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This chapter addresses quantum information transmission and processing. Quantum noise is dominant in long-haul transmission lines, even for strong signals. Amplifier noise is avoided by using a noise-free amplifier. Injecting a strongly squeezed state into an unused port of a coupler reduces branching noise. The next section explains the no-cloning theorem and the theory and experimental evidence for (imperfect) quantum cloning machines. The use of single photons for secure quantum key distribution in cryptography is then discussed. Entanglement as a quantum resource first appears in the explanation of quantum dense coding and the inverse process of quantum teleportation. The chapter ends with a brief discussion of quantum computing, including quantum parallelism, quantum logic gates, and quantum circuits. A survey of experiments in quantum computing is followed by a study of proposals for combining linear optics with local measurements to construct quantum computers.

This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.

The brain is exposed to continuous sensory input from the environment (and the body). How does the brain encode such continuous sensory input and translate it into neural activity, e.g., stimulus-induced activity? Results from cellular recordings show that single neurons and a population of neurons represent the stimulus in a rather sparse way so that many stimuli are represented in one neuron’s (or one population of neurons’) activity. This amounts to a many-to-one relationship between stimuli and neurons entailing sparse coding. As such, sparse coding must be distinguished from other coding strategies like dense and local coding that propose a one-to-many and one-to-one relationship between stimuli and neurons. How is such sparse coding possible? The neurons’ (and population of neurons’) activity seems to encode the statistical frequency distribution of stimuli across their different discrete points in physical time and space; that is, their natural statistics. However, this is possible only when presupposing that differences between the stimuli’s different discrete points in physical space and time are encoded into neural activity. In other words, spatial and temporal differences (between the different discrete points in physical time and space) must be encoded into neural activity in order for sparse coding as a many-to-one relationship between stimuli and neurons to be possible.

This chapter discusses some of the ways in which quantum entanglement can be used for quantum communication. Suppose Alice and Bob share an entangled state: they can use it for dense coding, in which Alice sends Bob just one qubit of an entangled pair, which delivers two bits of classical information. Alice and Bob can also use an entangled state for teleportation, in which a qubit in a unknown quantum state is teleported from Alice to Bob when Alice sends Bob two classical bits. Entanglement swapping is an example of a tripartite communication protocol. If Alice and Bob share entanglement, and Bob and Charlie share entanglement, entanglement swapping can be used to entangle Alice and Charlie. Even if Alice and Bob are entangled, Alice cannot communicate with Bob without sending him classical bits or qubits, which prevents Alice and Bob from breaking the laws of relativity. This chapter also discusses Pauli matrices and the extension of Hilbert spaces.

We have seen, in Section 2.5, how the superposition principle leads to the existence of entangled states of two or more quantum systems. Such states are characterized by the existence of correlations between the systems, the form of which cannot be satisfactorily accounted for by any classical theory. These have played a central role in the development of quantum theory since early in its development, starting with the famous paradox or dilemma of Einstein, Podolsky, and Rosen (EPR). No less disturbing than the EPR dilemma is the problem of Schrödinger’s cat, an example of the apparent absurdity of following entanglement into the macroscopic world. It was Schrödinger who gave us the name entanglement; he emphasized its fundamental significance when he wrote, ‘I would call this not one but the characteristic trait of quantum mechanics, the one that enforces the entire departure from classical thought’. The EPR dilemma represents a profound challenge to classical reasoning in that it seems to present a conflict between the ideas of the reality of physical properties and the locality imposed by the finite velocity of light. This challenge and the developments that followed have served to refine the concept of entanglement and will be described in the first section of this chapter. We start by recalling that a state of two quantum systems is entangled if its density operator cannot be written as a product of density operators for the two systems, or as a probability-weighted sum of such products. For pure states, the condition for entanglement can be stated more simply: a pure state of two quantum systems is not entangled only if the state vector can be written as a product of state vectors for the two systems. In the discipline of quantum information, entanglement is viewed as a resource to be exploited. We shall find, both here and in the subsequent chapters, that our subject owes much of its distinctive flavour to the utilization of entanglement.

A remarkable application of quantum mechanical concepts of coherent superposition and quantum entanglement is a quantum computer which can solve certain problems at speeds unbelievably faster than the conventional computer. In this chapter, the basic principles and the conditions for the implementation of the quantum computer are introduced and the limitations imposed by the probabilistic nature of quantum mechanics and the inevitable decoherence phenomenon are discussed. Next the basic building blocks, the quantum logic gates, are introduced. These include the Hadamard, the CNOT, and the quantum phase gates. After these preliminaries, the implementation of the Deutsch algorithm, quantum teleportation, and quantum dense coding in terms of the quantum logic gates are discussed. It is also shown how the Bell states can be produced and measured using a sequence of quantum logic gates.