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This chapter reviews a collection of other philosophical and conceptual consequences of the Everett interpretation: the popular-science idea that chaos theory makes the future sensitively dependent upon our present actions; some exotic situations involving quantum probability, such as the infamous ‘quantum suicide’ thought experiment; the proposal by Deutsch that other worlds are directly observed in quantum-mechanical interference processes; the ontologica status of mixed states; and the nature of time travel in an Everettian universe.

This book explains nuclear structure b building on a few elementary physical ideas. It discusses the shell structure of nuclei, starting with the independent particle model and going on to the shell model for multiparticle configurations. Collective models for even-even nuclei are also considered, starting with macroscopic models of vibrational and rotational motion. It covers the evolution of nuclear structure with nucleon number, as well as odd-mass deformed nuclei, the Nilsson model and its consequences, and exotic nuclei and radioactive beams. This book presents two principal facts: namely, the beautiful richness and variety of nuclear physics and the extent to which we can understand nuclear data and models by invoking a few extremely basic ideas and drawing upon arguments that are physically transparent and intuitive. Many of the arguments presented in the book are based on a few basic simple ideas, including the short-range nature of the nuclear force, the effects of the Pauli principle, and two-state mixing.

This chapter discusses how entangled two states are. One obvious way is to consider the distillation procedure, in which the entanglement of any state is quantified by how many fully entangled qubits can be distilled from an entangled state. However, it is not easy to see how to distil singlets out of n copies of a general mixed state. Hence, it is necessary to first see what can be done with pure states. This chapter discusses the distillation of multiple copies of a pure state, analogy between quantum entanglement and the Carnot cycle, properties of entanglement measures, entanglement of pure states, entanglement of mixed states, measures of entanglement derived from relative entropy, relationship between classical information and entanglement, and links between entanglement and thermodynamics.

Covered are Hilbert space, vector, and operator representations of pure and mixed states, measurable physical quantities (observables), Gleason's theorem, Lueders’ Rule, unitary operators, and Schroedinger's Equation, symmetries of the Hamiltonian and the corresponding conservation laws, and superselection rules.

This chapter highlights other surprising aspects of QM_∞: unlike theories of ordinary QM, theories of QM_∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM_∞.

Quantum error correction is a very advanced field, and researchers have invested a great deal of time in trying to find the best ways of combating errors in quantum computers, winch occur when the computer is not properly isolated from its environment. Theoretically, quantum computation can be stabilised against any kind of influence of the environment. Classical error correction has been generalised to quantum mechanics to show that a reliable quantum computer can be constructed out of unreliable simple elements, or quantum gates. This chapter discusses the basic techniques of quantum error correction and looks at some general conditions which a quantum error correction code has to satisfy, explains how to protect qubits against general errors and achieve fault-tolerant computing, compares quantum error correction with Maxwell's demon, and performs error correction for pure states and mixed states.

The Mach–Zehnder interferometer experiment described earlier shows why and how quantum mechanics is different from classical mechanics. A photon sent through a beam splitter behaves like a particle when it is observed by only one of the two detectors. When two beam splitters are used, the photon “interferes with itself” and behaves like a wave. This is the so-called wave-particle duality of quantum mechanics which leads to quantum entanglement. This chapter discusses quantum superpositions when two or more particles are present. Understanding and analysing entanglement is one of the most interesting directions in the field of quantum information. First, a historical background of quantum entanglement is given, followed by a discussion on Bell's inequalities, separable states that do not violate Bell's inequalities, pure states that violate Bell's inequalities, mixed states that do not violate Bell's inequalities, and entanglement in second quantisation.

This chapter reviews some of the basic concepts coming from the theory of entanglement, their applications to many-body systems, and some of the new theoretical methods that have come up in this context. Section 4.2 and 4.3 introduce entanglement for pure and mixed states, respectively. Two quantities are highlighted: the entropy of entanglement and the quantum mutual information. The first one measures the bipartite entanglement for pure states, whereas the second one measures correlations for both pure and mixed states. Section 4.4 concentrates on many-body states in thermal equilibrium that interact with short-range interactions in lattices. It shows that there is a common property of thermal states in lattice systems with short-range interactions, namely the area law. This fact can guide us in finding an efficient language to describe many-body quantum systems in which the number of parameters only scales polynomially with the number of sites. Under certain conditions, this is indeed possible in terms of so-called tensor network states. Section 4.5 considers how the area law leads to such descriptions and briefly reviews some of them while Section 4.6 concludes.

This chapter reviews all the rules of quantum mechanics and their mathematical notation. It also considers quantum entanglement, a fundamental resource in quantum information. There are four basic postulates of quantum mechanics, that indicate how to represent physical systems, how to represent observations, how to carry out measurements, and how systems evolve when “not measured”. Before describing the laws of quantum mechanics, their mathematical background is first described, along with a Mach Zehnder interferometer experiment in which the (strange) readings on the detectors can be explained by quantum mechanics. This chapter also discusses the Dirac notation, qubits, Hilbert spaces, projective measurements and operations, unitary operations, eigenvectors and eigenvalues, spectral decomposition, applications of the spectral theorem, Dirac notation shorthands, and mixed states.

From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.

This chapter discusses the dual nature of small particles, focusing on electrons. It begins by describing how the classical laws of physics, which were mostly established by the start of the twentieth century, did not work very well when applied on the atomic scale. In order to explain the atomic-scale phenomena, a new kind of physics was invented in the 1920s. This new physics was called quantum physics, or quantum mechanics. It applied to the world on the scale of the atom—the microscopic world, or microworld. The discussions then turn to an experiment involving ‘quantum’ coins; the difference between superposition state and mixed state; light, waves, and interference; and interference with electrons.

This chapter describes how the phenomenological understanding of superconductors was vastly enhanced by the Ginzburg–Landau theory. Its success rested on the fact that it could quantitatively yield the two characteristic lengths λ and ξ and address the question of boundary energy between superconducting and normal phases in terms of the Ginzburg–Landau parameter κ. for κ <1/√2, the sign of the boundary is positive and these are type I superconductors (including pure elements) where superconductivity is abruptly lost at H_c. When κ ≥1/√2, the boundary energy is negative and such materials (including impure metals and alloys) are type II superconductors. Field penetration now begins at a lower critical field H_c1 (<H_c) and continues until an upper critical field H_c2 (>H_c), where the bulk of the superconductivity is lost. Between the two fields, the sample is in a mixed state. Basic features of type II superconductors are discussed.

In this chapter, it is noted that a large upper critical field H_c2 is a basic requirement of a type II superconductor for high-field applications, but that alone does not confer upon the superconductor the ability to carry a high electrical current density without resistance in a large magnetic field. In the mixed state, the flow of electrical current in the presence of a magnetic field results in a driving force that causes the vortex lines to move, and moving vortices give rise to dissipation. The presence of various kinds of crystal defects and inhomogeneities is mandatory for high J_c. Various interesting approaches followed to stabilise conductors against the adverse effects of flux instabilities are described. The relevant basic aspects of vortex motion and its hindrance by pinning centres are discussed in terms of the concepts of critical state, flux creep, flux flow, and pinning interactions.

If a quantum state is prescriptive then what state should an agent assign, what expectations does this justify, and what are the grounds for those expectations? I address these questions and introduce a third important idea—decoherence. A subsystem of a system assigned an entangled state may be assigned a mixed state represented by a density operator. Quantum state assignment is an objective matter, but the correct assignment must be relativized to the physical situation of an actual or hypothetical agent for whom its prescription offers good advice, since differently situated agents have access to different information. However this situation is described, it is true, empirically significant magnitude claims that make the description correct, while others provide the objective grounds for the agent’s expectations. Quantum models of environmental decoherence certify the empirical significance of these magnitude claims while also licensing application of the Born rule to others without mentioning measurement.

For many aspects of device design, an exact solution to Schrödinger’s equation is not needed. However, it may simultaneously be required that all of the physical features are clearly understood. The most important technique for approaching these problems is perturbation theory, since it is difficult to develop physical intuition by just numerical means. For the case of solutions to the time independent Schrödinger equation, such as where an electric or magnetic field is applied, time independent perturbation theory is very useful, and is typically adequate for many problems. In some cases, problems may need an exact solution, but it may not be necessary to consider all the levels, leading to the approximation of using just a few levels. If the Hamiltonian is time dependent, we use time dependent perturbation theory which leads to Fermi’s golden rule. The result leads to a Dirac delta-function which can be eliminated by using the density of states.