Abstract Information is often considered classical in a definite state rather than in a superposition of states. It seems rather strange to consider information in superpositions. Some people would, on the basis of this argument, conclude that quantum information can never exist and we can only have access to classical information. It turns out, however, that quantum information can be quantified in the same way as classical information using Shannon's prescription. There is a unique measure (up to a constant additive or multiplicative term) of quantum information such that S (the von Neumann entropy) is purely a function of the probabilities of outcomes of measurements made on a quantum system (that is, a function of a density operator); S is a continuous function of probability; S is additive. This chapter discusses the fidelity of pure quantum states, Helstrom's discrimination, quantum data compression, entropy of observation, conditional entropy and mutual information, relative entropy, and statistical interpretation of relative entropy.
Abstract Information is often considered classical in a definite state rather than in a superposition of states. It seems rather strange to consider information in superpositions. Some people would, on the basis of this argument, conclude that quantum information can never exist and we can only have access to classical information. It turns out, however, that quantum information can be quantified in the same way as classical information using Shannon's prescription. There is a unique measure (up to a constant additive or multiplicative term) of quantum information such that S (the von Neumann entropy) is purely a function of the probabilities of outcomes of measurements made on a quantum system (that is, a function of a density operator); S is a continuous function of probability; S is additive. This chapter discusses the fidelity of pure quantum states, Helstrom's discrimination, quantum data compression, entropy of observation, conditional entropy and mutual information, relative entropy, and statistical interpretation of relative entropy.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumman entropy. It is the quantum analog of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to a notion of the information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state in an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is in a quantum system. The beginning definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical spatial correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations.
Abstract This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.
This chapter sets the basis of quantum information theory (QIT) . The central purpose of QIT is to qualify the transmission of either classical or quantum information over quantum channels . The starting point of the QIT description is von Neumann entropy , S (ρ), which represents the quantum counterpart of Shannon's classical entropy, H ( X ). Such a definition rests on that of the density operator (or density matrix ) of a quantum system, ρ, which plays a role similar to that of the random-events source X in Shannon's theory. As we shall see, there also exists an elegant and one-to-one correspondence between the quantum and classical definitions of the entropy variants relative entropy , joint entropy , conditional entropy , and mutual information . But such a similarity is only apparent. Indeed, one becomes rapidly convinced from a systematic analysis of the entropy's additivity rules that fundamental differences separate the two worlds. The classical notion of information correlation between two event sources for quantum states shall be referred to as quantum entanglement . We then define a quantum communication channel, which encodes and decodes classical information into or from quantum states. The analysis shows that the mutual information H ( X ; Y ) between originator and recipient in this communication channel cannot exceed a quantity χ, called the Holevo bound , which itself satisfies χ ≤ H ( X ), where H ( X ) is the entropy of the originator's classical information source.
From a classical perspective, a quantum state is a surprising mixture of information and ignorance. If a system is sitting in an eigenstate of an observable, we can predict with confidence that a measurement result will yield the appropriate eigenvalue. With a superposition, certainty is not granted to us. Unlike the closest possible classical case, where a superposition represents a genuine blending of classical states, the quantum superposition is a strange overlapping of states simultaneously occupied by the system. On measurement, this ontologically curious animal collapses into one of the overlapping states, a process that is not well understood to this day. The collapse is random, a-causal, but governed by quantum amplitudes. Heading into a measurement, we do not know what we are going to get out. Given the evident link between entropy and information, the probabilities and uncertainties baked into a quantum state may well be a fertile ground for another form of entropy. The density matrix gives us a flexible tool for studying the properties of quantum ensembles of any kind. In this chapter, we will see how von Neumann helped develop and use the density matrix to study the problem of measurement and the information/entropy content of quantum states .
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.