to the equations of motion for the density matrix. We next show that the kernel preserves the normalization, positive-definiteness, and Hermiticity of the system density matrix and causes the system to approach equilibrium monotonically. In the limit of a classical reservoir our results include a derivation, without the repeated use of random phasing, of Bloch's equations of motion for the density matrix of a spin system imbedded in a thermal reservoir.
A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.