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We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information., 8 Pages, RevTeX 4. Associated information (including a Java simulation) can be found at http://www.imaph.tu-bs.de/qi/monty

We consider situations in which i) Alice wishes to send quantum information to Bob via a noisy quantum channel, ii) Alice has a classical description of the states she wishes to send and iii) Alice can make use of a finite amount of noiseless classical information. After setting up the problem in general, we focus attention on one specific scenario in which Alice sends a known qubit down a depolarizing channel along with a noiseless cbit. We describe a protocol which we conjecture is optimal and calculate the average fidelity obtained. A surprising amount of structure is revealed even for this simple case which suggests that relationships between quantum and classical information could in general be very intricate., RevTeX, 5 pages, 2 figures Typo in reference 9 corrected

In this chapter, we will study the particles moving along a line from another perspective: How well do we know where a particle is? The answer depends naturally on the accuracy of our measurement. There is, however, also a question about how well the position of a particle is defined. A high-precision measurement cannot be made on a diffuse object.

More than half century ago, Shannon's identification of information as the reduction of entropy greatly clarified the meaning of information and established information theory as a science. Since all human activities represent extraction and transformation of low entropy from the environment, it is natural to relate economic value to low entropy. However, some conceptual difficulties prevented the development of an entropy theory of value. In this work, we identify economic value as the reduction of entropy mathematically and explore the relation between physical entropy and economic value. In the process, we show how the detailed investigation of information theory, thermodynamic theory and the theory of evolution resolves the conceptual difficulties that confound us for many years. The entropy theory of value, by establishing the theory of value on the firm foundation of thermodynamics, greatly clarified many fundamental issues in economic theory and human activities.

The theory of noncommutative dynamical entropy and quantum symbolic dynamics for quantum dynamical systems is analised from the point of view of quantum information theory. Using a general quantum dynamical system as a communication channel one can define different classical capacities depending on the character of resources applied for encoding and decoding procedures and on the type of information sources. It is shown that for Bernoulli sources the entanglement-assisted classical capacity, which is the largest one, is bounded from above by the quantum dynamical entropy defined in terms of operational partitions of unity. Stronger results are proved for the particular class of quantum dynamical systems -- quantum Bernoulli shifts. Different classical capacities are exactly computed and the entanglement-assisted one is equal to the dynamical entropy in this case., Comment: 6 pages

In this paper we discuss the problem of splitting the total correlations for a bipartite quantum state described by the Von Neumann mutual information into classical and quantum parts. We propose a measure of the classical correlations as the difference between the Von Neumann mutual information and the relative entropy of entanglement. We compare this measure with different measures proposed in the literature., Comment: 5 pages, 1 figure

A new paradigm for distributed quantum systems where information is a valuable resource is developed. After finding a unique measure for information, we construct a scheme for it's manipulation in analogy with entanglement theory. In this scheme instead of maximally entangled states, two parties distill local states. We show that, surprisingly, the main tools of entanglement theory are general enough to work in this opposite scheme. Up to plausible assumptions, we show that the amount of information that must be lost during the protocol of concentration of local information can be expressed as the relative entropy distance from some special set of states., Comment: 4 pages, revtex4, presentation improved, to appear in Phys. Rev. Lett

A scheme is presented for protecting one-qubit quantum information against decoherence due to a general environment and local exchange interactions. The scheme operates essentially by distributing information over two pairs of qubits and through error prevention procedures. In the scheme, quantum information is encoded through a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs; leakage out of the encoding space due to amplitude damping is reduced by quantum Zeno effect. In addition, how to construct decoherence-free states for n-qubit information against phase and exchange errors is discussed., Comment: accepted for publication in Physical Review A

We define a "nit" as a radix n measure of quantum information which is based on state partitions associated with the outcomes of n-ary observables and which, for n>2, is fundamentally irreducible to a binary coding. Properties of this measure for entangled many-particle states are discussed. k particles specify k nits in such a way that k mutually commuting measurements of observables with n possible outcomes are sufficient to determine the information., 4 pages, 2 figures

We describe a universal information compression scheme that compresses any pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalisation procedure that enables any classical compression algorithm to be utilised in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel-Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a by-product we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel-Ziv method, it can be applied also to target sequences that are not i.i.d., Comment: 17 pages, no figures. A preliminary version of this work was presented at EQIS '02, Tokyo, September 2002

Quantum correlations have long been recognized as an informational resource for quantum communication and computation. It now seems that they can also do physical work. See Letter p.61 Landauer's erasure principle, a widely accepted part of classical information theory first proposed by Rolf Landauer in 1961, asserts that it is necessary to perform work in order to erase data. This occurs when carrying out irreversible operations, thus releasing heat to the environment. For example, in electronics, heat generation is a major obstacle to circuitry miniaturization. Del Rio et al. show that the situation is completely different in the presence of quantum information about the system, and the implications of Landauer's principle are invalid. The more that is known about a system, the less it costs to erase it. An observer who is strongly correlated with a system may even gain work while erasing it, therefore cooling the environment. The quantum systems needed to experimentally demonstrate these results are, in principle, accessible with current technology.

Decoding Reality: The Universe as Quantum Information, by Vlatko Vedral, Oxford, Oxford University Press, 2010, 240 pp., £16.99 (hardback), ISBN 9780199237692. Scope: monograph. Level: general read...

A general prescription is given for finding uncertainty relations that dictate the lower bounds on the measures of spread corresponding to two different representations of a partially coherent wave field or mixed quantum state, for a given measure of overall coherence or purity. In particular it is shown that the coherent modes of the fields/states that achieve the lower bounds are independent of the measure of purity being used, and that this measure determines only the amount in which these modes contribute. Our results are important in the design of optical systems with partially coherent light and in quantum mixed states, for which maximal joint localization is desired. These ideas are illustrated for the case of optical beams, pulses propagating in dispersive media and quantum phase.

von Neumann entropy represents the amount of information in the quantum state, and this was extended by Ohya for general quantum systems [10]. Umegaki first defined the quantum relative entropy for -finite von Neumann algebras, which was extended by Araki, and Uhlmann, for general von Neumann algebras and *-algebras, respectively. In 1983 Ohya introduced the quantum mutual entropy by using compound states; this describes the amount of information correctly transmitted through the quantum channel, which was also extended by Ohya for general quantum systems. In this paper, we briefly explain Ohya?s -mixing entropy and the quantum mutual entropy for general quantum systems. By using structure equivalent class, we will introduce entropy type functionals based on quantum information theory to improve treatment for the Gaussian communication process.

Relevance of key quantum information measures for analysis of quantum systems is discussed. It is argued that possible ways of measuring quantum information are based on compatibility/incompatibility of the quantum states of a quantum system, resulting in the coherent information and introduced here the compatible information measures, respectively. A sketch of an information optimization of a quantum experimental setup is proposed., 10 pages, 5 figures, submitted to the Procs of 17th Int'l Conf. on Coherent and Nonlinear Optics (ICONO-2001), June 26-July 1, 2001, Minsk, Belarus

We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A consequence of our results is that bound entangled states cannot increase the capacity of a noiseless quantum channel., 9 pages, 2 figures, uses qic.sty, submitted to QIC journal. Improved presentation and typographical errors corrected

The coherent information concept is used to analyze a variety of simple quantum systems. Coherent information was calculated for the information decay in a two-level atom in the presence of an external resonant field, for the information exchange between two coupled two-level atoms, and for the information transfer from a two-level atom to another atom and to a photon field. The coherent information is shown to be equal to zero for all full-measurement procedures, but it completely retains its original value for quantum duplication. Transmission of information from one open subsystem to another one in the entire closed system is analyzed to learn quantum information about the forbidden atomic transition via a dipole active transition of the same atom. It is argued that coherent information can be used effectively to quantify the information channels in physical systems where quantum coherence plays an important role.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A continuous key distribution scheme is proposed that relies on a pair of canonically conjugate quantum variables. It allows two remote parties to share a secret Gaussian key by encoding it into one of the two quadrature components of a single-mode electromagnetic field. The resulting quantum cryptographic information vs disturbance tradeoff is investigated for an individual attack based on the optimal continuous cloning machine. It is shown that the information gained by the eavesdropper then simply equals the information lost by the receiver., Comment: 5 pages, RevTex

An entangled two-mode coherent state is studied within the framework of $2\times 2$ dimensional Hilbert space. An entanglement concentration scheme based on joint Bell-state measurements is worked out. When the entangled coherent state is embedded in vacuum environment, its entanglement is degraded but not totally lost. It is found that the larger the initial coherent amplitude, the faster entanglement decreases. We investigate a scheme to teleport a coherent superposition state while considering a mixed quantum channel. We find that the decohered entangled coherent state may be useless for quantum teleportation as it gives the optimal fidelity of teleportation less than the classical limit 2/3., 7 pages, 4 figures. The content has been modified and the title has been changed. Accepted for publication in Phys. Rev. A

The author calls attention to previous work with related results, which has escaped scrutiny before the publication of the article "Nonlinear quantum evolution with maximal entropy production", Phys.Rev.A63, 022105 (2001)., Comment: RevTex-latex2e, 2pgs., no figs.; brief report to appear in the May 2001 issue of Phys.Rev.A

We show how to compute or at least to estimate various capacity-related quantities for bosonic Gaussian channels. Among these are the coherent information, the entanglement-assisted classical capacity, the one-shot classical capacity, and a quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation or amplification and classical noise.

We show that a qubit can be used to substitute for an arbitrarily large number of classical bits. We consider a physical system S interacting locally with a classical field phi(x) as it travels directly from point A to point B. The field has the property that its integrated value is an integer multiple of some constant. The problem is to determine whether the integer is odd or even. This task can be performed perfectly if S is a qubit. On the otherhand, if S is a classical system then we show that it must carry an arbitrarily large amount of classical information. We identify the physical reason for such a huge quantum advantage, and show that it also implies a large difference between the size of quantum and classical memories necessary for some computations. We also present a simple proof that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement., Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement simulation with classical communication; v3: minor correction to main proof, change of title, added reference

Basic concepts in probability and information theory have been used to examine some longstanding problem in quantum physics. In particular, a new concept called the conservation of information appears to offer insight into laser interferometer experiments done at the University of Rochester and at the University of California at Berkeley. These interferometer experiments demonstrate that information controls quantum interference and presumably the wave function. New “gedanken” experiments proposed here deal explicitly with the collapse of the quantum mechanical wave function by information alone. Use would be made of laser fluorescence and resonance ionization spectroscopy to explore more fully the control of the wave function with information; as well as the temporal relationship of information generated locally with information in a distant system. If one assumes the conservation of information at all time and in any reference frame, teleportation of atoms emerges as a natural consequence.

Publisher Summary This chapter discusses the relationship between entropy information and optics. Recent development of optical communication, signal processing, and computing, among other discoveries, the relationship between optics and entropy information has grown more profound. From the viewpoint of mathematic formalism, entropy information is basically a probabilistic concept. In other words, without probability theory there would be no entropy information. Information transmission can be in fact represented by spatial and temporal information. Information channels are described according to the type of input/output ensemble and are considered discrete or continuous. In practice, all communication channels are band limited. An information channel can be a low-pass, bandpass, or discrete bandpass channel. A problem of considerable importance in signal processing is the extraction of signal from random noise and distortion. There are, however, two major approaches to this issue; the extraction of signals that have been lost in random noise as examples, called signal detection, and the reconstruction of unknown signals that have been distorted as examples, called signal recovery or restoration.

Quantum mechanics and information theory are among the most important scientific discoveries of the last century. Although these two areas initially developed separately it has emerged that they are in fact intimately related. In this review I will show how quantum information theory extends traditional information theory by exploring the limits imposed by quantum, rather than classical mechanics on information storage and transmission. The derivation of many key results uniquely differentiates this review from the "usual" presentation in that they are shown to follow logically from one crucial property of relative entropy. Within the review optimal bounds on the speed-up that quantum computers can achieve over their classical counter-parts are outlined using information theoretic arguments. In addition important implications of quantum information theory to thermodynamics and quantum measurement are intermittently discussed. A number of simple examples and derivations including quantum super-dense coding, quantum teleportation, Deutsch's and Grover's algorithms are also included., Comment: 40 pages, 11 figures

Like the coherent information, the entanglement of formation may be used as a measure of the quantum information sent over a channel. These measures are related, and share the property of giving necessary and sufficient conditions for perfect quantum error correction.

“Can we truly compute, until we understand what information really is?” Gordon Scarrott. A new, mathematically described understanding of physically meaningful information, quantum holography, concerning actual knowledge of the 3 dimensional physical world in natural systems, is proposed. It is based on demonstrably proven anticipatory quantum mechanical laws and the new awareness in quantum theory. This understanding concerns a form of information, which holography shows, almost certainly existed before the origination of living systems and even from the beginning of the cosmos. It produces physically realisable mathematical definitions of the concepts of information, knowledge, learning, intelligence, perception, cognition, etc. Some of its other many advantages are cited. In particular, being quite distinct from bits, which are simply physically realisable mental models for the carriage/transmission of symbolic data (dependent for its meaning on human interpretation) it is not, its mathematical theory indicates, subject to the processing limitations of the combinatorial explosion governing algorithmic complexity, or to the known processing limitations of formal systems, such as the Halting Problem, as they are thought to apply to classical digital computing systems.

We present theoretical and experimental evidences, which show that the material phase of the state vector is causally related with the dynamic of the quantum system and becomes carrier of physical information., Comment: 3 pages, 1 figure

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as $$tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } $$ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states $$\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|$$ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and $$pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1$$

We present an approximate inference approach based on the Information Theory and the Pseudoinverse Formalism that allows to obtain all the accessible micro-states of a quantum system compatible with a given information. We assume that the Hamiltonian of the system is unknown and the only available information is restricted to a set of expectation values of noncommuting operators. With this information the density matrix is inferred and, exploring the Null space of the input operators, one can define the physical regions for which the density matrix is positive definite. The method is applied for the inference of entangled quantum states.

The generalized measurements that decrease the entropy of the measured system in average are characterized in terms of their completely positive map valued measures. A lower bound for the entropy decrease is also obtained.

The communication of directions using quantum states is a useful laboratory test for some basic facts of quantum information. For a system of spin-1/2 particles there are different quantum states that can encode directions. This information can later be decoded by means of a generalized measurement. In this talk we present the optimal strategies under different assumptions., Comment: Contribution to the ICQI, Rochester (NY) June 2001

Although it has been mainly during the last 15years that the number of scientific investigationsrelatingtoquantuminformationhasbecomegreatenoughthatQuantum Information Science (QIS) could be identified as a distinct field of research encompassing quantum computing and cryptography, its roots can be seen as extending back to the early decades of quantum theory itself. That is, QIS can be viewed as resulting from the on-going study of fundamental questions in quantum mechanics, now often referred to as simply Quantum Foundations. It relates mechanics and information in the microscopic realm, offering new insights by connecting the concepts and constraints of communications theory to information theory in a way that provides new methods for probing the elements of fundamental mechanics. It is to such probing of the implications of quantum correlations, measurements, and information theory that this Special Issue of Quantum Information Processing is devoted, focusing on foundational ones. Despite a tremendous increase in activity surrounding the technological aspects of quantum information and entanglement, which is to say surrounding applications of the qubit and the e-bit in communication and information processing, foundational issues relating to quantum correlation, measurement, and information theory remain in need of further clarification and development (cf., e.g. [1]). This is evident here, for example, in the contribution of Mauro D’Ariano and Alessandro Tosini, who consider tests of postulates of the “fair operational framework” in which quantum mechanics is understood as a set of rules allowing experimenters to predict the results of future measurement events that can be used toyieldinformation on thebasis of suitabletests

How much information about the original state preparation can be extracted from a quantum system which has already been measured? That is, how many independent (noncommunicating) observers can measure the quantum system sequentially and give a nontrivial estimation of the original unknown state? We investigate these questions, and show from a simple example that information about the original preparation is not entirely lost as a result of the measurement-induced collapse of the quantum state, and that an infinite number of independent observers who have no prior knowledge about the initial state can gain partial information about the original preparation of the quantum system.

This paper discusses a discretization problem for quantum continuous channels. We show that the binary discretization asymptotically realizes the capacity and the maximum mutual information; the classical continuous channel does not have such an asymptotic property. We further clarify the significance of Gordon's result from the aspect of binary discretization for the maximum mutual information.

We generalize the Shannon's information theory in a nonadditive way by focusing on the source coding theorem. The nonadditive information content we adopted is consistent with the concept of the form invariance structure of the nonextensive entropy. Some general properties of the nonadditive information entropy are studied, in addition, the relation between the nonadditivity $q$ and the codeword length is pointed out., 9 pages, no figures, RevTex, accepted for publication in Phys. Rev. E(an error in proof of theorem 1 was corrected, typos corrected)

The cloning of conjugate continuous quantum variables is analyzed based on the concept of Gaussian cloning machines, i.e., transformations that yield copies that are Gaussian mixtures centered on the state to be copied. The optimality of Gaussian cloning machines that transform N identical input states into M output states is investigated, and bounds on the fidelity of the process are derived via a connection with quantum estimation theory. In particular, the optimal N-to-$M$ cloning fidelity for coherent states is found to be equal to $MN/(MN+M\ensuremath{-}N).$

The coherent information concept is used to analyze a variety of simple quantum systems. Coherent information was calculated for the information decay in a two-level atom in the presence of an external resonant field, for the information exchange between two coupled two-level atoms, and for the information transfer from a two-level atom to another atom and to a photon field. The coherent information is shown to be equal to zero for all full-measurement procedures, but it completely retains its original value for quantum duplication. Transmission of information from one open subsystem to another one in the entire closed system is analyzed to learn quantum information about the forbidden atomic transition via a dipole active transition of the same atom. It is argued that coherent information can be used effectively to quantify the information channels in physical systems where quantum coherence plays an important role., Comment: 24 pages, 7 figs; Final versiob after minor changes, title changed; to be published in Phys. Rev. A, September 2000

A new measure of information in quantum mechanics is proposed which takes into account that for quantum systems the only feature known before an experiment is performed are the probabilities for various events to occur. The sum of the individual measures of information for mutually complementary observations is invariant under the choice of the particular set of complementary observations and conserved if there is no information exchange with an environment. That operational quantum information invariant results in N bits of information for a system consisting of N qubits., 4 pages, 1 figure

How much information about an unknown quantum state can be obtained by a measurement? We propose a model independent answer: the information obtained is equal to the minimum entropy of the outputs of the measurement, where the minimum is taken over all measurements which measure the same ``property'' of the state. This minimization is necessary because the measurement outcomes can be redundant, and this redundancy must be eliminated. We show that this minimum entropy is less or equal than the von Neumann entropy of the unknown states. That is a measurement can extract at most one meaningful bit from every qubit carried by the unknown states.

This paper briefly considers the notion of a biology of quantum information from a number of complementary points of view. We begin with a very brief look at some of the biomolecular systems that are thought to exploit quantum mechanical effects and then turn to the issue of measurement in these systems and the concomitant generation of information. This leads us to look at the internalist stance and the exchange interaction of quantum particles. We suggest that exchange interaction can also be viewed using ecological ideas related to apparatus-object. This can also help develop the important notion of complementarity in biosystems in relation to the nature and generation of information at the microphysical scale.

Basing on unified approach to {\it all} kinds of quantum capacities we show that the rate of quantum information transmission is bounded by the maximal attainable rate of coherent information. Moreover, we show that, if for any bipartite state the one-way distillable entanglement is no less than coherent information, then one obtains Shannon-like formulas for all the capacities. The inequality also implies that the decrease of distillable entanglement due to mixing process does not exceed of corresponding loss information about a system., Comment: 4 pages, Revtex

An information theoretic measure is derived that quantifies the statistical coherence between systems evolving in time. The standard time delayed mutual information fails to distinguish information that is actually exchanged from shared information due to common history and input signals. In our new approach, these influences are excluded by appropriate conditioning of transition probabilities. The resulting transfer entropy is able to distinguish driving and responding elements and to detect asymmetry in the coupling of subsystems., Comment: 4 pages, 4 Figures, Revtex

The discipline of information theory was founded by Claude Shannon in a truly remarkable paper [Sh] which laid down the foundations of the subject. We begin with a quote from this paper which is an excellent summary of the main concern of information theory: The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.

We show that the recently discovered universal upper bound on the thermal conductance of a single channel comprising particles obeying arbitrary fractional statistics is in fact a consequence of a more general universal upper bound, involving the averaged entropy and energy currents of a single channel connecting heat reservoirs with arbitrary temperatures and chemical potentials. The latter upper bound in turn leads, via Holevo's theorem, to a universal (i.e., statistics independent) upper bound on the optimum capacity for classical information transmission down a single, wideband quantum channel., Comment: 4 figures added; minor changes to text. To appear in Phys. Rev. A, Vol. 62, Nov. 2000

A concern has been expressed that ``the Jaynes principle can produce fake entanglement'' [R. Horodecki et al., Phys. Rev. A {\bf 59}, 1799 (1999)]. In this paper we discuss the general problem of distilling maximally entangled states from $N$ copies of a bipartite quantum system about which only partial information is known, for instance in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principle of maximum entropy to more than one copy of a system, but the nature of this problem is classical and was discussed extensively by Jaynes. Under the additional assumption that the state $\rho^{(N)}$ of the $N$ copies of the quantum system is exchangeable, one can write down a simple general expression for $\rho^{(N)}$. We show how to modify two standard entanglement purification protocols, one-way hashing and recurrence, so that they can be applied to exchangeable states. We thus give an explicit algorithm for distilling entanglement from an unknown or partially known quantum state., Comment: 20 pages RevTeX 3.0 + 1 figure (encapsulated Postscript) Submitted to Physical Review A

A counterexample to the 'additivity question', the most celebrated open problem in the mathematical theory of quantum information, casts doubt on the possibility of finding a simple expression for the information capacity of a quantum channel.

We present optimal and minimal measurements on identical copies of an unknown state of a qubit when the quality of measuring strategies is quantified with the gain of information (Kullback of probability distributions). We also show that the maximal gain of information occurs, among isotropic priors, when the state is known to be pure. Universality of optimal measurements follows from our results: using the fidelity or the gain of information, two different figures of merits, leads to exactly the same conclusions. We finally investigate the optimal capacity of $N$ copies of an unknown state as a quantum channel of information., Revtex, 5 pages, no figures

This chapter has two goals. The first is to develop the quantum version of multiplicity. That will show us how entropy is expressed in quantum theory. The second goal is to develop a general quantitative definition of temperature. Entropy plays a key role in that definition. The density of states When a physical system has reached thermal equilibrium, its macroscopic properties do not change with time. In quantum theory, the energy eigenstates of an isolated system provide predictions and estimates that are constant in time; therefore such states are appropriate for a quantum description of thermal equilibrium. To be sure, the information at hand will not enable us to select a single state as uniquely the correct state to use. We will be driven to consider many states and to form sums over them. This section develops a mathematical technique for working with such sums. When the system is both isolated and of finite size (as we shall specify here), the energy eigenstates form a discrete set, whose members we can arrange and label in order of increasing energy. (If any two distinct states happen to have the same energy, we just assign them consecutive labels.) Typically, the states will be densely spaced in energy, and so a sum over a range of states can often be approximated adequately by an integral with respect to energy, provided that we have constructed an appropriate density of states : a function that specifies the number of energy eigenstates per unit energy interval.