Quantum Communication, Quantum Networks, and Quantum Sensing

By Ivan B. Djordjevic

Quantum Communication, Quantum Networks, and Quantum Sensing represents a self-contained introduction to quantum communication, quantum error-correction, quantum networks, and quantum sensing. It starts with basic concepts from classical detection theory, information theory, and channel coding fundamentals before continuing with basic principles of quantum mechanics including state vectors, operators, density operators, measurements, and dynamics of a quantum system. It continues with fundamental principles of quantum information processing, basic quantum gates, no-cloning and theorem on indistinguishability of arbitrary quantum states. The book then focuses on quantum information theory, quantum detection and Gaussian quantum information theories, and quantum key distribution (QKD). The book then covers quantum error correction codes (QECCs) before introducing quantum networks. The book concludes with quantum sensing and quantum radars, quantum machine learning and fault-tolerant quantum error correction concepts.

• Integrates quantum information processing fundamentals, quantum communication, quantum error correction, quantum networks, QKD, quantum sensing, and quantum machine learning
• Provides in-depth exposition on the design of quantum error correction circuits, quantum communications systems, quantum networks, and quantum sensing systems
• Shows how to design the information processing circuits, stabilizer codes, CSS codes, entanglement-assisted quantum error correction codes
• Describes quantum machine learning

• Language: English
• Publisher: Academic Press
• Release date: Jul 17, 2022
• ISBN: 9780128231005

Ivan B. Djordjevic

Ivan B. Djordjevic is a Professor of Electrical and Computer Engineering and Optical Sciences, Director of the Optical Communications Systems Laboratory and the Quantum Communications Laboratory, and co-Director of the Signal Processing and Coding Lab at the University of Arizona. He is a fellow of IEEE and the Optical Society. Prof. Djordjevic has authored or co-authored seven books and more than 530 journal and conference publications. He presently serves as a Senior Editor and member of the Editorial Board on the OSA/IEEE Journal of Optical Communications and Networking; the IOP Journal of Optics; IEEE Communications Letters; the Elsevier Physical Communication Journal, PHYCOM; Optical and Quantum Electronics; and Frequenz. As of August 2020, he holds 53 U.S. patents.

Book preview

Quantum Communication, Quantum Networks, and Quantum Sensing

Ivan B. Djordjevic

Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, United States

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1. Basics of quantum information, quantum communication, quantum sensing, and quantum networking

1.0. Overview

1.1. Photon polarization

1.2. The concept of qubit

1.3. Quantum gates and quantum information processing

1.4. Quantum teleportation

1.5. Quantum error correction concepts

1.6. Quantum sensing

1.7. Quantum key distribution

1.8. Quantum networking

1.9. Organization of the book

Chapter 2. Information theory, error correction, and detection theory

2.1. Classical information theory fundamentals

2.2. Channel coding preliminaries

2.3. Linear block codes

2.4. Cyclic codes

2.5. Bose–Chaudhuri–Hocquenghem codes

2.6. Reed–Solomon, concatenated, and product codes

2.7. Detection and estimation theory fundamentals

2.8. Concluding remarks

Chapter 3. Quantum information processing fundamentals

3.1. Quantum information processing features

3.2. State vectors, operators, projection operators, and density operators

3.3. Measurements, uncertainty relations, and dynamics of quantum systems

3.4. Superposition principle, quantum parallelism, and quantum information processing basics

3.5. No-cloning theorem

3.6. Distinguishing quantum states

3.7. Quantum entanglement

3.8. Operator-sum representation

3.9. Decoherence effects, depolarization, and amplitude damping channel models

3.10. Summary

Chapter 4. Quantum information theory fundamentals

4.1. Introductory remarks

4.2. Von Neumann entropy

4.3. Holevo information, accessible information, and Holevo bound

4.4. Data compression and Schumacher's noiseless quantum coding theorem

4.5. Quantum channels

4.6. Quantum channel coding and Holevo–Schumacher–Westmoreland theorem

4.7. Summary

Chapter 5. Quantum detection and quantum communication

5.1. Density operators (revisited)

5.2. Quantum detection theory fundamentals

5.3. Binary quantum detection

5.4. Coherent states, quadrature operators, and uncertainty relations

5.5. Binary quantum optical communication in the absence of background radiation

5.6. Field coherent states, P-representation, and noise representation

5.7. Binary quantum detection in the presence of noise

5.8. Gaussian states, transformation, and channels, squeezed states, and Gaussian state detection

5.9. Generation of quantum states

5.10. Multilevel quantum optical communication

5.11. Summary

Chapter 6. Quantum key distribution

6.1. Cryptography basics

6.2. Quantum key distribution basics

6.3. No-cloning theorem and distinguishing quantum states

6.4. Discrete variable quantum key distribution protocols

6.5. Quantum key distribution security

6.6. Decoy-state protocols

6.7. Measurement-device-independent quantum key distribution protocols

6.8. Twin-field quantum key distribution protocols

6.9. Information reconciliation and privacy amplification

6.10. Continuous variable quantum key distribution

6.11. Summary

Chapter 7. Quantum error correction fundamentals

7.1. Pauli operators (revisited)

7.2. Quantum error correction concepts

7.3. Quantum error correction

7.4. Important quantum coding bounds

7.5. Quantum operations (superoperators) and quantum channel models

7.6. Summary

Chapter 8. Quantum stabilizer codes and beyond

8.1. Stabilizer codes

8.2. Encoded operators

8.3. Finite geometry representation

8.4. Standard form of stabilizer codes

8.5. Efficient encoding and decoding

8.6. Nonbinary stabilizer codes

8.7. Subsystem codes

8.8. Topological codes

8.9. Surface codes

8.10. Entanglement-assisted quantum codes

8.11. Summary

Chapter 9. Quantum low-density parity-check codes

9.1. Classical low-density parity-check codes

9.2. Dual-containing quantum low-density parity-check codes

9.3. Entanglement-assisted quantum low-density parity-check codes

9.4. Iterative decoding of quantum low-density parity-check codes

9.5. Spatially coupled quantum low-density parity-check codes

9.6. Summary

Chapter 10. Quantum networking

10.1. Quantum communications networks and the quantum Internet

10.2. Quantum teleportation and quantum relay

10.3. Entanglement distribution

10.4. Engineering entangled states and hybrid continuous-variable–discrete-variable quantum networks

10.5. Cluster state-based quantum networking

10.6. Surface code-based and quantum low-density parity-check code-based quantum networking

10.7. Entanglement-assisted communication and networking

10.8. Summary

Chapter 11. Quantum sensing and quantum radars

11.1. Quantum phase estimation

11.2. Quantum Fisher information and quantum Cramér–Rao bound

11.3. Distributed quantum sensing

11.4. Quantum radars

11.5. Summary

Chapter 12. Quantum machine learning

12.1. Machine learning fundamentals

12.2. The Ising model, adiabatic quantum computing, and quantum annealing

12.3. Quantum approximate optimization algorithm and variational quantum eigensolver

12.4. Quantum boosting

12.5. Quantum random access memory

12.6. Quantum matrix inversion

12.7. Quantum principal component analysis

12.8. Quantum optimization-based clustering

12.9. Grover algorithm-based global quantum optimization

12.10. Quantum K-means

12.11. Quantum support vector machines

12.12. Quantum neural networks

12.13. Summary

Chapter 13. Fault-tolerant quantum error correction

13.1. Fault-tolerance basics

13.2. Fault-tolerant quantum information processing concepts

13.3. Fault-tolerant quantum error correction

13.4. Summary

Index

Copyright

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Preface

Quantum information is related to using quantum mechanical concepts to perform information processing and information transmission. Quantum information processing (QIP) is an exciting research area with numerous applications, including quantum key distribution (QKD), quantum teleportation, quantum and distributed quantum computing, quantum sensing, quantum networking, quantum radar, and quantum machine learning. Quantum information science (QIS) is experiencing rapid development, as judged by the number of published books on QIP, quantum computing, and quantum communications and the numerous conferences devoted solely to QIS. Given the novelty of underlying QIS concepts, this topic is expected to be a subject of interest to many scientists, engineers, governmental agencies, industries, and investors, not just those involved in QIP research. The entanglement represents a unique QIP feature enabling quantum computers to solve problems that are numerically intractable for classical computers, enabling quantum-enhanced sensors with measurement sensitivities exceeding classical limits and providing certifiable data transmission security guaranteed by the laws of quantum mechanics rather than the unproven assumptions used in conventional cryptography for computational security. Preshared entanglement can be used to (1) improve classical channel capacity, (2) enable secure communications, (3) improve sensor sensitivity, (4) enable distributed quantum computing, (5) enable entanglement-assisted distributed sensing, and (6) enable provably secure quantum computer access, as well as perform many other tasks.

Because of the interdisciplinary nature of QIS, this book aims to provide the right balance among QIP, quantum error correction, quantum communication, quantum sensing, and quantum networking. The main objectives of the book can be summarized as follows:

(1) Book describes trends in QIP, quantum error correction, quantum communication, quantum sensing, quantum networking, and quantum machine learning.

(2) Book represents a self-contained introduction to quantum communication, quantum error correction, quantum sensing, and quantum networking.

(3) Book targets a wide range of readers: electrical engineers, communication engineers, optical engineers, applied mathematicians, computer scientists, and physicists.

(4) Book does not require prior knowledge of quantum mechanics or QIP. The basic concepts are provided in Chapter 3.

(5) The book does not require any prerequisite background, except for an understanding of the basic concepts of vector algebra at the undergraduate level.

(6) For readers not familiar with information theory, channel coding, and detection theory, Chapter 2 provides basic concepts and definitions in the respective fields. The chapter only covers information, coding, and detection theory concepts relevant in later chapters.

(7) Readers not interested in quantum error correction can still approach the book’s other chapters.

(8) Readers solely interested in quantum error correction do not need to read other chapters, except Chapters 3 and 4, and can concentrate on just the quantum error correction chapters. Namely, an effort has been made to create independent chapters while ensuring a proper flow between chapters.

(9) The book starts with intuitive and basic concepts before moving to medium-difficulty topics and then to complex, mathematically involved topics.

(10) Several courses can be offered using this book. Examples include courses on (1) quantum error correction, (2) QIP and quantum machine learning, (3) quantum communication, (4) quantum sensing, (5) quantum networking, and (6) integration of the concepts of quantum communication, quantum sensing, and quantum networking.

(11) The reader who completes this book will be able to perform independent research in various fields of QIS.

This book represents a self-contained introduction to quantum information, quantum communication, quantum error correction, quantum sensing, and quantum networking. The successful reader of the book will be ready for further study in these areas and will be prepared to perform independent research. The reader who has completed the book will be able to design QIP circuits, stabilizer codes, Calderbank–Shor–Steane (CSS) codes, subsystem codes, topological codes, surface codes, and entanglement-assisted quantum error correction codes. The reader will also be proficient in fault-tolerant quantum error correction. Further, the reader will be proficient in quantum communication, QKD, and quantum machine learning. The reader will have gained the required knowledge in quantum sensing and quantum radars to perform independent research in these fields. Readers of this book will also be able to propose quantum transceivers suitable for quantum communication and quantum sensing applications. Moreover, book readers will be able to propose quantum communication networking protocols and quantum networking architectures.

Some unique features of the book can be summarized as follows:

(1) This book integrates QIP, communication, error correction, sensing, networking, and machine learning.

(2) The book does not require knowledge of quantum mechanics.

(3) This book does not require any prerequisite material except for basic concepts of vector algebra at the undergraduate level.

(4) This book offers in-depth exposition on the design of QIP, error correction, communication and networking, and sensing circuits and modules.

(5) The successful student will be prepared for further study in these areas and qualified to perform independent research.

(6) The reader who has completed this book will be able to design information-processing circuits, stabilizer codes, CSS codes, topological codes, surface codes, subsystem codes, quantum low-density parity-check (LDPC) codes, and entanglement-assisted quantum error correction codes and will be proficient in fault-tolerant design.

(7) The successful student will be able to propose quantum transceivers for quantum communication and quantum sensing applications.

(8) The successful student will be able to propose quantum communication networking protocols and quantum networking architectures.

(9) Students who have completed the book will be proficient in quantum machine learning.

(10) Finally, students will be proficient in QKD and quantum teleportation.

(11) Extra material to support the book will be provided on an accompanying website to be developed after publication.

The author would like to acknowledge the support of the National Science Foundation.

Finally, special thanks are extended to Prem Kumar Kaliamoorthi, Kyle Gravel, and Tim Pitts of Elsevier for their tremendous efforts in organizing the logistics of the book, including editing and promotion, which are indispensable to making this book happen.

Chapter 1: Basics of quantum information, quantum communication, quantum sensing, and quantum networking

Abstract

In this chapter, we provide an overview of quantum communications (QuComs), quantum error correction (QEC), quantum networks, and quantum sensing (QuSen) and discuss their relevance. We then discuss the basic principles of QuComs, QEC, QuNets, and QuSen. The last section of the chapter is devoted to the organization of the book, with a detailed description of each chapter.

Keywords

Quantum communications; Quantum error correction; Quantum information processing; Quantum key distribution; Quantum networking; Quantum sensing; Quantum teleportation

1.0 Overview

1.1 Photon polarization

1.2 The concept of qubit

1.3 Quantum gates and quantum information processing

1.4 Quantum teleportation

1.5 Quantum error correction concepts

1.6 Quantum sensing

1.7 Quantum key distribution

1.8 Quantum networking

1.9 Organization of the book

References

1.0. Overview

This chapter is devoted to introducing quantum information processing (QIP), quantum communication, quantum networking, quantum sensing, and quantum error correction coding (QECC) [1–18]. Quantum information is related to quantum mechanical concepts used in information processing or the transmission of quantum information. QIP is an exciting research area with numerous applications, including quantum key distribution (QKD), quantum teleportation, quantum and distributed quantum computing, quantum sensing, quantum networking, and quantum radar.

Fundamental features of QIP are different from those of classical signal processing and can be summarized as three characteristics [1,2,6,7,10]: (1) linear superposition, (2) entanglement and (3) quantum parallelism. The following are some basic details of these features:

(1) Linear superposition. Contrary to the classical bit, a quantum bit or qubit can take not only two discrete values, 0 and 1, but also all possible linear combinations of them. This is a consequence of a fundamental property of quantum states—it is possible to construct a linear superposition of quantum state |0〉 and quantum state |1〉.

(2) Entanglement. At a quantum level, it appears that two quantum objects can form a single entity even when they are well separated from each other. Any attempt to consider this entity as a combination of two independent quantum objects given by the tensor product of quantum states fails unless the possibility of signal propagation at a superluminal speed is allowed. These quantum objects that cannot be decomposed into the tensor product of individual independent quantum objects are commonly referred to as entangled quantum objects. Given that arbitrary quantum states cannot be copied, which is a consequence of the no-cloning theorem, communication at superluminal speed is not possible, and as a consequence, entangled quantum states cannot be written as the tensor product of independent quantum states. Moreover, it can be shown that the amount of information contained in an entangled state of N qubits grows exponentially instead of linearly as is the case for classical bits.

(3) Quantum parallelism. Quantum parallelism is the possibility of performing a large number of operations in parallel, which represents its key difference from classical information processing and computing. Namely, in classical computing, it is possible to know the internal status of the computer. On the other hand, because of the no-cloning theorem, it is not possible to know the current state of a quantum computer. This property has led to the development of Shor factorization algorithm, which can be used to crack the Rivest–Shamir–Adleman (RSA) encryption protocol. Some other important quantum algorithms include the Grover search algorithm, which is used to search for an entry in an unstructured database; the quantum Fourier transform, which is the basis for a number of algorithms; and Simon's algorithm. The quantum computer can encode all input strings of length N simultaneously in a single computation step. In other words, the quantum computer can simultaneously pursue 2N classical paths, indicating that the quantum computer is significantly more powerful than the classical one.

Although QIP has opened up some fascinating perspectives, certain limitations must be overcome before QIP becomes a commercial reality. The first is related to the number of existing quantum algorithms, which is significantly lower than that of classical algorithms. The second problem is related to physical implementation issues. Another problem, which can be considered problem number one, relates to decoherence. Decoherence is related to the interaction of qubits with the environment that blurs the fragile superposition states. Decoherence also introduces errors, indicating that the quantum register should be sufficiently isolated from the environment so that only a few random errors occasionally occur and can be corrected by QECC techniques. One of the most powerful applications of quantum error correction is protecting quantum information as it dynamically undergoes quantum computation. Imperfect quantum gates affect quantum computation by introducing errors in computed data. Moreover, the imperfect control gates introduce errors in processed sequences, since the wrong operations can be applied. The QECC scheme now must deal not only with errors introduced by the quantum channel but also with errors introduced by imperfect quantum gates during the encoding/decoding process.

Nevertheless, QIP opens new avenues for various applications, including high-performance computing, high-precision sensing, and secure communications [1,2]. The entanglement represents a unique QIP feature enabling quantum computers capable of solving the problems that are numerically intractable for classical computers [19], enabling quantum-enhanced sensors with measurement sensitivities exceeding the classical limit [20]; and providing certifiable security for data transmissions whose security is guaranteed by the laws of quantum mechanics, known as the quantum key distribution (QKD), rather than unproven assumptions used in cryptography based on computational security [1,2]. The preshared entanglement can be used to (1) improve the classical channel capacity [21–27], (2) enable secure communications, (3) improve sensor sensitivity, (4) enable distributed quantum computing [28], (5) enable entanglement-assisted distributed sensing, and (6) enable provably secure quantum computer access [29], to mention a few.

The quantum network represents the network of nodes interconnected by quantum links. The quantum links could be heterogeneous, including fiber-optics links, free-space optical (FSO) links, and satellite links, to mention a few. The quantum communication network (QCN) can be defined as a system of quantum nodes in which an uninterrupted quantum channel can be established between any two nodes. When the quantum nodes are equipped with quantum computers capable of exchanging the qubits (or qudits) over the QCN by teleportation, the corresponding QCN is commonly referred to as the quantum Internet. Several QKD testbeds have been reported, including the DARPA QKD network [30], the Tokyo QKD network [31], and secure communication based on the quantum cryptography (SECOQC) network [32]. QKD can also be used to establish a QKD-based campus-to-campus virtual private network employing the IPsec protocol [33], as well as to establish a network setup to use transport-layer security based on QKD [34]. However, all these networks employ a dark fiber infrastructure. Classical communication networks have nodes connected by various channel types, including FSO, optical fiber, ground-satellite links, wireless RF, and coaxial cables. Such a heterogeneous architecture will be equally important for quantum nodes to access a QCN. Indeed, quantum communications (QuComs) are individually validated in free-space, optical fiber, and between a satellite and a ground station, but a heterogeneous QCN with multiple channel types remains currently elusive.

Quantum sensing is an application of QIP that can find practical realization in the near term. When we use nonclassical sources to improve the sensitivity of measuring a certain parameter in a sensing application, we refer to such a methodology as quantum metrology [20,35,36]. When we repeat the measurement of a certain parameter N times and average out the individual measurement results, we can achieve the precision that scales as N −¹/², which is commonly referred to as the standard quantum limit (SQL). By employing nonclassical sources such as the entangled source or squeezed light, we can achieve better precision than SQL. As an illustration, in Laser Interferometer Gravitational-Wave Observatory (LIGO) experiments [35,36], squeezed light is injected into a Michelson interferometer to overcome the SQL (dictated by the shot noise). Alternatively, in distributed quantum sensing [20], multiple entangled sensors are employed to obtain a certain global property of an interrogated object and beat the SQL. In addition to LIGO and target detection/quantum radars [37,38], quantum metrology has also been applied in biological sensing [39], microscopy [40], and phase-tracking applications [41].

This introductory chapter is organized as follows. In Section 1.1, photon polarization is described as representing a simple and the most natural connection to the QuComs. In the same section, we introduce some basic quantum mechanics concepts, such as the state concept. We also introduce the Dirac notation, which will be used throughout the book. In Section 1.2, we formally introduce the concept of qubit and provide its geometric interpretation. Section 1.3 is devoted to basic quantum gates and QIP fundamentals. The basic concepts of quantum teleportation are introduced in Section 1.4. Section 1.5 is devoted to the basic QECC concepts. Section 1.6 introduces the quantum sensing concept. Section 1.7 is devoted to the quantum key distribution (QKD), also known as quantum cryptography. Section 1.8 is related to the quantum networks basics. The organization of the book is described in Section 1.9.

1.1. Photon polarization

The electric/magnetic field of plane linearly polarized waves is described as follows [42]:

(1.1)

where E ( H ) denotes electric (magnetic) field, p denotes the polarization orientation, r = x e x + y e y + z e z is the position vector, and k = k x e x + k y e y + k z e z denotes the wave propagation vector whose magnitude is k =2π/λ, with λ being the operating wavelength. For the x-polarization waves ( p = e x, k = k e z), Eq. (1.1) becomes

(1.2)

while for y-polarization ( p = e y, k = k e z), it becomes

(1.3)

where δ is the relative phase difference between the two orthogonal waves. The resultant wave can be obtained by combining Eqs. (1.2) and (1.3) as follows:

(1.4)

The linearly polarized wave is obtained by setting the phase difference to an integer multiple of 2π, namely

:

(1.5)

We can represent the linear polarization by ignoring the time-dependent term as shown in Fig. 1.1A. On the other hand, if

the elliptical polarization is obtained. From Eqs. (1.2) and (1.3), by eliminating the time-dependent term, we obtain the following equation of ellipse:

(1.6)

which is shown in Fig. 1.1B. By setting

the equation of the ellipse becomes

(1.7)

Figure 1.1  Various forms of polarization: (A) linear polarization, (B) elliptic polarization, and (C) circular polarization.

Further, by setting

the equation of the ellipse becomes the circle

(1.8)

and the corresponding polarization is known as circular polarization (see Fig. 1.1C). A right circularly polarized wave is obtained for δ=π/2+2mπ:

(1.9)

Otherwise, for δ= −π/2+2mπ, the polarization is known as left circularly polarized.

Quite often, the Jones vector representation of the polarization wave is used:

(1.10)

where κ is the power-splitting ratio between states of polarization (SOPs), with the complex phasor term typically omitted in practice.

Another interesting representation is the Stokes vector representation:

(1.11)

where the parameter S 0 is related to the optical intensity by

(1.12a)

The parameter S 1 >0 is related to the preference for horizontal polarization and is defined by

(1.12b)

The parameter S 2 >0 is related to the preference for the π/4 SOP:

(1.12c)

Finally, the parameters S 3 >0 is related to the preference for right-circular polarization, and it is defined by

(1.12d)

The parameter S 0 is related to other Stokes parameters by

(1.13)

The degree of polarization is defined by

(1.14)

For p =1, the polarization does not change with time. The Stokes vector can be represented in terms of Jones vector parameters as

(1.15)

After normalization with respect to S 0, normalized Stokes parameters are given by

(1.16)

The polarization state can be represented as a point on a Poincaré sphere if normalized Stokes parameters are used, as shown in Fig. 1.2. The points at the opposite sides of the line crossing the center represent orthogonal polarizations.

The polarization ellipse is often represented by ellipticity and azimuth, which are illustrated in Fig. 1.3. Ellipticity is defined by the ratio of the half-axes lengths. The corresponding angle is called the ellipticity angle and is denoted ε. A small ellipticity means that the polarization ellipse is highly elongated, while for zero ellipticity, polarization is linear. For ε=±π/4, polarization is circular. For ε >0, polarization is right-elliptical. On the other hand, the azimuth angle η defines the orientation of the main axis of the ellipse with respect to E x .

Figure 1.2  Representation of polarization state as a point on a Poincaré sphere.

Figure 1.3  The ellipticity and azimuth of the polarization ellipse.

The parameters of polarization ellipse can be related to the Jones vector parameters by

(1.17)

Finally, the parameters of the polarization ellipse can be related to the Stokes vector parameters by

(1.18)

and the corresponding geometrical interpretation is provided in Fig. 1.2.

Let us now observe the polarizer–analyzer ensemble, shown in Fig. 1.4. When an electromagnetic wave passes through the polarizer, it can be represented as a vector in the xOy plane transversal to the propagation direction, as given by Eq. (1.5), where the angle θ depends on the filter orientation. By introducing the unit vector

 Eq. (1.5) can be rewritten as

(1.19)

If θ =0rad, the light is polarized along the x-axis, while for θ =π/2rad, it is polarized along the y-axis. Natural light is unpolarized, as it represents an incoherent superposition of 50% of light polarized along the x-axis and 50% along the y-axis. After the analyzer, the polarization vector makes an angle ϕ with respect to the x-axis, which can be represented by a unit vector

, and the output electric field is given by

(1.20)

Figure 1.4  The polarizer–analyzer ensemble for the study of photon polarization.

The intensity of the analyzer output field is given by

(1.21)

which is commonly referred to as the Malus law.

Polarization decomposition by a birefringent plate is studied in Fig. 1.5. Experiments indicate that photodetectors PDx and PDy are never triggered simultaneously, meaning that an entire photon reaches either PDx or PDy (a photon never splits). Therefore, the corresponding probabilities that photodetector PDx and PDx detect a photon can be determined by

(1.22)

If the total number of photons is N, the number of detected photons in x-polarization will be N x ≅ Ncos² θ, and the number of detected photons in y-polarization will be N y ≅ Nsin² θ. In the limit as N→∞, we would expect the Malus law to be obtained.

Let us now study the polarization decomposition and recombination by means of birefringent plates, as illustrated in Fig. 1.6. Classical physics prediction of total probability of a photon passing the polarizer-analyzer ensemble is given by

(1.23)

which is inconsistent with the Malus law, given by Eq. (1.21). To reconstruct the results from wave optics, it is necessary to introduce the concept into quantum mechanics of the probability amplitude that α is detected as β, which is denoted

which is a complex number. The probability is obtained as the squared magnitude of probability amplitude:

Figure 1.5  Polarization decomposition by a birefringent plate. PD: photodetector.

Figure 1.6  Polarization decomposition and recombination by a birefringent plate.

(1.24)

The relevant probability amplitudes related to Fig. 1.6 are summarized as

(1.25)

The basic principle of quantum mechanics is to sum up the probability amplitudes for indistinguishable paths to obtain

(1.26)

The corresponding total probability is

(1.27)

and this result is consistent with the Malus law!

Based on the previous discussion, the state vector of the photon polarization is given by

(1.28)

where ψ x is related to x-polarization, and ψ y is related to y-polarization, with normalization condition as follows:

(1.29)

In this representation, the x- and y-polarization photons can be represented by

(1.30)

and the right and left circular polarization photons are represented by

(1.31)

In Eqs. (1.28)–(1.31), we used Dirac notation to denote the column-vectors (kets). In Dirac notation, with each column vector (ket) |ψ〉, we associate a row vector (bra) 〈ψ| as follows:

(1.32)

The scalar (dot) product of ket |ϕ〉 bra 〈ψ| is defined by bracket as follows:

(1.33)

The normalization condition can be expressed in terms of a scalar product by

(1.34)

Based on Eqs. (1.30) and (1.31), it is evident that

Because the vectors |x〉 and |y〉 are orthogonal, their dot product is zero:

(1.35)

and they form the basis. Any state vector |ψ〉 can be written as a linear superposition of basis kets as follows:

(1.36)

We can now use Eqs. (1.34) and (1.36) to derive an important relation in quantum mechanics, known as completeness relation. The projections of state vector |ψ〉 along basis vectors |x〉 and |y〉 are given by

(1.37)

By substituting Eq. (1.37) into Eq. (1.36), we obtain

(1.38)

and from the right side of Eq. (1.38), we derive the completeness relation:

(1.39)

where denotes the identity operator.

The probability that the photon in state |ψ〉 will pass the x-polaroid is given by

(1.40)

and the probability amplitude of the photon in state |ψ〉 to pass the x-polaroid is

(1.41)

Let |ϕ〉 and |ψ〉 be two physical states; the probability amplitude of finding ϕ in ψ, denoted as a(ϕ→ψ), is given by

(1.42)

and the probability for ϕ to pass ψ test is given by

(1.43)

1.2. The concept of qubit

Based on the previous section, it can be concluded that the quantum bit, also known as the qubit, lies in a two-dimensional Hilbert space H isomorphic to the C²-space, where C is the complex number space, and can be represented as

(1.44)

where the |0〉 and |1〉 states are the computational basis (CB) states, and |ψ〉 is a superposition state. If we measure a qubit, we will obtain |0〉 with a probability |α|² and |1〉 with a probability of |β|². Measurement changes the state of a qubit from a superposition of |0〉 and |1〉 to the specific state consistent with the measurement result. If we parametrize the probability amplitudes α and β as follows:

(1.45)

where θ is a polar angle, and ϕ is an azimuthal angle. We can geometrically represent the qubit by Bloch sphere (or the Poincaré sphere for the photon) as illustrated in Fig. 1.7. (Note that the Bloch sphere from Fig. 1.7, commonly used in QIP, is different from that of the Poincaré sphere from Fig. 1.2, commonly used in optics.) Bloch vector coordinates are given by (cosϕ sinθ, sinϕ sinθ, cosθ). This Bloch vector representation is related to CB by

(1.46)

where 0≤θ≤π and 0≤ ϕ <2π. The north and south poles correspond to computational |0〉 (|x〉-polarization) and |1〉 (|y〉-polarization) basis kets, respectively. Other important bases are the diagonal basis {|+〉, |−〉}, very often denoted as

related to CB by

(1.47)

and the circular basis {|R〉,|L〉}, related to the CB as follows:

Figure 1.7  Bloch (Poincaré) sphere representation of a single qubit.

(1.48)

1.3. Quantum gates and quantum information processing

In quantum mechanics, the primitive, undefined concepts are physical system, observable, and state. The concept of state has been introduced in previous sections. An observable, such as momentum and spin, can be represented by an operator, such as A, in the vector space in question. An operator, or gate, acts on a ket from the left: (A) ⋅ |α〉=A |α〉, and results in another ket. A linear operator (gate) B can be expressed in terms of eigenkets {|a (n)〉} of an Hermitian operator A. (An operator A is said to be Hermitian if ) The operator X is associated with a square matrix (albeit infinite in extent), whose elements are

(1.49)

and can explicitly be written as

(1.50)

where we use the notation to denote that operator X is represented by the matrix above.

Very important single-qubit gates are the Hadamard gate H, the phase shift gate S, the π/8 (or T) gate, the controlled-NOT (or CNOT) gate, and Pauli operators X, Y, and Z. The Hadamard gate H, phase shift gate, T gate, and CNOT gate have the following matrix representation in computational basis (CB) {|0〉,|1〉}:

(1.51)

The Pauli operators, on the other hand, have the following matrix representation in CB:

(1.52)

The action of Pauli gates on an arbitrary qubit

is given as follows:

(1.53)

So the action of the X-gate is to introduce the bit-flip, the action of the Z-gate is to introduce the phase-flip, and the action of the Y-gate is to simultaneously introduce the bit- and phase-flips.

Figure 1.8  Important quantum gates and their action: (A) single-qubit gate, (B) controlled-U gate, (C) CNOT-gate, (D) SWAP-gate, and (E) Toffoli gate.

Several important single-, double-, and three-qubit gates are shown in Fig. 1.8. The action of the single-qubit gate is to apply the operator U on qubit |ψ〉, which results in another qubit. Controlled-U gate conditionally applies the operator U on target qubit |ψ〉, when the control qubit |c〉 is in |1〉-state. One particularly important controlled U-gate is the controlled-NOT (CNOT) gate. This gate flips the content of target qubit |t〉 when the control qubit |c〉 is in |1〉-state. The purpose of SWAP-gate is to interchange the positions of two qubits and can be implemented using three CNOT-gates, as shown in Fig. 1.8D. Finally, the Toffoli gate represents the generalization of CNOT-gate, where two control qubits are used.

The minimum set of gates that can perform an arbitrary quantum computation algorithm is known as the universal set of gates [43,44]. The most popular sets of universal quantum gates are {H, S, CNOT, Toffoli} gates, {H, S, π/8 (T), CNOT} gates, the Barenco gate [45], and the Deutsch gate [46]. With these universal quantum gates, more complicated operations can be performed. As an illustration, Fig. 1.10 shows the Bell state [Einstein–Podolsky–Rosen (EPR) pair] preparation circuit, which is highly important in quantum teleportation and QKD applications.

So far, single-, double-, and triple-qubit quantum gates have been considered. An arbitrary quantum state of K qubits has the form ∑ s  α s  | s 〉, where s runs over all binary strings of length K. Therefore, there are 2 K complex coefficients, all independent except the normalization constraint:

(1.54)

For example, the state α00|00〉+ α01|01〉+ α10|10〉+ α11|11〉 (with |α00|²+ |α01|²+ |α10|²+ |α11|² =1) is the general 2-qubit state (we use |00〉 to denote the tensor product |0〉⊗|0〉). The multiple qubits can be entangled, so they cannot be decomposed into two separate states. For example, the Bell state or EPR pair (|00〉+ |11〉)/√2 cannot be written in terms of tensor product |ψ 1〉 |ψ 2〉= (α1|0〉+β1|1〉)⊗(α2|0〉+β2|1〉)=α1α2|00〉+ α1β2|01〉+ β1α2|10〉+ β1β2|11〉, because in that case, the following conditions need to be satisfied α1α2 =β1β2 =1/√2 and α1β2 =β1α2 =0, which a priori has no reason to be valid. This state can be obtained using the circuit shown in Fig. 1.9, for a two-qubit input state |00〉.

Figure 1.9  Bell state (EPR pair) preparation circuit.

1.4. Quantum teleportation

Quantum teleportation [47] is a technique to transfer quantum information from source to destination by employing the entangled states. Namely, in quantum teleportation, the entanglement in the Bell state (EPR pair) is used to transport arbitrary quantum state |ψ〉 between two distant observers A and B (often called Alice and Bob), as illustrated in Fig. 1.10. The quantum teleportation system employs three qubits, qubit 1 is an arbitrary state to be teleported, while qubits 2 and 3 are in Bell state |B 00〉=(|00〉+|11〉)/√2. Let the state to be teleported be denoted by |ψ〉= a|0〉+b|1〉. The input to the circuit shown in Fig. 1.10 is, therefore, |ψ〉|B 00〉, and can be rewritten as

(1.55)

The CNOT-gate is then applied with the first qubit serving as the control and the second qubit as the target, which transforms Eq. (1.55) into

(1.56)

In the next stage, the Hadamard gate is applied to the first qubit, which maps |0〉 to (|0〉+|1〉)/√2 and |1〉 to (|0〉-|1〉)/√2, so the overall transformation of the quantum state of Eq. (1.55) is as follows:

(1.57)

Figure 1.10  Illustration of quantum teleportation principle.

The measurements are performed on qubits 1 and 2, and based on the results of measurements, denoted respectively as a and b, the controlled-X (CNOT) and controlled-Z gates are applied conditionally to lead to the following content on qubit 3:

(1.58)

indicating that the arbitrary state |ψ〉 is teleported to the remote destination and can be found at the qubit 3 position.

1.5. Quantum error correction concepts

QIP relies on delicate superposition states that are sensitive to environmental interactions, resulting in decoherence. Moreover, quantum gates are imperfect, and quantum error correction coding (QECC) is necessary to enable fault-tolerant computing and deal with quantum errors [48–53]. QECC is also essential in quantum communication and quantum teleportation applications. The elements of quantum error correction codes are shown in Fig. 1.11A. The (N,K) QECC code performs encoding of the quantum state of K qubits specified by 2 K complex coefficients α s into a quantum state of N qubits in such a way that errors can be detected and corrected, and all 2 K complex coefficients can be perfectly restored, up to the global phase shift. Namely, we know from quantum mechanics that two states |ψ〉 and ejθ|ψ〉 are equal up to a global phase shift as the results of measurement on both states are the same. A quantum error correction consists of four major steps: encoding, error detection, error recovery, and decoding. The sender (Alice) encodes quantum information in state |ψ〉 with the help of local ancilla qubits |0〉, and then sends the encoded qubits over a noisy quantum channel (say free-space optical channel or fiber-optics channel). The receiver (Bob) performs multi-qubit measurement on all qubits to diagnose the channel error and performs a recovery unitary operation R to reverse the action of the channel. The quantum error correction is essentially more complicated than classical error correction, with key differences being summarized as follows: (1) the no-cloning theorem indicates that it is impossible to make a copy of an arbitrary quantum state, (2) quantum errors are continuous, and a qubit can be in any superposition of the two bases states, and (3) the measurements destroy the quantum information. The quantum error correction principles will be more evident after a simple example below.

Figure 1.11  (A) A quantum error correction principle. (B) Bit-flipping channel model. (C) Three-qubit flip code encoder.

Assume we want to send a single qubit |Ψ〉=α|0〉+β|1〉 through the quantum channel in which during transmission the transmitted qubit can be flipped to X |Ψ〉=β |0〉+α |1〉 with probability p. Such a quantum channel is called a bit-flip channel, and it can be described as shown in Fig. 1.11B. Three-qubit flip code sends the same qubit three times and therefore represents the repetition code equivalent. The corresponding codewords in this code are The three-qubit flip code encoder is shown in Fig. 1.11C. One input qubit and two ancillae are used at the input encoder, which can be represented by

. The first ancilla qubit (the second qubit at the encoder input) is controlled by the information qubit (the first qubit at encoder input), so its output can be represented by

(if the control qubit is |1〉 the target qubit gets flipped, otherwise it stays unchanged). The output of the first CNOT gate is used as input to the second CNOT gate in which the second ancilla qubit (the third qubit) is controlled by the information qubit (the first qubit), so the corresponding encoder output is obtained as

, which indicates that basis codewords are indeed and . With this code, we can correct a single qubit-flip error that occurs with probability (1− p)³ +3p(1− p)² =1− 3p ² +2 p ³. Therefore, the probability of an error remaining uncorrected or wrongly corrected with this code is 3p ² − 2 p ³. It is clear from Fig. 1.11C that a three-qubit bit-flip encoder is a systematic encoder in which the information qubit is unchanged, and the ancilla qubits are used to impose the encoding operation and create the parity qubits (the output qubits 2 and 3).

Let us assume that a qubit flip occurred on the first qubit, leading to received quantum word |Ψ r 〉=α|100〉+β|011〉. To identify the error, measurements must be performed on the observables Z1Z2 and Z2Z3, where the subscript denotes the index of the qubit on which a given Pauli gate is applied. The measurement result is eigenvalue ±1, and the corresponding eigenvectors are two valid codewords, namely |000〉 and |111〉. The observables can be represented as follows:

(1.59)

It can be shown that

. The first syndrome indicates that an error occurred on either the first or second qubit while the second syndrome indicates that error is not located on the second or third qubits. The intersection reveals that the first qubit was in error. Using this approach, we can create the three-qubit look-up table (LUT), given as Table 1.1. Three-qubit flip code error detection and error correction circuits are shown in Fig. 1.12. The results of measurements on ancillae, as illustrated in Fig. 1.12A, will determine the error syndrome [±1±1], and based on the LUT given by Table 1.1, we identify the error event and apply the corresponding X i gate on the ith qubit being in error, and the error is corrected since X ² = I. The control logic operation is described in Table 1.1. For example, if both outputs at the measurement circuits are −1, the operator X 2 is activated. The last step is decoding, as shown in Fig. 1.12B, by simply reversing the order of elements in the corresponding encoder.

Table 1.1

Figure 1.12  (A) Three-qubit flip code error detection and error correction circuit. (B) Decoder circuit configuration.

1.6. Quantum sensing

As an illustration of quantum sensing applications, the conceptual schematic of the distributed quantum sensing (DQS) concept [20,54,55] is provided in Fig. 1.13. Based on initial quantum state ρ 0 =|ψ 0〉〈ψ 0|, the source quantum module generates a multipartite entangled probe state that is shared by M sensors to scan an object of interest. The measurement results obtained by M sensors are (classically) postprocessed to determine a certain global parameter of the scanned object. On the other hand, in distributed classical sensing, the quantum state of M sensors can be separated; that is, it can be written as a tensor product

. The distributing sensing scenario can be described by a unitary operator being the product of M unitary operators, that is,

, wherein each unitary measures one parameter. The output quantum state can be described by the following transformation:

(1.60)

where

is the input entangled probe state. In DQS, we are interested in estimating the global parameter rather than estimating all unknown parameters. This can be achieved by calculating the weighted average as follows:

(1.61)

Figure 1.13  Illustrating the distributed quantum sensing concept for entangled inputs.

In the special case where w m =1/M and θ m = θ, using the entangled probe state, we can beat the SQL and achieve the Heisenberg limit. For additional details, an interested reader is referred to Chapter 11.

1.7. Quantum key distribution

QKD exploits the principle of quantum mechanics to enable the provably secure distribution of a private key between remote destinations. Private key cryptography is much older than the public key cryptosystems commonly used today. In a private key cryptosystem, Alice (sender) must have an encoding key, while Bob (receiver) must have a matching decoding key to decrypt the encoded message. The simplest private key cryptosystem is the Vernam cipher (one-time pad), which operates as follows [8]: (1) Alice and Bob share n-bit key strings, (2) Alice encodes her n-bit message by adding the message and the key together, and (3) Bob decodes the information by subtracting the key from the received message. The are several drawbacks of this scheme: (1) secure distribution of the key as well as the key length must be at least as long as the message length, (2) the key bits cannot be reused, and (3) the keys must be delivered in advance, securely stored until use, and destroyed after use.

On the other hand, if the classical information, such as key, is transmitted over the quantum channel, thanks to the no-cloning theorem, which states that a device cannot be constructed to produce an exact copy of an arbitrary quantum state, the eavesdropper cannot get an exact copy of the key in a QKD system. Namely, in an attempt to distinguish between two nonorthogonal states, information gain is only possible at the expense of introducing the disturbance to the signal. This observation can be proved as follows. Let |ψ 1〉 and |ψ 2〉 be two nonorthogonal states Eve is interested to learn about and |α〉 be the standard state prepared by the Eve. The Eve tries to interact with these states without disturbing them, and as the result of this interaction, the following transformation is performed [8]:

(1.62)

The Eve hopes that the states |α〉 and |β〉 would be different, in an attempt to learn something about the states. However, any unitary transformation must preserve the dot product, so from Eq. (1.60), we obtain

(1.63)

which indicates that

and consequently |α〉=|β〉. Therefore, an attempt to distinguish between |ψ 1〉 and |ψ 2〉, Eve will disturb them. The key idea of QKD, therefore, is to transmit nonorthogonal qubit states between Alice and Bob, and by checking for disturbances in the transmitted state, they can establish an upper bound on the noise/eavesdropping level in their communication channel.

One of the simplest QKD protocols is the BB84 protocol, named after Bennett and Brassard, who proposed it in 1984 [56,57]. The BB84 protocol can be implemented using different degrees of freedom (DOF), including the polarization DOF [58] or the phase of the photons [59]. Experimentally, the BB84 protocol has been demonstrated over both fiber-optics and free-space optical (FSO) channels [58,60]. The polarization-based BB84 protocol over the fiber-optics channel is affected by polarization mode dispersion (PMD), polarization-dependent loss (PDL), and fiber loss, which affect the transmission distance. To extend the transmission distance, phase encoding is employed as in Ref. [60]. Unfortunately, the secure key rate over 405km of ultra-low-loss fiber is extremely low, only 6.5b/s. In an FSO channel, the polarization effects are minimized; however, the atmospheric turbulence can introduce the wavefront distortion and random phase fluctuations. Previous QKD demonstrations include a satellite-to-ground FSO link demonstration and a demonstration over an FSO link between two locations in the Canary Islands [58].

Two bases that are used in the BB84 protocol are the computational basis CB={|0〉, |1〉} and the diagonal basis:

(1.64)

These bases belong to the class of mutually unbiased bases (MUBs) [61–64]. The keyword unbiased means that if a system is prepared in a state belonging to one of the bases, all measurement outcomes with respect to the other basis are equally likely. Alice randomly selects the MUB, followed by random selection of the basis state. The logical 0 is represented by |0〉, |+〉; while the logical 1 is represented by |1〉, |−〉. Bob measures each qubit by randomly selecting the basis, computational or diagonal. In a sifting procedure, Alice and Bob announce the bases used for each qubit and keep only instances when they have used the same basis. The polarization-based BB84 QKD protocol employs four corresponding states as shown in Fig. 1.14. The bulky optics implementation of the BB84 protocol is illustrated in Fig. 1.15.

Figure 1.14  The four states being employed in the BB84 protocol.

Figure 1.15  Illustrating the polarization-based BB84 protocol. PBS: polarization beam splitter. PD: single-photon detector.

Once the QKD protocol is completed, transmitter A and receiver B perform a series of classical steps, as illustrated in Fig. 1.16. As the transmission distance increases and for higher key distribution speeds, error correction becomes increasingly important. By performing the information reconciliation by low-density parity-check (LDPC) codes, the higher input bit-error rates (BERs) can be tolerated compared to other coding schemes. The privacy amplification is further performed to eliminate any information obtained by an eavesdropper. Privacy amplification must be performed between Alice and Bob to distill a smaller set of bits from the generated key whose correlation with Eve's string is below the desired threshold. One way to accomplish privacy amplification is through universal hash functions. The simplest family of hash functions is based on multiplication with a random element of the Galois field GF(2 m ) (m > n), where n denotes the number of bits remaining after information reconciliation is completed. The threshold for the maximum tolerable error rate is dictated by the efficiency of the implemented information reconciliation and privacy amplification protocols. The reader interested in learning about QKD with more details is referred to Chapter 6.

Figure 1.16  Classical postprocessing steps.

1.8. Quantum networking

The QuCom is the cornerstone to fully exploit the power of QIP to distribute entanglement at a distance to interconnect quantum devices. However, the distribution of entanglement over long distances has been an outstanding challenge due to photon losses—i.e., quantum signals cannot be amplified without introducing additional noise that degrades or destroys the transmitted entanglement. Hence, QuCom calls for fundamentally distinct loss-mitigation mechanisms to establish long-range entanglement. In this regard, quantum repeaters have been pursued over the last decade to overcome the exponentially low entanglement distribution rate versus transmission distance in optical fiber. Despite encouraging advances of using silicon vacancy (SiV) defects in diamond quantum memory to surpass the repeaterless key-generation bound [65], several technological hurdles, including the scalability of quantum devices, the indistinguishability of emitted photons, and quantum error correction, have to be overcome before the development of fully functional quantum repeaters for long-distance QuCom. As an alternate solution, satellites have been leveraged as relays for QuCom over thousands of kilometers by virtue of the more favorable quadratic scaling of photon loss versus the distance in FSO links. Remarkably, QKD [59], quantum teleportation [66], and entanglement distribution [67] have been demonstrated in a ground-to-satellite QuCom testbed across several thousand kilometers, showing tremendous potential for building up satellite-mediated wide-area QuCom networks via FSO links. As another ubiquitous use case, FSO links would grant ad hoc quantum devices access to QuCom networks, akin to the internet connecting billions of mobile devices. As such, the envisioned future quantum internet will interconnect various genres of quantum devices through heterogeneous fiber, ground-to-satellite links, and FSO links as illustrated in Fig. 1.17.

At present, while QuCom is individually validated over a specific type of quantum link, heterogeneous quantum interconnect that enables full integration of diverse quantum modules into a unified QuCom network remains elusive. To implement such a quantum Internet infrastructure, various quantum interconnects and quantum interfaces are needed. The quantum interface can be defined as a quantum module or subsystem that connects the stationary qubits, typically located in quantum memories, and flying (mobile) qubits to establish a quantum channel between any two remote nodes. There are two outstanding challenges to be solved before such a global QCN (or quantum Internet) becomes realizable. First, photon loss fundamentally limits the entanglement distribution rate because quantum signals cannot be amplified without introducing additional noise that degrades or destroys the transmitted entanglement. Second, QIP devices rest upon stationary quantum information carried by, e.g., solid-state spin states, superconducting qubits, and ultracold atoms, whereas entanglement distribution hinges on flying quantum information encoded on photons. To interconnect QIP devices, the quantum interconnect modules must convert back and forth between stationary and flying quantum information. Some possible solutions to the various challenges that current heterogeneous QCNs are facing are described in Chapter 10.

Figure 1.17  Envisioned wide-area quantum communication network empowered by heterogeneous quantum interconnects. EA: entanglement assisted communication.

1.9. Organization of the book

This section is related to the organization of the book. This book covers various topics of quantum information processing, quantum communication, quantum error correction, quantum sensing, and quantum networking. The chapters of the book are organized as described below.

Chapter 2 is devoted to the basic concepts of information theory, coding theory, and detection and estimation theory. Section 2.1 provides the definitions of entropy, joint entropy, conditional entropy, relative entropy, mutual information, and channel capacity, followed by the information capacity theorem. We also briefly describe the source coding and data compaction concepts. Further, we discuss the channel capacity of discrete memoryless, continuous, and optical channels. The standard FEC schemes that belong to the class of hard-decision codes are described as well. More powerful FEC schemes belong to the class of soft iteratively decodable codes, but their description is beyond the scope of this chapter. Iteratively decodable codes, such as LDPC codes, are described in Chapter 9. In Section 2.2 we introduce classical channel coding preliminaries. Section 2.3 is devoted to the basics of linear block codes (LBCs), such as the definition of the generator and parity-check matrices, syndrome decoding, distance properties of LBCs,