This series of introductory lectures consists of two parts. The first part reviews the basic notions of quantum physics and some primitives of quantum information. The latter are notions that one must somehow be familiar with in the field: quantum cloning, teleportation and entanglement swapping, entanglement distillation, state estimation (both single shot and multi-copy), and quantum coding. The second part is devoted to a detailed introduction to the topic of quantum correlations. The evidence for failure of alternative theories is presented: the violation of Bell’s inequalities is dealt with in detail, other tests like Leggett’s inequalities and the before-before experiment are discussed. The last two lectures introduce the formalism of no-signaling probability distributions and explain the possibility of device-independent quantum information.
This series of introductory lectures consists of two parts. The first part reviews the basic notions of quantum physics and some primitives of quantum information. The latter are notions that one must somehow be familiar with in the field: quantum cloning, teleportation and entanglement swapping, entanglement distillation, state estimation (both single shot and multi-copy), and quantum coding. The second part is devoted to a detailed introduction to the topic of quantum correlations. The evidence for failure of alternative theories is presented: the violation of Bell’s inequalities is dealt with in detail, other tests like Leggett’s inequalities and the before-before experiment are discussed. The last two lectures introduce the formalism of no-signaling probability distributions and explain the possibility of device-independent quantum information.
Introduction We will now turn to quantum entanglements and their contemporary, possibly practical, interest. Entanglements, first discussed in the E.P.R. paradox (Einstein, Podolsky, Rosen, 1935) are a perplexing nonlocal feature of quantum mechanics. This was immediately and succinctly discussed by Schrödinger (1935). The long history of this apparent paradox is outlined in the wonderful book of Jammer (1974). We will not focus on the central issue of “hidden variables” and their resolution by the Bell inequalities (Bell, 1964) and the test, nor the failure, of these in experiment (see Fry, 1998). The distinctive quantum nature of entanglements has led to two quantum effects which we will discuss: quantum information teleportation (Zeilenger, 1998) and quantum computation by means of entangled states. A nice, recent, elementary introduction to the latter is in the Los Alamos reports of James and Kwiat (2002). The quantum correlations or entanglements are sensitive to environmental destruction. This was pointed out early by Zurek, who termed this “decoherence.” The loss of coherence may occur on a short time scale. Recent discussions have been given by Zurek (2002, 2003). We have already given a theoretical example early in Chapter 2. It is pertinent to discuss this here, since it is a property of opensystem quantum master equations that is a central part of our study in this book. In a sense, the decoherence of correlations has turned out to be a “practical” application of these theoretical notions. Are there remedies for unwanted decoherence? Quantum error correction is a possibility. This will be mentioned also. Entanglements: foundations Following the reading of the E.P.R. paper, Schrödinger (1935b) quickly repeated the argument from quantum theory, but more generally. He introduced the term “entanglement” to describe what he seemed to agree was a curious, if not unacceptable, property. Let us follow his point of view. Given a state ψ ( x, y ) of a composite system of particles x and y formed in their mutual interaction. ψ ( x , y ) is not a product state.
Why yet another book on quantum information theory? Like many lecturers we began writing this text because none of the alternatives seemed quite right. This book is aimed squarely at undergraduate physics students who want a brief but reasonably thorough introduction to the exciting ideas of quantum information, including its applications in computation and communication. It is based on a short course we have taught to fourth-year students at Oxford University since 2004; for the most part it only assumes knowledge of elementary quantum mechanics and linear algebra, and so could even be taught to third-year undergraduates. A brief revision guide to quantum mechanics is provided as an appendix, which should cover any minor points that have been missed. As the title implies the book is structured in three parts, starting with the basics of quantum information and then applying this to quantum computation and quantum communication. Part I is self-contained, but contains only the barest hints of the exciting applications which attract many people to this field and so might prove unsatisfying on its own. Parts II and III draw heavily on Part I, but are largely independent of each other, and it would be perfectly possible to study only one of these two without the other. As this text is aimed at physics undergraduates, we believe that it is vital to cover experimental techniques, rather than merely presenting quantum information as a series of abstract quantum operations. We have, however, concentrated on the basic ideas underlying each approach, rather than worrying about particular experimental details.
Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
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.
We explore quantum nonlocality in one of the simplest bipartite scenarios. Several new facet-defining Bell inequalities for the scenario are obtained with their quantum violations analyzed in details. Surprisingly, all these inequalities involving only genuine ternary-outcome measurements can be violated maximally by some two-qubit entangled states, such as the maximally entangled two-qubit state. This gives further evidence that in analyzing the quantum violation of Bell inequalities, or in the application of the latter to device-independent quantum information processing tasks, the commonly held wisdom of equating the local Hilbert space dimension of the optimal state with the number of measurement outcomes is not necessarily justifiable. In addition, when restricted to the minimal qubit subspace, it can be shown that one of these Bell inequalities requires non-projective measurements to attain maximal quantum violation, thereby giving the first example of a facet-defining Bell inequality where a genuine positive-operator-valued measure is relevant. We experimentally demonstrate the quantum violation of this and two other Bell inequalities for this scenario using energy–time entangled photon pairs. Using the obtained measurement statistics, we demonstrate how characterization of the underlying resource in the spirit of device-independence, but supplemented with auxiliary assumptions, can be achieved. In particular, we discuss how one may get around the fact that, due to finite-size effects, raw measurement statistics typically violate the non-signaling condition.
The scope of this thesis is to present and discuss some experiments related to the quantum phenomenon of entanglement. We start by introducing the two forms of entanglement that we use: energy-time entanglement and single-photon entanglement. We explain how entanglement can be created through the process of spontaneous parametric down conversion in non-linear waveguides. We explain the importance of Bell inequalities and how we can use them to test the nonlocal nature of quantum correlations. The first experiment is a Bell test with spacelike separation, where we explore the possibility that a change in the gravitational field around one of the detection devices could trigger the collapse of the wavefunction, ending the entangled state. In the second one, we test the hypothesis that a superluminal signal defined in a universal privileged reference frame travelling between the detectors could influence the results. Last, we present the first experimental implementation of an entanglement purification protocol using single-photon entanglement and linear optics elements.