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Chaos An Interdisciplinary Journal of Nonlinear Science

Modeling quantum measurement probability as a classical stochastic process

Daniel T. Gillespie, William O. Alltop, Jorge M. Martin

Published Sep 1, 2001• 3 citations
Publisher

Abstract

The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (c) 2001 American Institute of Physics.

DOI
10.1063/1.1378791

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The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (c) 2001 American Institute of Physics.
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes. The well-established definition for non-Markovianity of a classical stochastic process represents a condition on the Kolmogorov hierarchy of the n-point joint probability distributions. Since this definition cannot be transferred to the quantum regime, quantum non-Markovianity has recently been defined and quantified in terms of the underlying quantum dynamical map, using either its divisibility properties or the behavior of the trace distance between pairs of initial states. Here, we investigate and compare these definitions and their relations to the classical notion of non-Markovianity by employing a large class of non-Markovian processes, known as semi-Markov processes, which admit a natural extension to the quantum case. A number of specific physical examples is constructed which allow to study the basic features of the classical and the quantum definitions and to evaluate explicitly the measures for quantum non-Markovianity. Our results clearly demonstrate several fundamental distinctions between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.
Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.
The study of non-Markovian quantum processes has attracted significant interest in recent decades, giving rise to several competing notions of quantum non-Markovianity. These notes serve as an introduction to the topic for graduate students familiar with quantum mechanics and probability theory. Owing to the vastness of the literature, we focus on two prominent characterizations of quantum Markovianity based on the divisibility of quantum channels and monotonically decreasing state distinguishability. The correspondence between classical concepts (stochastic matrices, Chapman-Kolmogorov equation) and their quantum analogs (dynamical maps, CP-divisibility) is emphasized throughout.
Quantum Markov models are employed ubiqui-tously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approxi-mations for a wide range of quantum optical and meso scopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
Markovian approximation is a widely-employed idea in the descriptions of the dynamics of open quantum systems. Although it is usually claimed to be a concept inspired by classical Markovianity, the physical meaning of this concept, however, is vague. In this thesis, I compare the descriptions of classical stochastic processes and open quantum systems, and show that those inherent differences lead to the non-trivial problem of characterizing quantum non-Markovianity. With various examples, I argue that the current most often used definitions of quantumMarkovianity in the literature do not fully capture the memoryless property of OQSs. To address this issue, I take a different approach by studying a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, while others are first proposed in this thesis. I define all these concepts under a unified framework, which allows one to rigorously build hierarchy relations among them. In this sense, quantum non-Markovianity is highly context-dependent. The results in this thesis, summarized as a hierarchy figure, are expected to bring clarity to the nature of quantumnon-Markovianity.
Abstract Although quantum coherence is a basic trait of quantum mechanics, the presence of coherences in the quantum description of a certain phenomenon does not rule out the possibility to give an alternative description of the same phenomenon in purely classical terms. Here, we give definite criteria to determine when and to what extent quantum coherence is equivalent to non-classicality. We prove that a Markovian multi-time statistics obtained from repeated measurements of a non-degenerate observable cannot be traced back to a classical statistics if and only if the dynamics generates coherences and subsequently turns them into populations. Furthermore, we show with simple examples that such connection between quantum coherence and non-classicality is generally absent if the statistics is non-Markovian.