Signals and Communication Technology

Gianfranco Cariolaro

Quantum
Communications


Signals and Communication Technology

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Gianfranco Cariolaro

Quantum Communications

123
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Gianfranco Cariolaro
Department of Information Engineering
University of Padova
Padova
Italy

ISSN 1860-4862
ISSN 1860-4870 (electronic)
Signals and Communication Technology
ISBN 978-3-319-15599-9
ISBN 978-3-319-15600-2 (eBook)
DOI 10.1007/978-3-319-15600-2
Library of Congress Control Number: 2015933147
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
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To David, Shu-Ning, Gabriele, and Elena


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Foreword

The birth of the original idea of Quantum Communications might be dated back to
the same age when Claude E. Shannon formulated the mathematical theory of
communications in 1948. In 1950, Dennis Gabor wrote a seminal paper on how to
revise Information Theory by considering Quantum Physics, introducing the term
“quantum noise”. Actually, as the carrier frequency goes up to few tens of a
terahertz, quantum noise rapidly becomes more dominant than thermal noise. In
1960, Theodore H. Maiman succeeded in producing the first beam of laser light,
whose frequency was at a few hundred terahertz. For a long time, it was a crucial
trigger for full-scale studies on Quantum Communications. It was not, however, a
straightforward task at all for researchers to establish the unification of the paradoxical aspects of Quantum Mechanics with the landmarks of Communications
Theory. It was only recently that the core of Quantum Communications, that is, the
theory of capacity for a lossy quantum-limited optical channel, was established.
Until now, many new ideas and schemes have been added to the original standard
scheme of Quantum Communications, represented by quantum key distribution,
quantum teleportation, and so on. Realizing a new paradigm of Quantum
Communications is now an endeavor in science and technology, because it requires
a grand sum of not only the latest Quantum Communications technologies but also
the basics of Information Theory and Signal Detection and Processing technologies.
Therefore, it is not easy for students and researchers to learn all the necessary
knowledge, to acquire techniques to design and implement the system, and to
operate it in practice. These tasks usually take a long time through a variety of
courses, and by reading many papers and several books.
This book is meant to achieve this very purpose. The author, Professor Gianfranco
Cariolaro, has been working for a long time in the fields of Communications and
Image Processing technologies, Deep Space Communications, and Quantum

Communications. From this book, readers can track a history of Quantum
Communications and learn its core concepts and very practical techniques. For this
decade, commercial applications of quantum key distribution have been taking place,
and in 2013, lunar laser communication was successfully demonstrated by the
National Aeronautics and Space Administration, where a novel photon counting
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Foreword

method was employed. This means that an era of Quantum Communications in
practice is around the corner. I am very excited to have this book at such a time.
Through this book, readers will also be able to see a future Communications Technology on the shoulder of a long history of Quantum Communications.
Tokyo, Japan, September 2014

Masahide Sasaki
Director of Quantum ICT Laboratory
National Institute of Information
and Communications Technology

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Preface

Quantum Mechanics represents one of the most successful theories in the history of

science. Born more than a hundred years ago, for several decades Quantum
Mechanics was confined to a revolutionary interpretation of Physics and related
fields, like Astronomy. Only in the last decades, after the discovery of laser with the
possibility of producing coherent light, did Quantum Mechanics receive a strong
interest in the area of information, with very innovative and promising applications
(Quantum Computer, Quantum Cryptography, and Quantum Communications).
In particular, the original ideas of Quantum Communications were developed by
Helstrom [10] and by scientists from MIT [11, 13] proving the superiority of
quantum systems with respect to classic optical systems. However, the research in
this specific field did not obtain the same spectacular expansion as the other fields
of quantum information. In our personal opinion, the reason is twofold. One is the
difficulty in the implementation of quantum receivers, which involves sophisticated
optical operations. The other reason, perhaps the most relevant, was due to the
advent of optical fibers, whose tremendous capacity annihilated the effort on the
improvement of performances of the other transmission systems. This may explain
the concentration of interest in the other fields of quantum information. Nevertheless, Quantum Communications deserve a more adequate attention for us to be
prepared for the future developments, being confident that a strong progress in
quantum optics will be surely achieved.
There is another motivation for considering Quantum Communications, especially for educational purposes in Information Engineering. In fact, continuing with
our personal viewpoint, Quantum Mechanics is a discipline that cannot be ignored
in the future curriculum of information engineers (electronics, computer science,
telecommunications, and automatic control). On the other hand, Quantum
Mechanics is a difficult discipline for its mathematical and also philosophical
impact, and cannot be introduced at the level of Physics and Mathematical Physics
because the study burden in information engineering is already quite heavy.
However, we realized (with some surprise) that the notions of Quantum Mechanics

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Preface

needed for Quantum Communications may be easily tackled by information
engineering students. In fact, the notions needed at this level (vector spaces and
probability theory) are already known to these students and require only an ad hoc
recall. Following these ideas, six years ago the author introduced a course on
Quantum Communications in the last year of the Telecommunications degree
(master level) at the Faculty of Engineering of the University of Padova, and, as
confirmed by students and colleagues, the conclusion was that the teaching
experiment has proved very successful.
At the same time, experience shows that the majority of students, who join
quantum optics and quantum information community after taking courses in
quantum mechanics with concentration on elementary particles and high-energy
physics, have very little feeling for the real notion of information transfer and
manipulation as it is known in practical telecommunications. The comprehensive
consideration of Quantum Communication concepts presented in this book serves
to establish this missing conceptual link between the formal Quantum Mechanics
theory formulated originally for particles and the quantum optical information
manipulation utilizing quantum mechanics along with optics and telecommunications tools.
It is difficult to predict in what direction quantum information will evolve or
when the quantum computer will arrive, but it will surely have a strong impact in
the future. Students and researchers that will have learned Quantum Communications, having acquired the methodology and language, will be open to any other
application in the field of Quantum Information.
Organization of the Book
The book is organized into three parts and 13 chapters.
Chapter 1 (Introduction) essentially describes the evolution of Quantum

Mechanics in the previous century, with special emphasis on the last part of the
evolution in the area of Quantum Information, with its promising and exciting
applications.
Part I: Fundamentals
Chapter 2 collects the mathematical background needed in the formulation and
development of Quantum Mechanics: mainly notions of linear vector spaces and
Hilbert spaces, with special emphasis on the eigendecomposition of linear
operators.
Chapter 3 introduces the fundamentals of Quantum Mechanics, in four postulates. Postulate 1 is concerned with the environment of Quantum Mechanics: a
Hilbert space. Postulate 2 formulates the evolution of a quantum system, according
to Schrödinger’s and Heisenberg’s visions. Postulate 3 is concerned with the
quantum measurements, which prescribes the possibility of extracting information
from a quantum system. Finally, Postulate 4 deals with the combination of two or
more interacting quantum systems. A particular emphasis is given to Postulate 3,

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Preface

xi

because it manages the information in a quantum system and will be the basis of
Quantum Communications and Quantum Information consideration.
Part II: Quantum Communications Systems
Chapter 4 deals with the general foundations of telecommunications systems and
the difference between Classical and Quantum Communications systems. In the
second part of the chapter the foundations of optical classical communications,
which is the necessary prologue to optical quantum communications, are
developed.

Chapter 5 develops the concept of optimal quantum decision, which establishes
the best criterion to perform the measurements of Postulate 3 in a quantum system
to extract information. Here a nontrivial effort is made to express the results within
the language of telecommunications, where the quantum decision is applied to the
receiver.
Chapter 6 develops suboptimization in quantum decision. Since optimization is
very difficult, and exact solutions are only known in few cases, suboptimization
techniques are considered, the most important of which is called square-root
measurements (SRM).
Chapter 7 deals with the general formulation of quantum communication systems, where the transmitter (Alice) prepares and launches the information in a
quantum channel and the receiver (Bob) extracts the information by applying the
quantum decision rules. Although, in principle, the transmission of analog information would be possible, according to the lines of present-day technology, only
digital information (data) is considered. In any case, we will refer to optical
communications, in which the information is conveyed through a coherent radiation
produced by a laser. The quantum formulation of coherent radiation is expressed
according to the universal and celebrated Glauber’s theory.
In the second part of the chapter, these basic ideas are applied to most popular
quantum communication systems, each one characterized by a specific modulation
format (OOK, PPM, PSK, and QAM). The performance of each specific system is
compared to that of the corresponding classical optical system, where the decision
is based on a simple photon counting. The comparisons will clearly state the
superiority of the quantum systems.
Chapter 8 reconsiders the analysis of Chap. 7 with the introduction of thermal
noise, in order to get a more realistic evaluation of the performance. Technically
speaking, the analysis in the absence of thermal noise is carried out using the
description of the system status made in terms of pure states, whereas the presence
of thermal noise requires a description in terms of density operators. Consequently,
the analysis becomes much more complicated (but challenging).
Chapter 9 deals with the implementation of coherent quantum communication
systems. The few implementations available in the literature and the difficulties

encountered in the realization are described. Also, some original ideas for an
improved implementation of quantum communication systems are described.

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Preface

Part III: Quantum Information
Chapter 10 begins by dealing with Quantum Information, which exhibits two
forms, discrete and continuous. Discrete quantum information is based on discrete
variables, the best known example of which is the quantum bit or qubit. Continuous
quantum information is based on continuous variables, the best known example of
which is provided by the quantized harmonic oscillator. An important remark is that
most of the operations in quantum information processing can be carried out both
with discrete and continuous variables (this last possibility is a quite recent
discovery).
Chapter 11, Quantum Mechanics fundamentals of Chap. 3 are confined to the
basic notions (relatively few) necessary to the development of Quantum Communications systems in Part II. In this chapter, for a full development of Quantum
Information, the above fundamentals are extended to continuous quantum variables,
to include Gaussian states and Gaussian transformations.
Chapter 12 deals with Information Theory, starting from Classical Shannon’s
Information Theory and then extending the concepts to Quantum Information
Theory. The latter is a relatively new discipline, which is based on quantum
mechanical principles and in particular on its intriguing resources, such as
entanglement.
Chapter 13 deals with the applications of Quantum Information, as quantum
random number generation, quantum key distribution, and teleportation. These

applications are developed with both discrete and continuous variables.
Suggested Paths
For the choice of the path one should bear in mind that the book is a combination
of Quantum Mechanics and Telecommunications, and perhaps students and
researchers in the area of Information Engineering have no preliminary knowledge
of Quantum Mechanics, whereas students and researchers in the area of Physics
may have no preliminary knowledge of Telecommunications (for which we recommend reading Chap. 4 on Telecommunications fundamentals).
As said above, the mathematics needed for the comprehension of the book is
confined to Linear Vector Spaces, as developed in Chap. 2. Hilbert spaces are
introduced for completeness, but they are not really used. The other mathematical
requirement is Probability Theory (probability fundamentals and random variables,
sometimes extended to random processes). These preliminaries must be known at a
good, but not too sophisticated level.
The book could be used by both graduate students (meaning people who have no
knowledge of Quantum Mechanics) and researchers (meaning people who have a
good knowledge of Quantum Mechanics, but not of classical Telecommunications)
following two different paths.
In the Introduction we will indicate in detail two different paths for “students”
and for “researchers”.

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xiii

Manuscript Preparation
To prepare the manuscript we used LATEX, supplemented with a personal
library of macros. The illustrations too are composed with LATEX, sometimes with

the help of Mathematica©.
Padova, October 2014

Gianfranco Cariolaro

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Acknowledgments

As an expert in traditional Telecommunications, in the twilight of my life, I decided
to tread on the unknown territory of Quantum Information. This required the help of
many people, without whom this book could never have been written. First of all, I
would like to mention Tommaso Occhipinti and Federica Fongher, students, with
whom I outlined the main ideas of the project.
I owe special thanks to Gianfranco Pierobon, who made fundamental contributions to many subjects in the book. In addition to sharing the same name,
Gianfranco and I shared a similar enthusiastic wonder toward Quantum Mechanics,
as we were both newcomers to the discipline. I still remember the trepidation and
scepticism with which we submitted our first paper to an international journal (on
the performance of Quantum Communications systems based on square root
measurements). But it turned out to be a success. Incidentally, I would like to
mention that the topic and the methodology of that paper were inspired by the work
of Professor Masahide Sasaki and his collaborators. Gianfranco’s help was so
valuable that I repeatedly offered him to co-author the book, but he always refused,
and, knowing his stubbornness, I had to give up. However, I hope he will accept
next time, with the next book.
Roberto Corvaja was very helpful by re-reading the manuscript, over and over
again, and integrating it with essential numeric computations. Nicola Laurenti
provided invaluable assistance in the development of Part III of the book, concerning the applications of Quantum Information, in particular, by helping me to
find my way in the jungle of this rapidly growing subject, as well as by proposing

alternative arguments. Tomaso Erseghe was kind enough to learn Quantum
Mechanics for the sole purpose of helping me, and he did so with great competence.
Several other dedicated readers offered me numerous, detailed, and insightful
suggestions: Antonio Assalini, Luigi Bellato, Cesare Barbieri, Gianpaolo Naletto,
Ezio Obetti, Stefano Olivares, Silvano Pupolin, Edi Ruffa, Lorenzo Sartoratti,
Giovanna Sturaro, Francesco Ticozzi, Paolo Villoresi, and the young students:
Nicola Dalla Pozza, Alberto Dall’Arche, Davide Marangon, and Giuseppe Vallone.

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xvi

Acknowledgments

I am particularly indebted to Nino Trainito, perhaps the only one to actually read
the whole manuscript!, who made several comments and considerably improved the
language.
I would like to mention that I was forced to interrupt my work for several months
due to an unfortunate, tragic event that affected my family. On that occasion, three
people in particular were crucial in helping me to cope with the situation and to
return to normal life, namely Consul Vincenzo De Luca, Prof. Renato Scienza, and
my friend Stefano Gastaldello. Other friends were very close to me in this difficult
period, namely, Cesare Barbieri, Peter Kraniauskas, Umberto Mengali, Marina
Munari, Silvano Pupolin, Romano Valussi, and Guido Vannucchi. I take the liberty
to mention all this, an unusual subject for an acknowledgments section, because
without the support of all these friends the book would have never been finished.
To all these people I owe a great debt of gratitude and offer heartfelt thanks.

Padova, October 2014

Gianfranco Cariolaro

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
A Brief History of Quantum Mechanics . . . . . .
1.2
Revolutionary Concepts of Quantum Mechanics
1.3
Quantum Information. . . . . . . . . . . . . . . . . . .
1.4
Content of the Book . . . . . . . . . . . . . . . . . . .
1.5
Suggested Paths . . . . . . . . . . . . . . . . . . . . . .
1.6
Conventions on Notation . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

1
5
7
10
13
14
16

Vector and Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Inner-Product Vector Spaces . . . . . . . . . . . . . . . . . .
2.4
Definition of Hilbert Space . . . . . . . . . . . . . . . . . . .
2.5
Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . .
2.7
Outer Product. Elementary Operators . . . . . . . . . . . .
2.8
Hermitian and Unitary Operators . . . . . . . . . . . . . . .
2.9
Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Spectral Decomposition Theorem (EID) . . . . . . . . . .
2.11 The Eigendecomposition (EID) as Diagonalization . .
2.12 Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . .

2.13 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Other Fundamentals Developed Throughout the Book
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I
2

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Fundamentals

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3

Contents

Elements of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
The Environment of Quantum Mechanics. . . . . . . . . . . . .
3.3
On the Statistical Description of a Closed Quantum System .
3.4
Dynamical Evolution of a Quantum System . . . . . . . . . . .
3.5
Quantum Measurements . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Measurements with Observables . . . . . . . . . . . . . . . . . . .
3.7

Generalized Quantum Measurements (POVM) . . . . . . . . .
3.8
Summary of Quantum Measurements. . . . . . . . . . . . . . . .
3.9
Combined Measurements . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Composite Quantum Systems . . . . . . . . . . . . . . . . . . . .
3.11 Nonunicity of the Density Operator Decomposition + . . . .
3.12 Revisiting the Qubit and Its Description . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II

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Quantum Communications

4


Introduction to Part II: Quantum Communications . . . . . . . . .
4.1
A General Scheme of a Telecommunications System . . . . .
4.2
Essential Performances of a Communication System . . . . .
4.3
Classical and Quantum Communications Systems . . . . . . .
4.4
Scenarios of Classical Optical Communications. . . . . . . . .
4.5
Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Filtered Poisson Processes . . . . . . . . . . . . . . . . . . . . . . .
4.7
Optical Detection: Semiclassical Model . . . . . . . . . . . . . .
4.8
Simplified Theory of Photon Counting and Implementation .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5

Quantum Decision Theory: Analysis and Optimization . . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Analysis of a Quantum Communications System. . . . . . .

5.3
Analysis and Optimization of Quantum Binary Systems . .
5.4
Binary Optimization with Pure States. . . . . . . . . . . . . . .
5.5
System Specification in Quantum Decision Theory . . . . .
5.6
State and Measurement Matrices with Pure States . . . . . .
5.7
State and Measurement Matrices with Mixed States + . . .
5.8
Formulation of Optimal Quantum Decision. . . . . . . . . . .
5.9
Holevo’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Numerical Methods for the Search for Optimal Operators.
5.11 Kennedy’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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5.12 The Geometry of a Constellation of States . . . . . . . .
5.13 The Geometrically Uniform Symmetry (GUS). . . . . .
5.14 Optimization with Geometrically Uniform Symmetry .
5.15 State Compression in Quantum Detection . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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221
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6

Quantum Decision Theory: Suboptimization . . . . . . .
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Square Root Measurements (SRM) . . . . . . . . . .
6.3
Performance Evaluation with the SRM Decision .
6.4
SRM with Mixed States . . . . . . . . . . . . . . . . .
6.5
SRM with Geometrically Uniform States (GUS) .
6.6
SRM with Mixed States Having the GUS. . . . . .
6.7
Quantum Compression with SRM . . . . . . . . . . .
6.8
Quantum Chernoff Bound . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

Quantum Communications Systems . . . . . . . . . . . . . .
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Overview of Coherent States . . . . . . . . . . . . . .
7.3
Constellations of Coherent States . . . . . . . . . . .
7.4
Parameters in a Constellation of Coherent States .
7.5
Theory of Classical Optical Systems . . . . . . . . .
7.6
Analysis of Classical Optical Binary Systems . . .

7.7
Quantum Decision with Pure States . . . . . . . . . .
7.8
Quantum Binary Communications Systems. . . . .
7.9
Quantum Systems with OOK Modulation. . . . . .
7.10 Quantum Systems with BPSK Modulation . . . . .
7.11 Quantum Systems with QAM Modulation . . . . .
7.12 Quantum Systems with PSK Modulation . . . . . .
7.13 Quantum Systems with PPM Modulation . . . . . .
7.14 Overview of Squeezed States . . . . . . . . . . . . . .
7.15 Quantum Communications with Squeezed States .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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281
281
282
287

292
296
304
314
316
318
320
323
331
337
348
354
358

8

Quantum Communications Systems with Thermal Noise. . . . .
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Representation of Thermal Noise. . . . . . . . . . . . . . . . . .
8.3
Noisy Coherent States as Gaussian States r . . . . . . . . . .
8.4
Discretization of Density Operators . . . . . . . . . . . . . . . .
8.5
Theory of Classical Optical Systems with Thermal Noise .
8.6
Check of Gaussianity in Classical Optical Detection . . . .
8.7

Quantum Communications Systems with Thermal Noise .

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361

361
363
367
369
373
376
381

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xx

Contents

8.8
Binary Systems in the Presence of Thermal Noise . . . . .
8.9
QAM Systems in the Presence of Thermal Noise . . . . .
8.10 PSK Systems in the Presence of Thermal Noise . . . . . .
8.11 PPM Systems in the Presence of Thermal Noise . . . . . .
8.12 PPM Performance Evaluation (Without Compression) . .
8.13 PPM Performance Evaluation Using State Compression .
8.14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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386
391
395
399
404
408
415
420

Implementation of QTLC Systems . . . . . . . . . . . . . . . . .
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Components for Quantum Communications Systems
9.3
Classical Optical Communications Systems . . . . . .
9.4
Binary Quantum Communications Systems. . . . . . .
9.5
Multilevel Quantum Communications Systems . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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421
421
423
431
433
443
446

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451
451
454
457
461
462

11 Fundamentals of Continuous Variables . . . . . . . . . . . . . . . .
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 From Discrete to Continuous in Quantum Mechanics . .
11.3 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . .
11.4 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Abstract Formulation of Continuous Quantum Variables

11.6 Phase Space Representation: Preliminaries . . . . . . . . . .
11.7 Phase Space Representation: Definitions for the N-Mode
11.8 Phase Space Representations in the Single Mode. . . . . .
11.9 Examples of Continuous States in the Single Mode . . . .
11.10 Gaussian Transformations and Gaussian Unitaries . . . . .
11.11 Gaussian Transformations in the N-Mode . . . . . . . . . . .
11.12 N-Mode Gaussian States . . . . . . . . . . . . . . . . . . . . . .
11.13 Normal Ordering of Gaussian Unitaries +. . . . . . . . . . .
11.14 Gaussian Transformations in the Single Mode. . . . . . . .
11.15 Single-Mode Gaussian States and Their Statistics . . . . .
11.16 More on Single-Mode Gaussian States . . . . . . . . . . . . .

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463
464
466
473
479
481

484
491
499
503
508
512
519
522
525
529
535

9

Part III

Quantum Information

10 Introduction to Quantum Information . . . . . . . . .
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
10.2 Partial Trace and Reduced Density Operators
10.3 Overview of Entanglement . . . . . . . . . . . . .
10.4 Purification of Mixed States . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

xxi

11.17 Gaussian States and Transformations in the Two-Mode
11.18 Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.19 Entanglement in Two-Mode Gaussian States. . . . . . . .
11.20 Gaussian States and Geometrically Uniform Symmetry
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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540
546
549
552
571

12 Classical and Quantum Information Theory . . . . . . . .
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Messages of Classical Information. . . . . . . . . . . .
12.3 Measure of Information and Classical Entropy . . .
12.4 Quantum Entropy . . . . . . . . . . . . . . . . . . . . . . .
12.5 Classical Data Compression (Source Coding) . . . .
12.6 Quantum Data Compression . . . . . . . . . . . . . . . .
12.7 Classical Channels and Channel Encoding . . . . . .
12.8 Quantum Channels and Open Systems . . . . . . . . .
12.9 Accessible Information and Holevo Bound . . . . . .
12.10 Transmission Through a Noisy Quantum Channel .

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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573
573
577
580
585
595
600
605
614
620
625
636

13 Applications of Quantum Information . . . . .

13.1 Introduction . . . . . . . . . . . . . . . . . . .
13.2 Quantum Random Number Generation.
13.3 Introduction to Quantum Cryptography
13.4 Quantum Key Distribution (QKD) . . . .
13.5 Teleportation . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . .

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639
639
640
645
646
659
662

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 A Brief History of Quantum Mechanics
A Few Milestones in Quantum Mechanics
1900: Black body radiation law (Max Planck)
1905: Postulation of photons to explain photoelectric effect (Albert Einstein)
1909: Interference experiments (Geoffrey Ingram Taylor)
1913: Quantization of angular momentum of hydrogen (Niels Bohr)
1923: Compton effect (Arthur Holly Compton)
1924: Wave–particle duality extended to incorporate matter (Louis de Broglie)
1925: Matrices as basis for Quantum Mechanics (Werner Heisenberg)
1926: Probabilistic interpretation of the wavefunction (Max Born)
1926: Gilbert Lewis coined the word photon
1926: Wave equation to explain the hydrogen atom (Erwin Schrödinger)
1927: Uncertainty principle (Werner Heisenberg)
1927: Copenhagen interpretation (Niels Bohr)
1928: First solution of Quantum Mechanics explaining spin (Paul Dirac)
1930: Principles of Quantum Mechanics (Paul Dirac)
1930: Interference, how quantized light interacts with atoms (Enrico Fermi)
1932: Mathematical foundations of Quantum Mechanics (John von Neumann)
1935: EPR paradox (Einstein, Podolsky, and Rosen)
1950s: Theory of photon statistic and counting (Hanbury Brown, and Twiss)
1960s: Quantum theory of coherence (Glauber, Wolf, Sudarshan, and others)
1970: (early 1970s) Tunable lasers

© Springer International Publishing Switzerland 2015
G. Cariolaro, Quantum Communications, Signals and Communication Technology,
DOI 10.1007/978-3-319-15600-2_1


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1


2

1 Introduction

1.1.1 The Dawn
In the last decade of the nineteenth century Newton’s mechanics, Maxwell’s electromagnetic theory, and Boltzmann’s statistical mechanics seemed capable of exhaustively explaining any relevant physical phenomenon. However, some phenomena,
initially deemed as marginal, did not completely fit in the structure of these classic
disciplines. It all began with the discoveries of a Physics student called Max Planck
(1858–1947).1 Planck’s research was triggered by the study of the emission and
absorption of light by physical bodies. At that time, the founding theory of radiation
emission by a black body was based on classical electromagnetism. Applying this
theory, the phenomenon was well explained for relatively low frequencies of the
emitted radiation (visible or near infrared and downwards); however, for high frequencies (ultraviolet and upwards) classical theory would predict an infinite increase
in the energy of the emitted radiation, which, as matter of fact, does not happen in
reality. To overcome such a problem, Planck formulated the hypothesis that the radiating energy could only exist in the form of discrete quantities, or “packets”, which
he called quanta. To set the framework of Planck’s problem, we must recall the
previous research of the physicist J.W. Strutt Lord Rayleigh (1842–1919), who
studied the radiation of the black body from a classical point of view, modeling it
as a collection of electromagnetic oscillators, and considering the presence of the
radiation at frequency ν as the consequence of the excitation of the oscillator at such
frequency. With some contribution by Sir James Hopwood Jeans (1877–1946), he
arrived at the formulation of the Rayleigh-Jeans Law, given by the expression
E(ν) =


8π kT
8π kT ν 4
=
,
c4
λ4

(1.1)

which gives the value E(ν) of energy density per frequency unit emitted by a black
body at frequency ν. In (1.1) k = 1.38 10−23 JK−1 is Boltzmann’s constant, T is
the absolute temperature of the black body, c is the speed of light, and λ = c/ν is
the wavelength. This law shows that the energy density irradiated by a black body
increases linearly with temperature and with the fourth power of the frequency of the
emitted radiation. Experimental measurements demonstrate that this law is perfectly
adequate at low frequencies: in fact, it is well known that, with increasing temperature,
the irradiated energy increases proportionally, at least up to the infrared. However,
measurements carried out at higher frequencies, for example in the ultraviolet range,
clearly show that the emitted energy values diverge considerably from those foreseen
by the theory. In addition, from a careful analysis of Eq. (1.1), one can see that
the expected result in this spectral interval has no physical meaning. In fact, this
equation states that, with increasing frequency, energy density increases indefinitely.
As a consequence, the equation asserts that the high-frequency oscillators (very
low wavelength, corresponding to the ultraviolet radiation, to the X-rays, and to
1

On December 14, 1900, Planck publishes his first paper on Quantum Theory in Verh. Deut. Phys.
Ges. 2,237–45.

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1.1 A Brief History of Quantum Mechanics

3

the γ -rays) should be excited even at room temperature. Such absurd result, which
posits the emission of a large amount of energy in the high-frequency region of the
electromagnetic spectrum, went under the name of ultraviolet catastrophe.
The solution of the problem was in fact due to Max Planck, who tackled it in mathematical terms. Instead of integrating the energies of the “elementary oscillators” (that
is, in practice, of the electrons “oscillating” around the nucleus) considering them as
continuous quantities, he performed a summation of the energies, hypothesizing that
they could assume only discrete values, proportional to the characteristic oscillation
frequency ν of the electrons, by an appropriate constant h
E = hν.

(1.2)

The relation discovered by Planck for the energy density per frequency unit of the
black body turns out to be (Planck’s relation)
E(ν) =

hν 3
8π
c3 ehν/kT − 1

and it appears to be in perfect agreement with the experimental distribution for each
temperature, assuming h = 6.63 10−34 Js; h is known as Planck’s constant.
Planck’s theoretical discovery on quanta became accepted by the classical physicists only when Albert Einstein (1879–1955)2 succeeded in explaining the photoelectric effect, speculating that light radiation was constituted by energy packets,
subsequently called “photons”. Einstein showed that, thanks to quanta, other physical phenomena could be explained, in addition to the black body emission proposed

by Planck, and at that point the discrete nature of electromagnetic radiation became
a fundamental and generally accepted concept.
Another problem that could not be explained by classical mechanics was the
regularity of the emission spectrum of an atom, that is, the fact that it always appeared
as formed by the same characteristic frequencies, independently of its origin and of
possible excitation processes it had undergone, a fact that could not be convincingly
explained by the model proposed by Ernest Rutherford (1871–1937) in 1911. The
first one to address the problem in mathematical terms was Niels Bohr (1885–1962)
in 1913. Bohr hypothesized that the lines of an atomic spectrum were originated
by the transition of an electron between two discrete states of an atom. This theory
correctly interpreted, for the first time, the emission and absorption properties of an
atom of hydrogen.
The next step in the development of Quantum Mechanics was due to LouisVictor Pierre Raymond de Broglie3 (1892–1987), who extended to the particles
with mass the wave–particle duality that had been evidenced for electromagnetic
2

In 1905 he published on the Annalen der Physik three articles, the first on light quanta, the second
on Brownian motion, which would definitely confirm the atomicity of matter, the third on the
foundations of restricted relativity.
3 After publishing a few papers, he developed in full form this original idea in his Ph.D. thesis
(1924): Recherches sur la théorie des quanta.

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4

1 Introduction

radiations. Louis de Broglie surmised that not only would light, generally modeled

as a wave, sometimes behave as a particle, but also electrons, usually modeled as
particles, could at times behave as waves. De Broglie suggested that the key for the
description of electrons in terms of wave–particle could be given by the relation
λ=

h
mv

(1.3)

where λ is the wavelength of the wave associated to the electron, and m e v are,
respectively, the mass and the velocity of the electron itself. For example, a wave is
associated to an electron moving along a closed orbit around the atomic nucleus. In
this particular case, the wave is stationary and its wavelength is linked to mass and
velocity by relation (1.3).
We can say that de Broglie’s contribution marks the end of the pioneering phase
of Quantum Mechanics, whose various phenomena were examined and explained
individually, without attempting to formulate a general theory.

1.1.2 The Maturity of Quantum Mechanics
Quantum Mechanics reached maturity in the 1920s and in the 1930s, moving from
Quantum Theory to Quantum Mechanics, thanks to the work of Schrödinger, Heisenberg, Dirac, Pauli, and others.
Shortly after de Broglie’s conjecture, almost simultaneously, Quantum Mechanics was presented by Erwin Schrödinger (1887–1961) and Werner Heisenberg
(1902–1976).4 Among the greatest physicists of the century, Schrödinger, stated the
fundamental equation of Undulatory Mechanics, known nowadays as Schrödinger’s
equation
H ψ = E ψ,
(1.4)
where ψ is an eigenfunction describing the state of the system, H is an operator,
called Hamiltonian, and E is the eigenvalue accounting for the system’s energy.5

This equation, stated for non relativistic energies, is the basis for the description of
the various phenomena of molecular, atomic, and quantum nuclear physics.
Heisenberg, instead, introduced into Physics the uncertainty of physical entities.
His Uncertainty Principle, in fact, asserts that it is impossible to know, simultaneously
and exactly, couples of physical entities, like position and velocity of a particle. In
essence, the more precisely we know the position of a particle, the less information
we have on momentum, and vice versa, according to:
4

In 1927, he published on Zeitschrift fur Physik his famous paper on the uncertainty principle,
entitled: Über den anschaulichen Inhalt der quanten theoretischen Kinematik und Mechanik.
5 Equation (1.4) is Schrödinger’s time-independent equation, where ψ is an eigenfunction.
Schrödinger’s equation can also include the time to take into account system evolution.

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1.1 A Brief History of Quantum Mechanics

ΔxΔp ≥

5

h
.
4π

(1.5)

This principle is of general validity, but it is particularly appreciable at the atomic or

subatomic scale.
The statistical laws related to the concept of probability became a reality: uncertainty is a fundamental fact, and the relations connected to the principle evidence an
insuperable limit to our knowledge of nature.
The more precisely the position is determined, the less precisely the momentum is known
in this instant, and vice versa. (Heisenberg, Uncertainty Paper, 1927)

To conclude this historical note, we find it appropriate to mention the fundamental contribution, albeit indirect, given by the mathematician David Hilbert (1862–
1943), since the modern version of Quantum Mechanics requires a Hilbert space as
mathematical context.

1.2 Revolutionary Concepts of Quantum Mechanics
In describing reality, Quantum Mechanics presents a few concepts that appear revolutionary with respect to Classical Physics, and even seem in contrast with common
sense. These concepts will be briefly summarized below.

1.2.1 Randomness
The fundamental difference between Classical Mechanics and Quantum Mechanics
lies in the fact that, while Classical Mechanics is a deterministic theory, Quantum
Mechanics envisages and formalizes indeterminate aspects of reality.
In the mathematical models of Classical Mechanics, once the initial state of
a system is known, and so are the forces acting on it, the system’s evolution is
perfectly predictable and deterministically measurable. Resort to probabilistic models is then justified exclusively by the need to account for lack of information on
entities characterizing the system.
In Quantum Mechanics, instead, randomness is an intrinsic element of the theory. In fact, it states that the measurements performed on a system, starting from
exactly the same initial conditions, may produce different results. This is not due to
measurement imprecision, but rather to the fact that the result of any measurement
is intrinsically random and must be dealt with the Theory of Probability.
Randomness in Quantum Mechanics is expressed by the fact that the measure
of an entity is described by a complex function (the wave function), whose squared
modulus gives the probability density of the result (intended as a random variable).


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