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Birkhäuser Boston eBooks

Complete Positivity and Open Quantum Systems

Rolando Rebolledo

Published Jan 1, 2001• 2 citations
Publisher

Abstract

The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0127-4_7

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The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The notion of a Quantum Markov Semigroup (QMS) is an extension of classical Markov semigroups motivated by the study of open quantum systems. Indeed, a number of physicists interested in open quantum systems introduced the notion of Quantum Dynamical Semigroups in the Seventies. Since then the subject has been extensively discussed and improved by both physicists and mathematicians, especially by the founders of Quantum Stochastic Analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.
The idea of an open quantum system was introduced in the 1950s as a response to the problems encountered in areas such as nuclear magnetic resonance and the decay of unstable atoms. Nowadays, dynamical models of open quantum systems have become essential components in many applications of quantum mechanics. This paper provides an overview of the fundamental concepts of open quantum systems. All underlying definitions, algebraic methods and crucial theorems are presented. In particular, dynamical semigroups with corresponding time-independent generators are characterized. Furthermore, evolution models that induce memory effects are discussed. Finally, measures of non-Markovianity are recapped and interpreted from a perspective of physical relevance.
After recalling the crucial role of first integrals (or constants of the motion) in classical and quantum dynamics, we show why the concept of martingale is their natural probabilistic counterpart. This is used in the framework of Euclidean Quantum Mechanics, a probabilistic approach to quantization. The relevant Theorem of Noether is shown; it is founded on a new probabilistic interpretation of the symmetry group of some heat equations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
We give a short survey of results related to the maximum entropy method. In this article we use the large deviations approach rather than the more direct convex analytical one. Indeed, the proposed applications are naturally stated in terms of large random particle systems, namely, the existence and construction problems for Schrödinger’s bridges and Nelson’s diffusion processes. These problems arise from probabilistic approaches to quantum mechanics. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.