Entropy in Classical and Quantum Information Theory
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On Entropy, Information, and Conservation of Information
entropy Article On Entropy, Information, and Conservation of Information Yunus A. Çengel Department of Mechanical Engineering, University of Nevada, Reno, NV 89557, USA; [email protected] Abstract: The term entropy is used in different meanings in different contexts, sometimes in contradic- tory ways, resulting in misunderstandings and confusion. The root cause of the problem is the close resemblance of the defining mathematical expressions of entropy in statistical thermodynamics and information in the communications field, also called entropy, differing only by a constant factor with the unit ‘J/K’ in thermodynamics and ‘bits’ in the information theory. The thermodynamic property entropy is closely associated with the physical quantities of thermal energy and temperature, while the entropy used in the communications field is a mathematical abstraction based on probabilities of messages. The terms information and entropy are often used interchangeably in several branches of sciences. This practice gives rise to the phrase conservation of entropy in the sense of conservation of information, which is in contradiction to the fundamental increase of entropy principle in thermody- namics as an expression of the second law. The aim of this paper is to clarify matters and eliminate confusion by putting things into their rightful places within their domains. The notion of conservation of information is also put into a proper perspective. Keywords: entropy; information; conservation of information; creation of information; destruction of information; Boltzmann relation Citation: Çengel, Y.A. On Entropy, 1. Introduction Information, and Conservation of One needs to be cautious when dealing with information since it is defined differently Information. Entropy 2021, 23, 779. -
ENERGY, ENTROPY, and INFORMATION Jean Thoma June
ENERGY, ENTROPY, AND INFORMATION Jean Thoma June 1977 Research Memoranda are interim reports on research being conducted by the International Institute for Applied Systems Analysis, and as such receive only limited scientific review. Views or opinions contained herein do not necessarily represent those of the Institute or of the National Member Organizations supporting the Institute. PREFACE This Research Memorandum contains the work done during the stay of Professor Dr.Sc. Jean Thoma, Zug, Switzerland, at IIASA in November 1976. It is based on extensive discussions with Professor HAfele and other members of the Energy Program. Al- though the content of this report is not yet very uniform because of the different starting points on the subject under consideration, its publication is considered a necessary step in fostering the related discussion at IIASA evolving around th.e problem of energy demand. ABSTRACT Thermodynamical considerations of energy and entropy are being pursued in order to arrive at a general starting point for relating entropy, negentropy, and information. Thus one hopes to ultimately arrive at a common denominator for quanti- ties of a more general nature, including economic parameters. The report closes with the description of various heating appli- cation.~and related efficiencies. Such considerations are important in order to understand in greater depth the nature and composition of energy demand. This may be highlighted by the observation that it is, of course, not the energy that is consumed or demanded for but the informa- tion that goes along with it. TABLE 'OF 'CONTENTS Introduction ..................................... 1 2 . Various Aspects of Entropy ........................2 2.1 i he no me no logical Entropy ........................ -
Lecture 4: 09.16.05 Temperature, Heat, and Entropy
3.012 Fundamentals of Materials Science Fall 2005 Lecture 4: 09.16.05 Temperature, heat, and entropy Today: LAST TIME .........................................................................................................................................................................................2� State functions ..............................................................................................................................................................................2� Path dependent variables: heat and work..................................................................................................................................2� DEFINING TEMPERATURE ...................................................................................................................................................................4� The zeroth law of thermodynamics .............................................................................................................................................4� The absolute temperature scale ..................................................................................................................................................5� CONSEQUENCES OF THE RELATION BETWEEN TEMPERATURE, HEAT, AND ENTROPY: HEAT CAPACITY .......................................6� The difference between heat and temperature ...........................................................................................................................6� Defining heat capacity.................................................................................................................................................................6� -
The Enthalpy, and the Entropy of Activation (Rabbit/Lobster/Chick/Tuna/Halibut/Cod) PHILIP S
Proc. Nat. Acad. Sci. USA Vol. 70, No. 2, pp. 430-432, February 1973 Temperature Adaptation of Enzymes: Roles of the Free Energy, the Enthalpy, and the Entropy of Activation (rabbit/lobster/chick/tuna/halibut/cod) PHILIP S. LOW, JEFFREY L. BADA, AND GEORGE N. SOMERO Scripps Institution of Oceanography, University of California, La Jolla, Calif. 92037 Communicated by A. Baird Hasting8, December 8, 1972 ABSTRACT The enzymic reactions of ectothermic function if they were capable of reducing the AG* character- (cold-blooded) species differ from those of avian and istic of their reactions more than were the homologous en- mammalian species in terms of the magnitudes of the three thermodynamic activation parameters, the free zymes of more warm-adapted species, i.e., birds or mammals. energy of activation (AG*), the enthalpy of activation In this paper, we report that the values of AG* are indeed (AH*), and the entropy of activation (AS*). Ectothermic slightly lower for enzymic reactions catalyzed by enzymes enzymes are more efficient than the homologous enzymes of ectotherms, relative to the homologous reactions of birds of birds and mammals in reducing the AG* "energy bar- rier" to a chemical reaction. Moreover, the relative im- and mammals. Moreover, the relative contributions of the portance of the enthalpic and entropic contributions to enthalpies and entropies of activation to AG* differ markedly AG* differs between these two broad classes of organisms. and, we feel, adaptively, between ectothermic and avian- mammalian enzymic reactions. Because all organisms conduct many of the same chemical transformations, certain functional classes of enzymes are METHODS present in virtually all species. -
A Mini-Introduction to Information Theory
A Mini-Introduction To Information Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case. arXiv:1805.11965v5 [hep-th] 4 Oct 2019 Contents 1 Introduction 2 2 Classical Information Theory 2 2.1 ShannonEntropy ................................... .... 2 2.2 ConditionalEntropy ................................. .... 4 2.3 RelativeEntropy .................................... ... 6 2.4 Monotonicity of Relative Entropy . ...... 7 3 Quantum Information Theory: Basic Ingredients 10 3.1 DensityMatrices .................................... ... 10 3.2 QuantumEntropy................................... .... 14 3.3 Concavity ......................................... .. 16 3.4 Conditional and Relative Quantum Entropy . ....... 17 3.5 Monotonicity of Relative Entropy . ...... 20 3.6 GeneralizedMeasurements . ...... 22 3.7 QuantumChannels ................................... ... 24 3.8 Thermodynamics And Quantum Channels . ...... 26 4 More On Quantum Information Theory 27 4.1 Quantum Teleportation and Conditional Entropy . ......... 28 4.2 Quantum Relative Entropy And Hypothesis Testing . ......... 32 4.3 Encoding -
Chapter 6 Quantum Entropy
Chapter 6 Quantum entropy There is a notion of entropy which quantifies the amount of uncertainty contained in an ensemble of Qbits. This is the von Neumann entropy that we introduce in this chapter. In some respects it behaves just like Shannon's entropy but in some others it is very different and strange. As an illustration let us immediately say that as in the classical theory, conditioning reduces entropy; but in sharp contrast with classical theory the entropy of a quantum system can be lower than the entropy of its parts. The von Neumann entropy was first introduced in the realm of quantum statistical mechanics, but we will see in later chapters that it enters naturally in various theorems of quantum information theory. 6.1 Main properties of Shannon entropy Let X be a random variable taking values x in some alphabet with probabil- ities px = Prob(X = x). The Shannon entropy of X is X 1 H(X) = p ln x p x x and quantifies the average uncertainty about X. The joint entropy of two random variables X, Y is similarly defined as X 1 H(X; Y ) = p ln x;y p x;y x;y and the conditional entropy X X 1 H(XjY ) = py pxjy ln p j y x;y x y 1 2 CHAPTER 6. QUANTUM ENTROPY where px;y pxjy = py The conditional entropy is the average uncertainty of X given that we observe Y = y. It is easily seen that H(XjY ) = H(X; Y ) − H(Y ) The formula is consistent with the interpretation of H(XjY ): when we ob- serve Y the uncertainty H(X; Y ) is reduced by the amount H(Y ). -
Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations
Article Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations Jen-Tsung Hsiang 1,*,† and Bei-Lok Hu 2,† ID 1 Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 2 Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, USA; [email protected] * Correspondence: [email protected]; Tel.: +86-21-3124-3754 † These authors contributed equally to this work. Received: 26 April 2018; Accepted: 30 May 2018; Published: 31 May 2018 Abstract: Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. -
Information Content and Error Analysis
43 INFORMATION CONTENT AND ERROR ANALYSIS Clive D Rodgers Atmospheric, Oceanic and Planetary Physics University of Oxford ESA Advanced Atmospheric Training Course September 15th – 20th, 2008 44 INFORMATION CONTENT OF A MEASUREMENT Information in a general qualitative sense: Conceptually, what does y tell you about x? We need to answer this to determine if a conceptual instrument design actually works • to optimise designs • Use the linear problem for simplicity to illustrate the ideas. y = Kx + ! 45 SHANNON INFORMATION The Shannon information content of a measurement of x is the change in the entropy of the • probability density function describing our knowledge of x. Entropy is defined by: • S P = P (x) log(P (x)/M (x))dx { } − Z M(x) is a measure function. We will take it to be constant. Compare this with the statistical mechanics definition of entropy: • S = k p ln p − i i i X P (x)dx corresponds to pi. 1/M (x) is a kind of scale for dx. The Shannon information content of a measurement is the change in entropy between the p.d.f. • before, P (x), and the p.d.f. after, P (x y), the measurement: | H = S P (x) S P (x y) { }− { | } What does this all mean? 46 ENTROPY OF A BOXCAR PDF Consider a uniform p.d.f in one dimension, constant in (0,a): P (x)=1/a 0 <x<a and zero outside.The entropy is given by a 1 1 S = ln dx = ln a − a a Z0 „ « Similarly, the entropy of any constant pdf in a finite volume V of arbitrary shape is: 1 1 S = ln dv = ln V − V V ZV „ « i.e the entropy is the log of the volume of state space occupied by the p.d.f. -
The Dynamics of a Central Spin in Quantum Heisenberg XY Model
Eur. Phys. J. D 46, 375–380 (2008) DOI: 10.1140/epjd/e2007-00300-9 THE EUROPEAN PHYSICAL JOURNAL D The dynamics of a central spin in quantum Heisenberg XY model X.-Z. Yuan1,a,K.-D.Zhu1,andH.-S.Goan2 1 Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P.R. China 2 Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan Received 19 March 2007 / Received in final form 30 August 2007 Published online 24 October 2007 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007 Abstract. In the thermodynamic limit, we present an exact calculation of the time dynamics of a central spin coupling with its environment at finite temperatures. The interactions belong to the Heisenberg XY type. The case of an environment with finite number of spins is also discussed. To get the reduced density matrix, we use a novel operator technique which is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. The expectation value of the central spin and the von Neumann entropy are obtained. PACS. 75.10.Jm Quantized spin models – 03.65.Yz Decoherence; open systems; quantum statistical meth- ods – 03.67.-a Quantum information z 1 Introduction like S0 and extend the operator technique to deal with the case of finite number environmental spins. Recently, using One of the promising candidates for quantum computa- numerical tools, the authors of references [5–8] studied the tion is spin systems [1–4] due to their long decoherence dynamic behavior of a central spin system decoherenced and relaxation time. -
Entanglement Entropy and Decoupling in the Universe
Entanglement Entropy and Decoupling in the Universe Yuichiro Nakai (Rutgers) with N. Shiba (Harvard) and M. Yamada (Tufts) arXiv:1709.02390 Brookhaven Forum 2017 Density matrix and Entropy In quantum mechanics, a physical state is described by a vector which belongs to the Hilbert space of the total system. However, in many situations, it is more convenient to consider quantum mechanics of a subsystem. In a finite temperature system which contacts with heat bath, Ignore the part of heat bath and consider a physical state given by a statistical average (Mixed state). Density matrix A physical observable Density matrix and Entropy Density matrix of a pure state Density matrix of canonical ensemble Thermodynamic entropy Free energy von Neumann entropy Entanglement entropy Decompose a quantum many-body system into its subsystems A and B. Trace out the density matrix over the Hilbert space of B. The reduced density matrix Entanglement entropy Nishioka, Ryu, Takayanagi (2009) Entanglement entropy Consider a system of two electron spins Pure state The reduced density matrix : Mixed state Entanglement entropy : Maximum: EPR pair Quantum entanglement Entanglement entropy ✓ When the total system is a pure state, The entanglement entropy represents the strength of quantum entanglement. ✓ When the total system is a mixed state, The entanglement entropy includes thermodynamic entropy. Any application to particle phenomenology or cosmology ?? Decoupling in the Universe Particles A are no longer in thermal contact with particles B when the interaction rate is smaller than the Hubble expansion rate. e.g. neutrino decoupling Decoupled subsystems are treated separately. Neutrino temperature drops independently from photon temperature. -
Chapter 4 Information Theory
Chapter Information Theory Intro duction This lecture covers entropy joint entropy mutual information and minimum descrip tion length See the texts by Cover and Mackay for a more comprehensive treatment Measures of Information Information on a computer is represented by binary bit strings Decimal numb ers can b e represented using the following enco ding The p osition of the binary digit 3 2 1 0 Bit Bit Bit Bit Decimal Table Binary encoding indicates its decimal equivalent such that if there are N bits the ith bit represents N i the decimal numb er Bit is referred to as the most signicant bit and bit N as the least signicant bit To enco de M dierent messages requires log M bits 2 Signal Pro cessing Course WD Penny April Entropy The table b elow shows the probability of o ccurrence px to two decimal places of i selected letters x in the English alphab et These statistics were taken from Mackays i b o ok on Information Theory The table also shows the information content of a x px hx i i i a e j q t z Table Probability and Information content of letters letter hx log i px i which is a measure of surprise if we had to guess what a randomly chosen letter of the English alphab et was going to b e wed say it was an A E T or other frequently o ccuring letter If it turned out to b e a Z wed b e surprised The letter E is so common that it is unusual to nd a sentence without one An exception is the page novel Gadsby by Ernest Vincent Wright in which -
Shannon Entropy and Kolmogorov Complexity
Information and Computation: Shannon Entropy and Kolmogorov Complexity Satyadev Nandakumar Department of Computer Science. IIT Kanpur October 19, 2016 This measures the average uncertainty of X in terms of the number of bits. Shannon Entropy Definition Let X be a random variable taking finitely many values, and P be its probability distribution. The Shannon Entropy of X is X 1 H(X ) = p(i) log : 2 p(i) i2X Shannon Entropy Definition Let X be a random variable taking finitely many values, and P be its probability distribution. The Shannon Entropy of X is X 1 H(X ) = p(i) log : 2 p(i) i2X This measures the average uncertainty of X in terms of the number of bits. The Triad Figure: Claude Shannon Figure: A. N. Kolmogorov Figure: Alan Turing Just Electrical Engineering \Shannon's contribution to pure mathematics was denied immediate recognition. I can recall now that even at the International Mathematical Congress, Amsterdam, 1954, my American colleagues in probability seemed rather doubtful about my allegedly exaggerated interest in Shannon's work, as they believed it consisted more of techniques than of mathematics itself. However, Shannon did not provide rigorous mathematical justification of the complicated cases and left it all to his followers. Still his mathematical intuition is amazingly correct." A. N. Kolmogorov, as quoted in [Shi89]. Kolmogorov and Entropy Kolmogorov's later work was fundamentally influenced by Shannon's. 1 Foundations: Kolmogorov Complexity - using the theory of algorithms to give a combinatorial interpretation of Shannon Entropy. 2 Analogy: Kolmogorov-Sinai Entropy, the only finitely-observable isomorphism-invariant property of dynamical systems.