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InTech eBooks

On Open Quantum Systems and Mathematical Modeling of Quantum Noise

Andrzej Jamiokowski

Published Jul 24, 2013• 1 citation
Publisher

Abstract

In modeling of physical systems for which time behaviour can be represented by stochastic processes one assumes that a system in question is described by certain mathematical model, for example, by random variables in classical case or by sets of non-commuting observables in quantum case, acting on an abstract probability space. In most cases our assumptions are not believed to be a fully realistic model of the physical reality, but nevertheless very often it turns out to be extremely useful.

DOI
10.5772/54537

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In modeling of physical systems for which time behaviour can be represented by stochastic processes one assumes that a system in question is described by certain mathematical model, for example, by random variables in classical case or by sets of non-commuting observables in quantum case, acting on an abstract probability space. In most cases our assumptions are not believed to be a fully realistic model of the physical reality, but nevertheless very often it turns out to be extremely useful.
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.
Introduction This chapter gives the physical and mathematical basis of decoherence in quantum systems, with particular emphasis on its importance in quantum information theory and quantum computation. We assume that the reader is already familiar with the basics of quantum mechanics, such as wave functions, the Schrödinger equation, and Hilbert spaces, which are covered in standard textbooks. Nevertheless, all basic concepts will be at least briefly defined and described. A working knowledge of graduate-level linear algebra and calculus is also assumed. In quantum information processing, the term decoherence is often used loosely to describe any kind of noise that can affect a quantum system. This can include sources of noise that seem, qualitatively, to be quite different from each other. These noise sources include: (i) Random driving forces from the environment (e.g., Brownian motion). (ii) Interactions that produce entanglement between the system and the environment. (iii) Statistical imprecision in the experimental controls on the system (e.g., timing errors, frequency fluctuations, etc.). Remarkably, all of these sources of noise can be described by an almost identical mathematical framework, in which pure quantum states can evolve into mixed quantum states, and the quantum effects necessary for information processing – interference, entanglement, and reversible evolution – are disrupted. The chapter begins by examining a highly schematic model of decoherence, involving systems of qubits, but uses this model to present the mathematical formalism used in describing more realistic systems as well.
Quantum mechanics must be regarded as open systems. On one hand, this is due to the fact that, like in classical physics, any realistic system is subjected to a coupling to an uncontrollable environment which influences it in a non-negligible way. The theory of open quantum systems thus plays a major role in many applications of quantum physics since perfect isolation of quantum system is not possible and since a complete microscopic description or control of the environment degrees of freedom is not feasible or only partially so [1]. Practical considerations therefore force one to seek for a simpler, effectively probabilistic description in terms of an open system. There is a close physical and mathematical connection between the evolution of an open system, the state changes induced by quantum measurements, and the classical notion of a stochastic process. The paper provides a bibliographic review of this interrelations, it shows the mathematical equivalence between markovian master equation and generalized piecewise deterministic processes [1] and it introduces the open system in an open observed environment model.
Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
This author has been engaged over the past several years in the effort to develop certain basic intuitions into a fully mathematical theory. These evolving intuitions first began with the idea that the economist's notion of utility might more truly be represented mathematically by a normed measure on the commodity space rather than by a point function, the value of the measure on a particular set of goods having the sense of a relative propensity of the individual to select a good in that set. With this idea there came in turn, quite naturally, the insight that utility and probability are one and the same thing. And accordingly it emerged that the theory we were seeking would not be restricted to the economic behavior of human beings, nor even to the behavior in general of human beings: it would pertain, in utmost generality, to all behavior of all systems. Thus, it must affirm that all behavior is the evolution of stochastic processes, and that all systems are stochastic processes. Consistent with this the structural elements of stochastic processes are seen as the fundamental, objective, real entities, the ultimate building-blocks of reality. These are eventualities and acts and-according to the traditional mathematical representation of stochastic processes, also-probabilities. This structural aspect of the theory under development puts into sharp evidence its wide divergence from the classical scientific conception of reality. Our intensive search failed to discover a dynamical law for the proposed new theory within the mathematical context of the traditional, σ-field-real-valued-measure representation of stochastic processes. We therefore have gone on to a higher level of possible mathematical representation. This particular representation would have collections of eventualities represented by lattices of certain configurations in a unitary space. Closed linear manifolds are among these configurations, and indeed the formalism of quantum mechanics falls into the context of this representation. But the general lattice element in the kinds of lattices being investigated is more complicated than a linear manifold, being heuristically comparable with an acute-angle cone of vectors. These configurations are formulated by means of a ƒlanking relation between linear manifolds. One such lattice has been developed. Others are currently being sought through variations on the notion of flanking. The fact that probabilities present themselves in the unitary space representation-as is familiar from quantum mechanics-as parameters of the relationship between linear manifolds suggests that probability is not a fundamental structural concept. It is consequently indicated that eventualities and acts alone are the basic elements of real structure. Accordingly, the dynamical law must be formulable in terms of these elements only. There is a natural suggestion of what the law might look like. Even at the present early stage of development of the new theory being sought it is possible to cite many pieces of empirical evidence vouching for its exclusive validity. Such evidence is particularly noteworthy in the domain of human behavior.