The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such
The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer-Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann definition of entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. But the latter has little to do with information theory. Consequently we introduce a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).
It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
The von Neumann entropy definition is -Tr ( d ln(d)) and is linked to: = Tr (d A) ((1)). Here d is a matrix as is A i.e Aij = . As a result d is often not diagonal. It seems further that the idea of entropy is linked to an operator (2). In other words, one might have certain probabilities linked to the operator P momentum and a different set to X. For a diagonal d, von Neumann’s form is the same as Shannon’s entropy, but uses the eigenvalues of d. For a pure quantum bound state Wn(x), it seems that one considers the two operators X and P and writes two entropies using Shannon’s entropy form: - Integral dx W(x)W(x) ln(WW) and -Integral dp a(p)a(p) ln(a(p)a(p)) where W(x)=wavefunction= Sum over p a(p)exp(ipx). Here P(x)=W(x)W(x) and P(p)=a(p)a(p). For example in (3), one may see calculations of these “distinct” entropies for a particle in a box with infinite potential walls. X and P are linked to measurables and for a quantum bound system one cannot know exactly x and p at the same time. One may write < KE > for a bound state in two ways: Integral dp a(p)a(p) pp/2m or Integral dx W(x) (-1/2m d/dx dW/dx). Using Shannon’s entropy approach one obtains two different entropy values Sx and Sp. With respect to von Neumann’s entropy, it is zero for a pure state. Although one cannot know x and p at the same time (in terms of a measurement) that does not prevent one from writing x and p together (i.e. both known) in exp(ipx) for the wavefunction of a free particle. In this note we suggest that one might consider product probabilities linked to more than one operator when considering entropy. For example, for <KE> one may write Integral dx P(x) Integral dp pp/2m P(p/x) thus acknowledging that both x and p play a role. In this case, P(x)P(p/x) is the probability to use in Shannon’s entropy as we have shown in (4). In (3) it is suggested that P(x)P(p/x) be used in order to incorporate both variables x and p. Using P(x)P(p) in Shannon’s entropy yields the same overall entropy when integrated over x and p i.e. S=Sx+Sp. One reason that the sum is useful is that it eliminates parameters such as L (box length) for a particle in a box with infinite walls (or k the spring constant for an oscillator. This allows for a quantum adiabatic transformation of a pure state to be isentropic. This follows from a Fourier transformation i.e. W(x)=Sum over p a(p)exp(ipx). The two variables p and x scale with L in the opposite manner i.e. x/L and pL (5). Thus we argue that entropy is not necessarily linked to only one operator, but rather for a pure state is linked to variables which appear in a Fourier transform. Usually a Fourier transform is simply a mathematical procedure, but not in the case of the quantum pure state wavefunction where the variable p is actually momentum. In this note we try to investigate how the quantum relationship between p and x emerges. We argue that it exists at the level of the classical action for a free particle (relativistic or nonrelativistic) A=-Et+px where dA/dx partial =p and dA/dt partial = -E. Thus p and x are variable pairs in classical dynamics and this carries over to classical statistical phase space linked to p and x. We argue that a quantum dynamical link between p and x is also present in A i.e. -id/dx exp(iA) = p exp(iA). In other words p and x are linked dynamically in quantum mechanics as well (for a free particle) with exp(-iEt+ipx) being part of this link. The Fourier series form is already present in this solution as one may create a momentum distribution i.e. Sum over p a(p)exp(ipx). This depends on p and x and both affect any operator based on p and x (or one of these). Thus we expect that probabilities linked to both p and x should appear in an expression of entropy for a quantum bound state.