Quantum Communications

The purpose of the chapter is to introduce Hilbert spaces, and more precisely the Hilbert spaces on the field of complex numbers, which represent the abstract environment in which Quantum Mechanics is developed. To arrive at Hilbert spaces, we proceed gradually, beginning with vector spaces, then considering inner-product vector spaces, and finally Hilbert spaces. The Dirac notation will be introduced and used. A particular emphasis is given to Hermitian and unitary operators and to the class of projectors. The eigendecomposition of such operators is seen in great detail. The final part deals with the tensor product of Hilbert spaces, which is the mathematical environment of composite quantum systems.

Quantum Mechanics will be formulated assuming four postulates: Postulate 1: gives the universal model of any physical system: a Hilbert space on the field of complex numbers. Postulate 2: models the temporal evolution of a closed physical system that is not influenced by other physical systems. Postulate 3: regards the information that can be extracted (through a quantum measurement) from a quantum system at a given time instant. Postulate 4: formalizes the interaction among physical systems through a combination of multiple Hilbert spaces into a single Hilbert space. The mathematical tool of this formulation is given by the tensor product. Considering that our final objective is Quantum Communications, some aspects of Quantum Mechanics will not be further expanded, such as the consequences of Postulate 2 on the evolution of a quantum system, while other points, in particular quantum measurements (Postulate 3) will be exhaustively investigated.

In this chapter the foundations of Telecommunications Systems are outlined and the differences between Classical and Quantum Communications are explained. Also, the leading concepts of the so-called digital revolution are outlined to justify the choice of dealing only with digital communications in this book. The second part of the chapter deals with the foundations of optical classical communications, which is the necessary prologue to optical quantum communications developed in the subsequent chapters. The mathematical framework for optical classical communications is given by Poisson processes, which are introduced in the final part and applied to classical photodetection.

Decision is the heart of any quantum digital communication system and is concerned with the formulation of how a measurement must be taken to argue on the transmitted message in the presence of uncertainties (due to the nature of any quantum measurement). The state of the quantum system is considered as assigned through a constellation of pure states or of density operators, whereas the measurement operators must be found with the goal of achieving the “best decision” (optimization), usually obtained by minimizing the error probability. The chapter, after a mathematical formulation of decision, moves on to optimization, where the guidelines are given by Holevo’s and Kennedy’s theorems. These theorems determine the conditions that must be fulfilled by an optimal system of measurement operators, but do not provide any clue on how to identify it. In any case, the problem of optimization is very difficult, and exact solutions (nonnumerical) are only known in few cases, mainly in the binary systems and in general when the state constellation has a symmetry. The specific symmetry that simplifies detection and optimization is called geometrically uniform symmetry (GUS).

The problem of quantum optimization is very difficult, and exact solutions are only known in few cases. To overcome the difficulty, suboptimal solutions are considered, the most important of which are the square-root measurements (SRM). This technique can be formulated both for pure and mixed states, and leads to a “pretty good” estimation of the performance of quantum communications systems. The chapter contains advanced (and also original) topics, concerning the SRM in general and the SRM in the presence of the geometrically uniform symmetry (GUS).

In this chapter the quantum decision theory, developed in the previous chapters, is applied to quantum communications systems by specifying the physical nature of the quantum states that carry the information. In practice, the states of a quantum communications system refer always to optical frequencies because at radio frequencies quantum phenomena are not appreciable. Specifically, quantum states represent coherent monochromatic radiations emitted by a laser and are called coherent states. Therefore the theory of coherent states, which is presented according to the formulation developed by Glauber in the 1980s of the last century, will play a fundamental role. Also squeezed states which represent an efficient form of optical radiation are considered. In the second part of the chapter, the most popular quantum communication systems are developed, starting from the simplest ones: The binary systems whose format may be the OOK (on–off keying) modulation and the 2-PSK (phase-shift keying) modulation. Then multilevel systems are developed with the formats: QAM (quadrature amplitude modulation), multilevel PSK modulation, and PPM (pulse position modulation). All these quantum systems will be compared with the classical counterparts, and the superiority in performance of quantum systems with respect to classical systems will be clearly stated.

Quantum Communications systems for the transmission of classical information are implemented at optical frequencies. The chapter begins by describing the main optical components, such as lasers, bemsplitters, optical modulators, and photodetectors. The implementation of transmitters and of the physical media (free space and fiber optics) do not differ substantially from their classical version. The difference is concentrated in the quantum receivers, where the extraction of information is based on quantum measurements. The chapter deals mainly with the implementation of quantum binary receivers, which have a long story, starting from Kennedy’s receiver proposed in 1973 and going on with its improvements as Dolinar’s receiver and Sasaki-Hirota’s receivers. Here the target is to reach Helstrom’s bound. The chapter concludes with some recent ideas on the implementation of quantum multilevel receivers.

The theory of Quantum Information deals with the information processing with quantum states. What is interesting is that in several cases the quantum information processing can have a great advantage with respect to classical information processing and its features often find no correspondence in the classical counterparts. The main examples of quantum information processing are the quantum computer, quantum communications, quantum key distribution (QKD), and quantum teleportation. Quantum Information exhibits two forms, discrete, as the qubit, and continuous, as coherent and more generally Gaussian states. An important remark is that most of the operations in quantum information processing can be carried out both with discrete and with continuous variables (this last possibility is a quite recent discovery). The comparison of these two possibilities should be made upon practical considerations. This chapter gives an introduction to Quantum Information, which will be developed in the last three chapters. Some advanced fundamentals, not sufficiently developed before, as entanglement, partial trace, purification, will be introduced in this chapter.

The fundamentals introduced in Chap. 3, based on the four postulates of Quantum Mechanics, are concerned with discrete variables, with operators and quantum measurements specified by enumerable sets. But Quantum Information makes use of both discrete and continuous variables. The extension of fundamentals to continuous variables is done starting from the quantum harmonic oscillator, which is the basis of the theory of the electromagnetic field, in which the electromagnetic radiation is represented as a combination of harmonic oscillators. In this context, position and momentum (canonical variables), and annihilator and the creation operator (bosonic variables) are the fundamental operators for the development of the theory of continuous variables. The environment is an infinite dimensional Hilbert space, but an alternative environment is given by the phase space, a two-dimensional real space where quantum states of infinite dimensions are simply represented (in the single mode) by two functions of two real variables, the Wigner and the characteristic functions. These functions allow for the introduction of Gaussian states and Gaussian transformations. In addition to offering an easy description in terms of Gaussian functions, Gaussian states and transformations are of great practical relevance and represent the main tool of Quantum Information processing, with applications to quantum computation, quantum cryptography, and quantum communications. Coherent states are notable examples of Gaussian states, but in this chapter several other examples of Gaussian states are also considered. The most interesting applications, in particular the ones based on the entanglement, are concerned with continuous variables in the multimode, where the Hilbert space is given by \(N\) replicas (tensor product) of the space of the single mode and the phase space becomes \(2N\)-dimensional. The extension to multimode continuous variables represents the main difficulty of the chapter.

This chapter presents some applications of Quantum Information Theory that deviate from the problem of reliably transmitting classical information. In fact, the inherent randomness in quantum measurements lends itself to devising methods for the fast automatic generation of true random numbers with quantum devices. Similarly, the possibility of detecting the presence of a measurement operation on a single quantum system, from another, nonorthogonal measurement on the same system, has opened the way to quantum cryptography. This constitutes an unconditionally secure replacement for the schemes that currently lie at the core of many protocols for securing the transmission and storing of information from a rational attacker. Eventually, we devote a paragraph to the topic of quantum teleportation, that is, the transfer of an unknown quantum state between two different locations that is achieved by making use of entanglement and only transmitting classical information.