Introduction.The theory of information, created by Shannon [23], is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g., cf.[10]), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g., cf.[8]).Through recent years, the theory is regarded as a new chapter in the theory of probability.Recently, Segal [22] gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics.Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki [16] and independently by Davis [3].Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf.for example [13~18] and [25 ~28]), and in the most recent paper [18] it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information.Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods.From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra.Davis [3] has independently studied on the almost same theme with Nakamura-Umegaki [16] and [18], in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy.Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper.Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p(aa*) ^ 0 for every a ε A. Such p is said to be state if p(I) = 1, to be normal in the terminology of Dixmier [4] if p(a a )tp(a) for a a ^a, and to be trace if p(ab) = p(ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p(p a ) = p(Σ Pa) for any disjoint family of projections {p a }aA (cf.Dixmier [4]).
Introduction.The theory of information, created by Shannon [23], is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g., cf.[10]), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g., cf.[8]).Through recent years, the theory is regarded as a new chapter in the theory of probability.Recently, Segal [22] gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics.Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki [16] and independently by Davis [3].Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf.for example [13~18] and [25 ~28]), and in the most recent paper [18] it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information.Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods.From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra.Davis [3] has independently studied on the almost same theme with Nakamura-Umegaki [16] and [18], in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy.Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper.Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p(aa*) ^ 0 for every a ε A. Such p is said to be state if p(I) = 1, to be normal in the terminology of Dixmier [4] if p(a a )tp(a) for a a ^a, and to be trace if p(ab) = p(ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p(p a ) = p(Σ Pa) for any disjoint family of projections {p a }aA (cf.Dixmier [4]).
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
Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.
A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.
As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.
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.
After a historical introduction, the most important classical and quantum entropies are introduced as constructions in classical and quantum probability theory. Classical entropies are studied from large deviation theory, including theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan, and are illustrated in some applications to both Boltzmannian and Gibbsian statistical physics. Quantum entropies superficially connect to classical entropies at the formula level, but more deeply do so via the crucial role of entropy in statistical hypothesis testing. The classical (relative) Kullback-Leibler entropy, its quantum counterpart introduced by Umegaki, as well as their deformations proposed by Renyi all fit naturally in this context. Quantum entropy faces the new problem of defining and computing the relative entropy of a pair of states on a subsystem, here formalized as a von Neumann algebra. This requires modular (aka Tomita-Takesaki) theory, which provides the framework for the relative quantum entropies introduced by Araki and Uhlmann (these encompass both the Kullback-Leibler and Umegaki entropies as special cases). To (re)define and compute these entropies in terms of density operators and traces, further constructions are needed, namely Haagerup's noncommutative L^p spaces. Von Neumann algebras also provide the setting for the Connes-Stormer-Narnhofer-Thirring entropy, which is a quantum version of the Kolmogorov-Sinai entropy in dynamical systems and ergodic theory, to which we also provide an introduction. This course was originally inspired by, and should be relevant to, black hole thermodynamics, although we discuss neither this application nor the second law. The course tries to be both mathematically rigorous and interesting to theoretical physicists. Prerequisites are undergraduate probability theory, functional analysis, and quantum theory.