The quantum state contains all accessible information about a system. However, for various reasons, the information is less complete than we might expect given the classical states we are used to. This suggests the possibility that the state itself contains a measure of entropy (lack of information), which could potentially arise in two ways. Firstly, quantum states contain an inherent uncertainty with respect to the measurement outcomes of conjugate variables. Secondly, due to limitations in the preparation process, we may not be sure which state a system is occupying. The best we can do in this situation is to assign each potential state a classical probability and construct a specialist mathematical representation to deal with the situation. This is known as the density matrix. It turns out to be a rather handy tool in general and crucial for the exploration of quantum entropy. So, in this chapter, we are going to extend our technical understanding of quantum theory and the representation of states. This will arm us sufficiently to venture into the physics of the measurement problem and the entropy of quantum systems, which we will tackle in Chapter 12 .
The quantum state contains all accessible information about a system. However, for various reasons, the information is less complete than we might expect given the classical states we are used to. This suggests the possibility that the state itself contains a measure of entropy (lack of information), which could potentially arise in two ways. Firstly, quantum states contain an inherent uncertainty with respect to the measurement outcomes of conjugate variables. Secondly, due to limitations in the preparation process, we may not be sure which state a system is occupying. The best we can do in this situation is to assign each potential state a classical probability and construct a specialist mathematical representation to deal with the situation. This is known as the density matrix. It turns out to be a rather handy tool in general and crucial for the exploration of quantum entropy. So, in this chapter, we are going to extend our technical understanding of quantum theory and the representation of states. This will arm us sufficiently to venture into the physics of the measurement problem and the entropy of quantum systems, which we will tackle in Chapter 12 .
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 astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
A classical state specifies the quantitative values of all the state variables relevant to a given system. In some situations, we can know the overall state of the system without being sure about the individual states of the constituent parts. On top of the problems inherent in picking out the crucial details of a system, there is another issue with trying to specify an initial state precisely: there is always some uncertainty connected with measurement. The quantum state of a system contains a series of complex numbers related to the probability that the system will collapse into a new state when a measurement takes place. Each of these new states represents a possible result of the measurement, which might be a path or a specific quantitative value of some physical variable.
All physical systems register bits of information, whether it be an atom, an electrical current, the location of a billiard ball, or a switch. Information can be classical, quantum, or a hybrid of both, depending on the system. For example, an atom or an electron or a superconducting system can register quantum information because the quantum theory applies to each of these systems, but we can safely argue that the location of a billiard ball registers classical information only. These atoms or electrons or superconducting systems can also register classical bits because it is always possible for a quantum system to register classical bits. The term information , in the context of information theory, has a precise meaning that is somewhat different from our prior “everyday” experience with it. Recall that the notion of the physical bit refers to the physical representation of a bit, and the information bit is a measure of how much we learn from the outcome of a random experiment. Perhaps the word “surprise” better captures the notion of information as it applies in the context of information theory. This chapter begins our formal study of classical information. Recall that Chapter 2 reviewed some of the major operational tasks in classical information theory. Here, our approach is somewhat different because our aim is to provide an intuitive understanding of information measures, in terms of the parties who have access to the classical systems. We define precise mathematical formulas that measure the amount of information encoded in a single physical system or in multiple physical systems. The advantage of developing this theory is that we can study information in its own right without having to consider the details of the physical system that registers it. We first introduce the entropy in Section 10.1 as the expected surprise of a random variable. We extend this basic notion of entropy to develop other measures of information in Sections 10.2–10.6 that prove useful as intuitive informational measures, but also, and perhaps more importantly, these measures are the answers to operational tasks that one might wish to perform using noisy resources. While introducing these quantities, we discuss and prove several mathematical results concerning them that are important tools for the practicing information theorist.
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.
Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.