In this chapter, we shall collect and generalize the results of the previous three chapters. We shall come back to the issue of quantum correlations in terms of information. We shall consider now the second aspect of quantum information [see Section 11.3], that is, information selection leading to a local growth of entropy (as it is the case during ameasurement). This will be the basis for examining again the whole process of measurement and finally assign an ontological status to fundamental entities of quantum theory like states, observables, and properties.
In this chapter, we shall collect and generalize the results of the previous three chapters. We shall come back to the issue of quantum correlations in terms of information. We shall consider now the second aspect of quantum information [see Section 11.3], that is, information selection leading to a local growth of entropy (as it is the case during ameasurement). This will be the basis for examining again the whole process of measurement and finally assign an ontological status to fundamental entities of quantum theory like states, observables, and properties.
Abstract The results of the experiments described in the last chapter all point fairly unambiguously to a reality that is decidedly non-local in nature, at least in the quantum domain. Of course, these are results that come as no surprise to those willing to accept one of the most fundamental principles of the Copenhagen interpretation — that quantum phenomena are describable only in terms of the dual classical physical concepts of waves and particles. The wave properties of quantum entities lend them inherently non-local characteristics, and it is only when we forcibly ‘collapse’ the wave function by making a measurement that we are taken aback by what appears to be contradictory or counter-intuitive behaviour.
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.
This book is not meant to contain a detailed analysis of the foundations of quantum mechanics. However, we felt an introduction to the general theory was required in order to make the presentation self-sufficient and clear the following chapters from possible ambiguities. We shall, wherever possible, avoid the intricacies of the mathematical description of infinite-dimensional spaces, as well as the conceptual subtleties related to the interpretation of the formalism (although the approach taken will necessarily betray aspects of the author's stand on these matters), and content ourselves with a ‘pragmatic’ introduction to the theory, as the chapter's title suggests. Throughout the chapter, we will even occasionally refer to a hypothetical ‘user’ of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), will be the subject of the second half of the chapter, as they are very relevant to the material covered in the book.
In this chapter, we will discuss the ontological status of the notion of quantuminformation. In the first part, we review the mathematical framework ofa generalized information theory. In this framework, probability theory will play a central role and, thus, we will devote Section 2 to it.We argue that the existence of probabilistic models that go beyond the classical and quantum realms, and the possibility of performing informational protocols in those models, allow us to claim that a generalized information theory can be conceived. Notice that this perspective is complementary to that of other research programs (see, e.g., Holik et al. 2015; Holik et al. 2016, and references therein). In the second part, we will consider the question about the ontology of the possible models satisfying the different instances of that generalized formalism. This will be the content of Section 3. Specifically, we want to study the role quantum mechanics plays in quantum information protocols; the question is whether there are other possible ontologies that can perform the same tasks. We will relate this problem to the existence of many examples of physical systems that are actually constructed using an essentially classical ontology but, on the other hand, are modeled by structures resembling the laws of quantum mechanics (e.g. Aerts 1998; Kwiat et al. 2000; Spreeuw 2001;Couder et al. 2005; Couder and Fort 2006; Qian et al. 2015). Their significancefor us relies in the fact that they can be used to perform quantum information protocols (though, in an inefficient way) (Man?ko et al. 2001; Bhattacharya et al. 2002; Francisco et al. 2006a, 2006b; Francisco and Ledesma 2008; Goldin et al. 2010). On this basis, we will explore the ontological implications of the existence of these simulations. Finally, in Section 4 we will draw our conclusions.
Nowadays, no one should doubt that quantum entanglement is (at least, one of) the \ncharacteristic traits of quantum mechanics. Furthermore, quantum information science \nis strongly influenced by the advantages provided by quantum entanglement, as the \nlatter can be regarded as a genuine resource for an efficient (better-than-classical) \nprocessing of information in certain computational tasks. The very basic notion of \ncorrelation is fundamental for the conceptualization of the physical communicational \nproblem and, as a consequence, for the notion of information itself. Thus, the question \nabout the nature of quantum information is close (and unavoidably linked) to the \nquestion about the nature of quantum correlations.
In this chapter, I summarize the general way in which the measurement process can be cast (by making use of effects and amplitude operators). Then, I show that there are two main problems with the state measurement: (i) how to avoid the disruptive back action on the state of the measured system during detection and (ii) how to extract complete information from this state. In order to deal with them I first introduce quantum non-demolition (QND) measurement and examine the problem whether the entire probability distribution of the measured observable is not altered by a QND measurement, which would allow repeated QND measurements with different observables to extract the whole information from the measured system. However, I show that this is not the case. Then, I deal with protective measurement as such and show that a reversible measurement is in fact not a measurement. However, by taking advantage of statistical methods (and therefore by renouncing measurement of the state of a single system), we can indeed reconstruct the wave function but only by partially recovering the information contained in the state. Two further prices to pay are to admit the existence of negative quasi-probabilities due to the interference terms and to make use of unsharp observables for guaranteeing informational completeness. Introduction The aim of this chapter is a review of the developments related to the measurement of the state vector and the problems that have been raised in this context. Traditionally, the state vector or the wave function describing quantum systems has been considered as a formal tool for calculating the probabilities associated with certain events like measurement outcomes, but few scholars have tried to attribute to it an ontological status. It is well known how many difficulties are related to an attempt at assigning a kind of reality to the state of quantum systems.