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Application of Statistical Mechanics to Control Theory and Information Theory.

Elaine R. Cohen, John M. Richardson

Published Jul 3, 1975• 0 citations
OpenAlex

Abstract

The results of four papers are summarized and additional preliminary results on problems of observability, controllability and entropy change in classical and quantum systems are discussed. Areas for future research, particularly in conjunction with the description of entropy change in quantum systems is discussed.

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Fundamental limits on the controllability of quantum mechanical systems are discussed in the light of quantum information theory. It is shown that the amount of entropy-reduction that can be extracted from a quantum system by feedback controller is upper bounded by a sum of the decrease of entropy achievable in open-loop control and the mutual information between the quantum system and the controller. This upper bound sets a fundamental limit on the performance of any quantum controllers whose designs are based on the possibilities to attain low entropy states. An application of this approach pertaining to quantum error correction is also discussed.
This thesis consists of four research papers and an introduction covering the most important concepts appearing in the papers. The papers deal with applications of gauge/gravity dualities in the study of various physical quantities and systems. Gauge/gravity dualities are equivalences between certain quantum field theories and classical theories of gravity. These dualities can be used as computational tools in a wide range of applications across different fields of physics, and as such they have garnered much attention in the last two decades. The great promise of these new tools is the ability to tackle difficult problems in strongly interacting quantum field theories by translating them to problems in classical gravity, where progress is much easier to make. Quantum information theory studies the information contained in quantum systems. Entanglement is the fundamental property of quantum mechanics that sets it apart from classical theories of physics. Entanglement is commonly quantified by entanglement entropy, a quantity which is difficult to compute in interacting quantum field theories. Gauge/gravity dualities provide a practical way for computing the entanglement entropy via the Ryu-Takayanagi formula. The primary focus of this thesis is to use this formula for computing various entanglement measures in strongly interacting quantum field theories via their gravity duals. The purpose of this thesis is to introduce quantum information theory concepts that have been important in our research. When applicable, quantities of interest are first introduced in the classical setting in order to build intuition about their behaviour. Quantum properties of entanglement measures are discussed in detail, along with their holographic counterparts, and remarks are made concerning their applications.
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.
Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. Furthermore, concurrent advances in experimental techniques and in the theory of quantum computation have led to a strong interest in questions of quantum information, in particular in the sense of the amount of information about unknown parameters in given observational data or accessible through various possible types of measurements. This scenery is outlined (with an audience of statisticians and probabilists in mind).