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Journal of Physics A Mathematical and General

Asymptotic operator algebras in quantum mechanics

Kay-Kong Wan, R. G. McLean

Published Mar 11, 1984• 8 citations
Publisher

Abstract

The asymptotic behaviour of quantum mechanical states at large times has recently been discussed by Wan and McLean. This paper deals with the corresponding behaviour of quantum mechanical operators. The concept of asymptotic operators is introduced and their mathematical properties in the weak, the uniform and the strong operator topologies studied. Results are presented in a series of theorems, lemmas and corollaries.

DOI
10.1088/0305-4470/17/4/023

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The asymptotic behaviour of quantum mechanical states at large times has recently been discussed by Wan and McLean. This paper deals with the corresponding behaviour of quantum mechanical operators. The concept of asymptotic operators is introduced and their mathematical properties in the weak, the uniform and the strong operator topologies studied. Results are presented in a series of theorems, lemmas and corollaries.
The asymptotic behaviour of quantum mechanical states at large times has been discussed in two recent papers by Wan and McLean (1983). In a third paper a detailed study of the corresponding behaviour of observables was made. The mathematical properties of asymptotic operators obtained are applied in this paper to establish an algebraic formulation of quantum mechanics which has the characteristic of being asymptotically separable. The age-old non-locality problem in quantum mechanics can then be effectively tackled.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Quantum mechanics is one of the principle pillars of modern physics. It also remains a topic of great interest to mathematicians. Since its discovery it has inspired and been inspired by many topics within modern mathematics, including functional analysis and operator algebras, Lie groups, Lie algebras and their representations, principle bundles, distribution theory, and much more. Written with beginning graduate students in mathematics in mind, this book provides a thorough treatment of (nonrelativistic) quantum mechanics in a style that is leisurely, without the usual theorem-proof grammar of pure mathematics, while remaining mathematically honest. The author takes the time to fully develop the required mathematics and employs a consistent mathematical presentation to clarify the often-confusing notation of physics texts. Along the way the reader encounters several topics requiring more advanced mathematics than found in many discussions of the subject, making for a fascinating course in how mathematics and physics interact.
This chapter presents a self-contained introduction to some recent developments in nonequilibrium quantum statistical mechanics. In the elementary framework of finite dimensional quantum systems it introduces the concept of entropy production and discusses various generating functionals which encode the statistical properties of its fluctuations. It explores the physical interpretations of these functionals and their mathematical properties. In particular, it investigates their relations to linear response theory, full counting statistics, and quantum hypothesis testing. The chapter discusses very briefly more technical issues linked to the thermodynamic limit as well as to the large time limit. These two limits turn the elementary, finite time fluctuation theory into a powerful machinery, which yields important information on the asymptotic behaviour of infinitely extended quantum systems, e.g., open systems coupled to infinite reservoirs. An essential mathematical tool to deal with such a system is Tomita–Takesaki's modular theory of von Neumann algebras. The power of this theory is somewhat shadowed by its technical aspects. Finite dimensional quantum systems are special since all the structures and results of this machinery can be described by elementary tools.