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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.

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.

No common cause model can fit the phenomena that violate Bell's Inequalities; what sorts of probability models could do so? To answer this, we need to broaden our concept of statistical or probability models, while not broadening it so much as to trivialize it. Introduced here are the distinctions between a surface (phenomenal) model and a theoretical model, and between the general class of geometric probability models and their subclass of quantum theoretical models, together with some elements of quantum logic, and the basic use of probability models to represent measurement situations.

During the 1960s, Bell produced his seminal work on the foundations of quantum theory, first showing that, contrary to von Neumann’s argument, hidden variable theories were allowed but then that they had to be non-local. Thus, he showed that quantum theory did not respect local causality, and he outlined experiments that could be used to test whether the assumption of local realism was true in quantum theory: Bell’s theorem, or Bell’s inequality, or Bell’s inequalities. His work was followed up by John Clauser, Abner Shimony, Michael Horne, and Richard Holt, who produced the CHSH inequality. Bell had moved to CERN, where he worked on the theory of neutrino experiments, and on nuclear and elementary particle physics, making crucial suggestions concerning the use of gauge theory for each type of physical force and also producing the Adler–Bell–Jackiw anomaly, or the ABJ anomaly.

How is the microscopic ontology of quantum mechanics to be understood according to the Everett interpretation? And how, in particular, are we to understand concepts like spatial locality in Everettian quantum mechanics? Building on joint work with Chris Timpson, the chapter develops a general approach to these questions (‘spacetime state realism’) and apply it to questions of locality and entanglement. After a digression on the general metaphysical problem of how we should think about ontology, the chapter compares this approach with others in the literature.

The first round of the Einstein–Bohr debates took place when Einstein challenged Bohr’s principle of complementarity at the Solvay conference in 1927 and Bohr successfully defended it. The most serious challenge, however, came in 1935 when a paper by Einstein, Podolsky, and Rosen (EPR) argued that quantum mechanics was incomplete through a gedanken experiment motivating an approach based on hidden variables. In this chapter, EPR’s arguments about the incompleteness of quantum mechanics and Bohr’s reply to them are presented. The ultimate answer came almost 30 years later, almost ten years after Einstein’s death, and was nothing that Einstein would have expected. Bell’s inequality and the subsequent Bell-CHSH inequality, that are satisfied by all theories based on the “self-evident truths” of reality and locality are discussed. The startling results that quantum mechanics violates Bell’s inequality and the experimental results are in agreement with the prediction of quantum mechanics are presented.

By 1935, the Copenhagen interpretation had become the orthodoxy. Einstein needed to find a situation in which it is possible in principle to acquire knowledge of the state of a quantum system without disturbing it in any way. Working with two young theorists, Boris Podolsky and Nathan Rosen, Einstein devised an extraordinarily cunning challenge based on entangled particles. We can discover the state of one particle with certainty by making measurements on its entangled partner. All we have to assume is that the particles are local: any measurement we make on one in no way affects or disturbs the other. Through the work of David Bohm and John Bell, the challenge posed by EPR became accessible to experiment, and Bell devised a simple test for all locally realistic theories. All the experiments performed to date suggest that the standard quantum formalism is correct: in any realistic interpretation, quantum particles are non-local.

Experiments on Bell’s theorem, or Bell’s inequality, were carried out by John Clauser and Stuart Freeman, whose results agreed with quantum theory, and by Richard Holt and Frank Pipkin, whose results disagreed with quantum theory but agreed with local causality. However, it came to be accepted that quantum theory was right: local causality was not obeyed. The locality loophole, the detector loophole, and the freedom of choice loophole were introduced. Bell encouraged the plans of Alain Aspect, who wanted to perform experiments where information about the directions of the polarizer in one wing of the experiment would not have time to reach the other wing. Bell refined his ideas on quantum theory, introducing the ‘beable’, in contrast to the usual ‘observable’, and commenting on many-worlds theories and the de Broglie–Bohm interpretation. He made important contributions to elementary particle physics through his studies of shadow scattering, and models of quarks.

Bell was a fine physicist. He did important work on accelerators and on nuclear and particle physics, but he will be mostly remembered for his seminal work on the foundations of quantum theory. It can be argued that Bell’s inequalities constitute both the fundamental mystery and the greatest opportunity of quantum theory. In work—taking an established component of physics and bringing in increased understanding and further challenges—he may be compared with Michael Faraday. Bell’s work was based on honesty, and honesty and integrity were also central to his personal life. John Bell’s honesty in his science, and his willingness to work steadily for its progress and to devote his time and efforts to help others to do the same, stand as guiding lights to those coming after him.

Chapter 14 examines entangled quantum systems, hidden variable theories and Bell’s inequality. In 1935, Einstein and collaborators, postulating the existence of elements of reality, analyzed an entangled system and concluded that quantum theory was incomplete. Their analysis is described using spin singlets, which are entangled states of two spin ½ particles. A possible avoidance of their conclusion is by using hidden variable theories. In analyzing a class of local hidden variable theories, John Bell derived an equality that could test them. This was done by experiments using entangled photons; their results violated the inequality, thereby establishing that quantum mechanics, not local hidden variable theories, is the correct description. Later theoretical analysis, and relevant experimental results, strongly supported this. A further theoretical analysis, involving just a single measurement, led to a pronounced conclusion: farewell to the elements of reality. Ditto as well to the spooky-action-at-a-distance problem that had so bothered Einstein.

The main thread of chapter 4 introduces some of the major mysteries and interpretational issues of quantum mechanics (QM). These mysteries and issues include: quantum superposition, quantum nonlocality, Bell’s inequality, entanglement, delayed choice, the measurement problem, and the lack of counterfactual definiteness. All these mysteries and interpretational issues of QM result from dynamical explanation in the mechanical universe and are dispatched using the authors’ adynamical explanation in the block universe, called Relational Blockworld (RBW). A possible link between RBW and quantum information theory is provided. The metaphysical underpinnings of RBW, such as contextual emergence, spatiotemporal ontological contextuality, and adynamical global constraints, are provided in Philosophy of Physics for Chapter 4. That is also where RBW is situated with respect to retrocausal accounts and it is shown that RBW is a realist, psi-epistemic account of QM. All the relevant formalism for this chapter is provided in Foundational Physics for Chapter 4.

Chapter 7 describes the fundamental role of randomness in quantum mechanics, in generating the first biomolecules, and in biological evolution. Experiments testing the Einstein–Podolsky–Rosen paradox have demonstrated, via Bell’s inequalities, that no local hidden variable theory can provide a viable alternative to quantum mechanics, with its fundamental randomness built in. Randomness presumably plays an equally important role in the chemical assembly of a wide array of polymer molecules to be sampled for their ability to store genetic information and self-replicate, fueling the sort of abiogenesis assumed in the RNA world hypothesis of life’s beginnings. Evidence for random mutations in biological evolution, microevolution of both bacteria and antibodies and macroevolution of the species, is briefly reviewed. The importance of natural selection in guiding the adaptation of species to changing environments is emphasized. A speculative role of cosmological natural selection for black-hole fecundity in the evolution of universes is discussed.