Advanced Quantum Communications E-Book

Sandor Imre

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 796

Veröffentlichungsjahr: 2012

Beschreibung

The book provides an overview of the most advanced quantum informational geometric techniques, which can help quantum communication theorists analyze quantum channels, such as security or additivity properties. Each section addresses an area of major research of quantum information theory and quantum communication networks. The authors present the fundamental theoretical results of quantum information theory, while also presenting the details of advanced quantum ccommunication protocols with clear mathematical and information theoretical background. This book bridges the gap between quantum physics, quantum information theory, and practical engineering.

Leseprobe

Table of Contents

Cover

IEEE PRESS

TITLE PAGE

COPYRIGHT PAGE

DEDICATION

EPIGRAPH

PREFACE

CHAPTER 1 INTRODUCTION

1.1 EMERGING QUANTUM INFLUENCES

1.2 QUANTUM INFORMATION THEORY

1.3 DIFFERENT CAPACITIES OF QUANTUM CHANNELS

1.4 CHALLENGES RELATED TO QUANTUM CHANNEL CAPACITIES

1.5 SECRET AND PRIVATE QUANTUM COMMUNICATION

1.6 QUANTUM COMMUNICATIONS NETWORKS

1.7 RECENT DEVELOPMENTS AND FUTURE DIRECTIONS

CHAPTER 2 INTRODUCTION TO QUANTUM INFORMATION THEORY

2.1 INTRODUCTION

2.2 BASIC DEFINITIONS AND FORMULAS

2.3 GEOMETRICAL INTERPRETATION OF THE DENSITY MATRICES

2.4 QUANTUM ENTANGLEMENT

2.5 ENTROPY OF QUANTUM STATES

2.6 MEASUREMENT OF THE AMOUNT OF ENTANGLEMENT

2.7 ENCODING CLASSICAL INFORMATION TO QUANTUM STATES

2.8 QUANTUM NOISELESS CHANNEL CODING

2.9 BRIEF SUMMARY

2.10 FURTHER READING

CHAPTER 3 THE CLASSICAL CAPACITIES OF QUANTUM CHANNELS

3.1 INTRODUCTION

3.2 FROM CLASSICAL TO QUANTUM COMMUNICATION CHANNELS

3.3 TRANSMISSION OF CLASSICAL INFORMATION OVER QUANTUM CHANNELS

3.4 THE HOLEVO-SCHUMACHER-WESTMORELAND THEOREM

3.5 CLASSICAL COMMUNICATION OVER QUANTUM CHANNELS

3.6 BRIEF SUMMARY OF CLASSICAL CAPACITIES

3.7 MULTILEVEL QUANTUM SYSTEMS AND QUDIT CHANNELS

3.8 THE ZERO-ERROR CAPACITY OF A QUANTUM CHANNEL

3.9 FURTHER READING

CHAPTER 4 THE QUANTUM CAPACITY OF QUANTUM CHANNELS

4.1 INTRODUCTION

4.2 TRANSMISSION OF QUANTUM INFORMATION

4.3 QUANTUM COHERENT INFORMATION

4.4 THE ASYMPTOTIC QUANTUM CAPACITY

4.5 RELATION BETWEEN CLASSICAL AND QUANTUM CAPACITIES OF QUANTUM CHANNELS

4.6 FURTHER READING

CHAPTER 5 GEOMETRIC INTERPRETATION OF QUANTUM CHANNELS

5.1 INTRODUCTION

5.2 GEOMETRIC INTERPRETATION OF THE QUANTUM CHANNELS

5.3 GEOMETRIC INTERPRETATION OF THE QUANTUM INFORMATIONAL DISTANCE

5.4 COMPUTATION OF SMALLEST QUANTUM BALL TO DERIVE THE HSW CAPACITY

5.5 ILLUSTRATIVE EXAMPLE

5.6 GEOMETRY OF BASIC QUANTUM CHANNEL MODELS

5.7 GEOMETRIC INTERPRETATION OF HSW CAPACITIES OF DIFFERENT QUANTUM CHANNEL MODELS

5.8 FURTHER READING

CHAPTER 6 ADDITIVITY OF QUANTUM CHANNEL CAPACITIES

6.1 INTRODUCTION

6.2 ADDITIVITY OF CLASSICAL CAPACITY

6.3 ADDITIVITY OF QUANTUM CAPACITY

6.4 ADDITIVITY OF HOLEVO INFORMATION

6.5 GEOMETRIC INTERPRETATION OF ADDITIVITY OF HSW CAPACITY

6.6 CLASSICAL AND QUANTUM CAPACITIES OF SOME CHANNELS

6.7 THE CLASSICAL ZERO-ERROR CAPACITIES OF SOME QUANTUM CHANNELS

6.8 FURTHER READING

CHAPTER 7 SUPERACTIVATION OF QUANTUM CHANNELS

7.1 INTRODUCTION

7.2 THE NON-ADDITIVITY OF PRIVATE INFORMATION

7.3 CHANNEL COMBINATION FOR SUPERADDITIVITY OF PRIVATE INFORMATION

7.4 SUPERACTIVATION OF QUANTUM CAPACITY OF ZERO-CAPACITY QUANTUM CHANNELS

7.5 BEHIND SUPERACTIVATION: THE INFORMATION THEORETIC DESCRIPTION

7.6 GEOMETRICAL INTERPRETATION OF QUANTUM CAPACITY

7.7 EXAMPLE OF GEOMETRIC INTERPRETATION OF SUPERACTIVATION

7.8 EXTENSION OF SUPERACTIVATION FOR MORE GENERAL CLASSES

7.9 SUPERACTIVATION OF ZERO-ERROR CAPACITIES

7.10 FURTHER READING

CHAPTER 8 QUANTUM SECURITY AND PRIVACY

8.1 INTRODUCTION

8.2 QUANTUM KEY DISTRIBUTION

8.3 PRIVATE COMMUNICATION OVER THE QUANTUM CHANNEL

8.4 QUANTUM CRYPTOGRAPHIC PRIMITIVES

8.5 FURTHER READING

CHAPTER 9 QUANTUM COMMUNICATION NETWORKS

9.1 LONG-DISTANCE QUANTUM COMMUNICATIONS

9.2 LEVELS OF ENTANGLEMENT SWAPPING

9.3 SCHEDULING TECHNIQUES OF PURIFICATION

9.4 HYBRID QUANTUM REPEATER

9.5 PROBABILISTIC QUANTUM NETWORKS

9.6 CONCLUSIONS

9.7 FURTHER READING

CHAPTER 10 RECENT DEVELOPMENTS AND FUTURE DIRECTIONS

10.1 INTRODUCTION

10.2 QUBIT IMPLEMENTATIONS

10.3 QUANTUM CPUs

10.4 QUANTUM MEMORIES

10.5 FURTHER READING

NOTATIONS AND ABBREVIATIONS

NOTATIONS

ABBREVIATIONS

REFERENCES

INDEX

Cover Image: Brian Smith/iStockphoto

Copyright © 2013 by the Institute of Electrical and Electronics Engineers. All rights reserved.

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ISBN 978-1-118-00236-0

To my teachers and professors of physics: Peter Edes, Laszlo Gorbe O. SchP, Prof. Laszlo Orosz.

Sandor Imre

To Peter Nagy and my family.

Laszlo Gyongyosi

Marvelous, what ideas the young people have these days. But I don’t believe a word of it.

–Albert Einstein (1927)

PREFACE

Navigare necesse est! (Shipping is a must.)

–Ancient Roman saying

Quantum computation and information is a new, rapidly developing, interdisciplinary field. According to Moore’s law, the physical limitations of classical physics–based technologies could be reached within a few years. The transistors on a chip will be squeezed to the atomic scale, and during the next decade we will step into the “Quantum Age.”

In order to support the reader as he or she is entering the jungle of quantum phenomena–driven communications, our book offers a concise and up-to-date introduction to the plain and the secret quantum communications, quantum networking.

The book is well suited to a very broad audience, and no prior knowledge of the various properties of quantum communications channels is assumed. However, readers should have a basic knowledge of complex numbers, vectors, and matrices. The topics discussed include quantum information theory, quantum cryptography, quantum communications, quantum informational geometry, computational geometrical methods in quantum computations, and the implementation of quantum networks of the future. Since advanced quantum communications methods will be one of the most important areas in experimental future communications, our book may be of interest to scientists in a wide range of fields. Therefore, we recommend this book to graduate students, researchers, and practitioners in electrical engineering, physics, and computer science.

In order to make the discussion easier to parse, we have included the following features:

Tables:

analyses and comparisons of the main properties of quantum communication channels;

Illustrations

: more than 260 figures and graphs with well-designed and clear interpretations;

Equations

: more than 700 equations; all of our mathematical derivations are based on clear physical images that make even the most involved results comprehensible and clear. Moreover, the mathematical formalism is kept to the minimum needed to understand the key results.

Innovative methods of presentation:

for example, geometrical representations of quantum channels and methods that facilitate the perception of complex problems.

The book explains the future’s advanced quantum communications schemes and gives an overview of the security of these systems from an engineering viewpoint. Each chapter addresses an area of major and current research in quantum communications. We tailored the chapters so that they include most recent developments and results in the field of advanced quantum communications.

The Further Reading sections at the end of each chapter provide the historical background of the discussed topics and should help the reader find useful information related to the given chapter. At the back of the book, the Notations and Abbreviations section should also prove useful to the reader. Further supporting material regarding the chapters of the book with the errata is available online at the following web address: http://www.mcl.hu/aqc/.

The authors wish the reader a challenging but pleasant journey at the dawn of Quantum Age among the islands of the “Quantum Communications” archipelago.

Sandor ImreLaszlo Gyongyosi

CHAPTER 1

INTRODUCTION

Nothing exists except atoms and empty space; everything else is opinion.

—Democritus of Abdera (ca. 400 BC)

Quantum computing has demonstrated its usefulness in the last decade with many new scientific discoveries. The quantum algorithms were under intensive research during the last quarter of the twentieth century. However, after Shor published the prime factorization method in 1994, and Grover introduced the quantum search method in 1996, results in the field of quantum algorithms tapered off somewhat. In the middle of the ’90s, there was silence in the field of quantum algorithms and this did not change until the beginning of the present century. This silence has been broken by the solution of some old number theoretic problems, which makes it possible to break certain—and not just those that are based on RSA—very strong cryptosystems. Notably, these hard mathematical problems can now be solved by polynomial-time quantum algorithms. Later, these results were extended to other number theoretic problems, and the revival of quantum computing has been more intensive than ever.

These very straightforward quantum algorithms can be used only if there is a stable framework of physical implementations standing behind them. Many new techniques have been developed in the last decade to implement a quantum computer in practice, using linear optics, adiabatic systems, and entangled physical particles. By the end of the twentieth century, many new practical developments had been realized, and many novel results introduced in the field of quantum computation and quantum information processing.

Another important research field related to the properties of the physical implementations of quantum information focused on the decoherence and the precision of the measurement outcomes. Many researchers started to analyze the question of whether entanglement could help to increase the precision of quantum computation and the probabilities of the right measurement outcomes.

The main task of quantum complexity theory is to clarify the limitations of quantum computation and to analyze the relationship between classical and quantum problem classes. As the quantum computer becomes a reality, the classical problem classes have to be regrouped and new subclasses have to be defined. The most important question is the description of the effects of quantum computational power on NP-Complete problems. According to our current knowledge, quantum computers cannot solve NP-Complete problems; hence if a problem is NP-Complete in terms of classical complexity theory, then it will remain NP-Complete in quantum complexity theory, too. On the other hand, there are still many open questions, such as the complexity of quantum computations or the error-bounds of the various quantum algorithms, and it is expected that new results will be born in the near future.

1.1 EMERGING QUANTUM INFLUENCES

Efficient quantum algorithms that have been developed for breaking classical cryptographic systems could become a reality in the next decade. According to Moore’s law, the physical limitations of classical semiconductor-based technologies could be reached by 2020, and we and you, dear reader, will step into the Quantum Age.

Public key classical cryptography relies heavily on the complexity of factoring integers (or similar problems such as discrete logarithms). Quantum computers can use the Shor algorithm to efficiently break today’s public key cryptosystems. We will need a new kind of cryptography in the future. Because classical cryptographic methods in wired and wireless systems are vulnerable, new methods based on quantum mechanical principles have been developed.

To break classical cryptosystems, several new different quantum algorithms (besides Shor’s algorithm) can be developed and used. After quantum computers are built, today’s encrypted information will no longer stay secure, because although the computational complexity of these classical schemes makes it hard for classical computers to solve them, they are not hard for quantum computers! Using classical computers, the efficiency of code breaking is restricted to polynomial time; however, with a quantum computer these tasks can be completed exponentially faster.

1.2 QUANTUM INFORMATION THEORY

The theoretical background of communication over quantum channels is based on the fundamental results of quantum information theory. The actual state of quantum information theory reflects our current knowledge of the quantum world, and it also determines the success of quantum communication protocols and techniques.

The phenomena of the quantum world cannot be described by the fundamental results of classical information theory. Quantum information theory is the natural extension of the results of classical information theory. But it brings something new into the global picture and helps to complete the missing, classically indescribable, and even unimaginable parts. Quantum information theory lays down the theoretical background of quantum information processing and synthesizes it with other aspects of quantum mechanics, such as experimental quantum communications, secure and private quantum channels, or quantum error correction techniques. This field is the cornerstone of quantum communications and quantum information processing.

The primary employment of quantum information theory is to describe quantum channel capacities, to measure entanglement, and to analyze the information-theoretic security of quantum cryptographic primitives. In Figure 1.1, we highlighted some important parts of quantum information theory.

Figure 1.1. Quantum information theory provides the theoretical background for various subjects in quantum information processing.

With the help of quantum information theory, information transmission through the quantum channel can be discussed for both classical and quantum information. The former can be defined by a formula very similar to the classical Shannon channel coding theorem. On the other hand, the latter challenge has opened new dimensions in information transmission. As we will see, there are still many open questions in quantum information theory. The various channel capacities of quantum channels have been proven to be nonadditive in general; however, there are many special cases for which strict additivity holds. These fundamental questions will be discussed in detail in this book.

As follows from the connection defined between classical and quantum information theory, every classical and quantum protocol can be described by using the elements of quantum information theory. The definitions and main results of quantum information theory, such as the density matrix, entanglement, measurement operators, quantum Shannon theory, von Neumann entropy, quantum relative entropy, Holevo bound, fidelity, and quantum informational distance, are discussed in Chapter 2.

1.3 DIFFERENT CAPACITIES OF QUANTUM CHANNELS

The concept of a quantum channel models communication at an abstract level, thus it does not require the deep analysis of the various physical systems. Instead it will be sufficient to distill their essence from the information transmission point of view.

The capacity of a quantum channel gives us the rate at which classical or quantum information increases with each use of the quantum channel. We can define the single-use and the asymptotic capacity of the quantum channel: the first quantifies the information that can be sent through a single use of the channel, the latter quantifies the information that can be transmitted if arbitrarily many uses of the quantum channel are allowed.

Many capacities can be defined for a quantum channel: it has a classical capacity, a quantum capacity, a private capacity, an entanglement assisted capacity, and a zero-error capacity (classical and quantum). Some of these have also been defined in classical information theory, but many of these are completely new.

The classical capacity of a quantum channel was first investigated by Holevo, who showed that from a two-level quantum state, or qubit, at most one bit of classical information can be extracted. This theory is not contradictory to the fact that the description of a quantum state requires an infinite number of classical bits. As we will see, this “one classical bit bound” holds just for two-level quantum states (the qubits) since, in the case of a d-level quantum state (called qudits) this bound can be exceeded.

The classical capacity of a quantum channel can be measured in different settings, depending on whether the input contains tensor product or entangled quantum states, and the output is measured by single or by joint measurement settings. These input and output combinations allow us to construct different channel settings, and the capacities in each case will be different. This is a completely new phenomenon in comparison to classical communication systems, where this kind of differentiation is not possible. The additivity of a quantum channel depends on the encoding scheme and on the measurement apparatus that is used for measuring the quantum states. If we use product input states, there is no entanglement among them, and if we do not apply joint measurement on the output, then the classical capacity of a quantum channel will be additive, which means that the capacity can be achieved by a single use. If we use joint measurement on the outputs, then such additivity is not guaranteed, which also suggests that in general the classical capacity is not additive. We note that many questions are still not solved in this field, as we will see later in Chapter 3 where the properties of classical capacity of quantum channels are discussed in detail.

The classical capacity of a quantum channel was formulated by Holevo, Schumacher, and Westmoreland [Holevo98], [Schumacher97], and it is known in quantum information theory as the HSW channel capacity. While the classical capacity measures classical information transmission over a noisy quantum channel, the quantum capacity of a quantum channel describes the amount of quantum information that can be transmitted through a noisy quantum channel. The formula of quantum capacity was introduced by Lloyd, Shor, and Devetak in [Lloyd97], [Shor02], [Devetak03], and after the inventors it is called the LSD channel capacity. Both the HSW and the LSD channel capacities provide lower bounds on the ultimate limit for a noisy quantum channel to transmit classical or quantum information. One of the most important applications of quantum capacity is the transmission of entanglement. The quantum error-correction techniques are developed for the optimization of quantum capacity in a noisy environment.

As in the case of classical channel capacity where we will use the Holevo information as measure, for quantum capacity we will introduce a completely different correlation measure, the concept of quantum coherent information. We note that the generalized quantum channel capacity cannot be measured by the single-use version of quantum coherent information (or at least, it works just for some special channels), hence we have to compute the asymptotic form. This fact also implies that the additivity of the quantum capacities will be violated, too. These questions and the still unsolved questions are described in Chapter 4, and we give a very nice implementation for their use in quantum communications.

Chapters 3 and 4 can be regarded as a “practical” application of the results described in the Chapter 2. While Chapter 2 provides a strong information theoretic background, Chapters 3 and 4 bring these results to reality, and converts them to a tangible format.

Exploiting the combination of the elements of quantum information theory and computational geometry, many still open questions regarding quantum channel capacities can be answered in a rather different way by comparison of the well-known methods. A plausible geometrical picture can be assigned to each channel model, and instead of numerical calculations on their capacities, one can utilize the much more straightforward geometric representation. This interesting and rather surprising field is introduced in Chapter 5.

1.4 CHALLENGES RELATED TO QUANTUM CHANNEL CAPACITIES

One very interesting and important problem regarding the capacity of quantum channels is whether entanglement between states can improve sending classical information through quantum channels. This problem is known as the additivity problem.

The question of additivity has emerged from an attempt to find an unambiguous formula for the capacity of a noisy quantum channel. The accessible classical information from continuous quantum degrees of freedom is limited. This limitation stands behind the additivity of quantum channel capacity. Up to this day, strict additivity for quantum channel capacity has been conjectured but not proven, and the additivity property of quantum channels is still an exciting subject of current research.

We are going to discover this field using an elegant geometrical interpretation in Chapter 6 where we also investigate whether entanglement across input states could help to enhance the transmission of information on quantum channels—as entanglement can help in other problems in quantum computation. To walk around this question different channel models will be studied.

The other, rather challenging current problem of quantum information theory is called superactivation. It makes it possible to use zero-capacity quantum channels to transmit information! The effect of superactivation was discovered in 2008, and later, in 2009, it was shown that both the classical and quantum zero-error capacities of a quantum channel can be superactivated in certain cases. The complete theoretical background of the superactivation is currently unsolved; however, we know that it is based on the nonadditivity (i.e., on the extreme violation of additivity) of the various quantum channel capacities and the entangled input states. Chapter 7 explains the theoretical background of superactivation of quantum capacity, quantum zero-error capacity, and the zero-error capacity of quantum channels, and we show how the various channel capacities of quantum channels can be superactivated.

1.5 SECRET AND PRIVATE QUANTUM COMMUNICATION

Using current classical computer architectures, the brute-force breaking of today’s public key cryptosystems could take an extremely long time, since the problem of factoring large integers in polynomial time is still not solved. On the other hand, if we use quantum computers instead of classical computer architectures, the factorization problem can be solved with polynomial complexity. This famous discovery of Shor’s was successfully demonstrated experimentally in 2001, and it revealed the fact that classical cryptographic methods will not be able to provide security in the future. However, the one-time pad (OTP) method could achieve perfect theoretical security in classical systems, but according to the challenges of key-distribution, these methods cannot be efficiently applied in practice. The perfect security of the OTP method was proved by Claude Shannon, but the problem of key-distribution in classical systems is not solvable according to the problem of copying information.

The status of the security of classical cryptosystems will change dramatically after the advent of quantum computers. Currently used and well-known cryptosystems, such as the RSA algorithm, Diffie-Hellman method, elliptic curve cryptography, Buchmann-Williams key exchange scheme, or the algebraically homomorphic cryptosystems, will be broken immediately when quantum computers become reality. On the other hand, not every problem can be solved by the exponential increase in speed. Currently, we conjecture that NP-Complete problems do not have efficient quantum algorithmic solution—at least, currently we have not found them—hence finding quantum mechanics-based solutions for exponential speedups of these problems is an important question and task in the future. However, currently all the classical cryptosystems are based on non NP-Complete problems, and the exponential speedup of quantum algorithms, and the theoretical weakness of these schemes, can be exploited quantum mechanically. We note that there have been some attempts to develop classical cryptosystems, which seem immune against quantum attacks, such as hash-, code- or lattice-based cryptography, the multivariate quadratic equation cryptosystems or secret-key cryptography, but these methods are neither efficiently implementable in practice, nor protected by NP-Complete problems. The proofs of the resistance of these classical schemes against quantum computers are based on the loose assumption that there will be no quantum algorithm in the future for an attack on these classical schemes better than Shor’s or Grover’s algorithm. As we have seen in history, cryptoanalysts could be easily disposed to believe that the best possible attack against the analyzed protocol has been found (see Enigma in WWII) or that it will not be possible in the future to attack the scheme with a better and faster method. This is very misleading and a dangerous belief.

As a general conclusion, all the classical cryptographic systems are resistant to the attacks of classical computers only. At this point, we have to raise the question: Does there exist a cryptographic scheme at all that is proved to guarantee unconditionally secure communication in the Quantum Age? The answer is definitely yes: it is called quantum cryptography.

In today’s communications networks, the widespread use of optical fiber and passive optical elements allows us to use quantum cryptography. In order to spread quantum cryptography, interfaces must be implemented that are able to manage together both quantum and classical channels. In practical implementations of quantum key distribution (QKD) protocols, Alice, the sender, uses weak coherent pulses (WCP) instead of a single photon source. As has been shown, WCP-based protocols have a security problem, since an eavesdropper can perform a photon number–splitting attack against the protocol. These kinds of attacks are based on the fact that some weak coherent pulses contain more than one photon in the same polarization state, which provides information to the eavesdropper without any disturbance. One of the main advantages of current practical QKD schemes is that the quantum communications in these methods can be implemented by using conventional optical devices, such as laser diodes, wave plates, beamsplitters, and detectors.

We will show that quantum cryptographic primitives can be extended to other types of secret message transmission. Quantum cryptography is just one possible application of the fundamental properties of quantum mechanics for secret information transmission; however, in the last decade many new, but not quantum cryptography–based, cryptographic primitives have been developed.

Some of the most important fields among the new, post-QKD results in secret quantum communications protocols are quantum authentication, quantum fingerprinting, and quantum privacy protocols. The theoretical background of quantum digital signature is based on classical public-key methods; however, classical keys cannot be used here. Although there have been many attempts to realize a practical quantum public key method, the complexity of the protocol is still so high that it is impossible to use it efficiently in practice. On the other hand, quantum fingerprinting is a much more achievable protocol. It makes it possible to generate a “hash” of a large data set, similar to classical hashing strategies. The hash of the quantum states can be computed in a relatively easy way, without extensive computational costs.

The development of quantum privacy is one of the most important results of the post-QKD research in quantum information processing. The privacy of the quantum channel can be ensured only if with every quantum state, the sender sends to the receiver two classical bits. The classical bits are derived from classical randomness, which randomness is shared between the parties. In the newer versions of quantum privacy protocols, the classical randomness can be changed to quantum randomness. We note that currently, the complexities of both the classical and the quantum-based privacy protocols are rather high. On the other hand, it has been proven that one classical bit per qubit is sufficient for an absolutely secure privacy communication, which allows using the protocol in various communications scenarios in future quantum networks. Privacy can be extended to remote database access, hence these protocols will have more importance in future quantum communications. In classical networks, private information retrieval is possible only if there is some shared randomness that can be encountered in the system. Recently, it has been shown that the privacy of “quantum servers” of a certain communication network can be ensured without shared information between them, and the privacy of the parties can be preserved if the quantum servers are cheating.

Digital signatures and the authentication of messages are well-known problems in classical communications methods, with many available protocols. These digital signatures and authentication schemes can be translated from classical to quantum systems; however, there are many differences. The quantum authentication can be realized in the case of quantum systems, too; however, the method requires both classical and quantum communications. An important result in this field is that the number of key bits required for the authentication is at least two times greater than the number of quantum states to be authenticated. In the case of a quantum digital signature schemes, the main task is not the integrity of the message, it is rather the validation of the personality of the sender. In quantum digital signature schemes, the public key consists of quantum states, hence the no-cloning theorem makes it impossible to distribute it among many parties. On the other hand, Alice can prepare the same state many times, hence these states can be used as public keys; however, the cost of the unconditional security of quantum public key methods is relatively high in comparison to classical schemes. These quantum protocols are currently still “under research,” hence it could have application in advanced quantum communications schemes of the future.

Quantum secret sharing, quantum data hiding, and quantum fingerprinting are also very new fields in secret quantum communications. The idea behind quantum secret sharing is that the parties of the communication get an incomprehensible secret message, and the secret can be recovered only if the parties start to communicate with each other. The secret quantum message is encoded in a joint quantum state, and the reduced density matrix computed by an individual party gives only zero bit information from the secret. In the quantum data hiding protocol, the parties get classical information; however, the decoding of the message is possible only if the parties have a quantum channel. This type of security scheme can also be applied in the quantum secret sharing protocol. In the quantum secret sharing scheme the parties receive quantum states, and after this reception, the parties have to use classical communication. The properties of private quantum communications and the most relevant quantum protocols will be discussed in Chapter 8. In the first part of the chapter we overview the possible attacks against the quantum key distribution protocols, then we study the quantum bit commitment protocol, quantum fingerprinting, and quantum public key cryptography.

1.6 QUANTUM COMMUNICATIONS NETWORKS

The hardest problem in future quantum communications is the long-distance delivery of quantum information. Since arbitrary unknown quantum states cannot be copied, the amplification of quantum bits is more complex compared with classical communications. The success of future long-distance quantum communications and global quantum key distribution systems strongly depends on the development of efficient quantum repeaters. It is not simply a signal amplifier, in contrast to the classical repeaters.

There are several differences between a classical and a quantum repeater. The quantum repeater nodes create highly entangled EPR states with high fidelity of entanglement. The entangled quantum states can be sent through the quantum channel as single quantum states or as multiple photons. In the first case the fidelity of the shared entanglement could be higher; however, it has lower probability of success in practice, since these quantum states can be lost easily on the noisy quantum channel. In the second case, the loss probability is lower; however, the fidelity will not as high as in the single photon case. In order to recover fidelity of entanglement from noisy quantum states, purification is needed.

Sharing of quantum entanglement plays a critical role in quantum repeaters. The fidelity of the entanglement decreases during the transmission through the noisy quantum channel. Therefore, in practical implementations, the quantum entanglement cannot be distributed over very long distances; instead, the EPR states are generated and distributed between smaller segments.

A practical approach of the quantum repeater is called the hybrid quantum repeater. It uses atomic-qubit entanglement and optical coherent state communication. In practice, the repeaters are connected by optical fibers, through which the entangled quantum states are sent. Quantum repeaters use the purification protocol to increase the fidelity of transmission. The rate of entanglement purification depends on the fidelity of the shared quantum states, since the purification step is a probabilistic process. Moreover, the success probability of the purification of the entangled quantum states depends on the fidelity of the entangled states: if the fidelity of entanglement of the shared state is low, then the success probability of its purification will be also low. Another important disadvantage of the purification algorithm is that it requires much classical information exchange between the quantum nodes.

Quantum computing offers fundamentally new solutions in the field of computer science. The classical biologically inspired self-organizing systems have increasing complexity and these constructions do not seem to be suitable for handling the service demands of the near future. Quantum probabilistic networks may be able to replace classical solutions with significantly higher efficiency. Using the quantum probabilistic nature, truly random behavior can be added to the self-organizing processes of biological networks.

The cell-organized, quantum mechanics–based cellular automata models have many advantages over classical models and circuits. For a quantum cellular machine, every cell is a finite-dimensional quantum system with unitary transformations, and there is a difference between the axiomatic structure of classical and quantum versions of cellular automata.

In Chapter 9, we give a brief overview of the possible solutions of future quantum-based networks and long-distance quantum communications. In the first part of the chapter, we describe long-distance quantum communications and the quantum repeater, while at the end of the chapter, we discuss the basic properties of quantum probabilistic networks.

1.7 RECENT DEVELOPMENTS AND FUTURE DIRECTIONS

In the last chapter of our book, we present an overview of the “experimental side” of quantum computation, the recent developments and the physical implementations. Quantum information processing uses the results of quantum mechanics and integrates them with the elements of information processing. Quantum communications may have an important role to play in the future’s secret quantum networks, in which truly unbreakable cryptographic schemes will be necessary. As an important future direction, quantum cryptographic schemes can be implemented to realize unconditionally secure communication. But the security of quantum cryptography cannot be the solution for every possible security problem.

In future telecommunications networks, practical quantum communications will be implemented in combination with classical systems, using the elements of classical data processing. These classical parts will be integrated into the less critical parts of the protocols, hence these solutions will not decrease the level of security. The currently implemented practical quantum networks all contain some classical elements, and in the future these schemes cannot be eliminated. In future quantum networks, the information will be protected by the no-cloning theorem.

The physical implementations of quantum communications networks will be based mainly on optical communications. Developments in the physical layer tend toward single photon sources, single photon detecting modules, and a reduction in the noise of the optical quantum channels. The loss due to the optical fibers determines the efficiency of the quantum communications, hence it will be an important task to develop implementable quantum repeaters in future. On the other hand, as opposed to the situation with classical bits, quantum bits cannot be copied, hence quantum memories will have an important role in the expansion of quantum repeaters. As the challenges of the physical layer become resolved, the next step could be the design of the communication between the physical and the higher layers, and the controlling and managing of the processes of the quantum layer by the classical one. All real life–based quantum communications networks are complex systems, with many degrees of freedom. The theoretical quantum protocols are just idealizations of the practical systems, without the imperfections caused by the environment. It is an important task in future developments to quantify experimentally the efficiency and the security that can be achieved in the noisy practical environment.

Another important direction is the development of a scalable quantum computer. Currently, only small scale implementations have been realized in the laboratory. The architecture of quantum computers can be based on various physical implementations, such as magnetic resonance, optical lattices, silicon-based approaches, electrons, and quantum cavities. In contrast to the laboratory environment, the development of scalable implementations is a more challenging problem. To realize quantum computers in practice, or to use the quantum Internet, more efforts will be needed. On the other hand, there is no other way. According to Moore’s law, we will step into the Quantum Age, very soon.

The ways and steps we just started could be different, but one thing is certain: quantum information will be the key to the revolution of the future’s information processing and telecommunications.

CHAPTER 2

INTRODUCTION TO QUANTUM INFORMATION THEORY

You don’t understand quantum mechanics, you just get used to it.

—John von Neumann

Having read the previous introductory chapter one can imagine that classical information theory must be extended fundamentally to cover the colorful phenomena of the quantum world. In this chapter, we introduce the reader to the definitions and main results of quantum information theory.

Quantum information processing exploits the quantum nature of information. It can help to resolve still open scientific problems and expand the boundaries of classical computation and communications systems. However, if one would like to use and control these promising quantum systems, he or she has to be familiar with the theoretical possibilities and limits revealed by quantum information theory. Its results allow designing quantum communications protocols, investigating the capacities of quantum channels, and constructing appropriate measurement strategies. Quantum information processing offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems.

This chapter is organized as follows. In the first part, we summarize the basic definitions and formulas of quantum information theory. We introduce the reader to the properties of density matrices, quantum entropy function, quantum mutual information, and quantum conditional entropy. Next, we discuss the measurement of the entanglement in quantum states. Then, we describe the encoding of quantum states and the meaning of Holevo information. Finally, we introduce the reader to the quantum noiseless channel coding theorem and the compression of quantum information.

The complete historical background with the survey of the most important works can be found in Section 2.10, Further Reading.

2.1 INTRODUCTION

The world of quantum information processing is describable with the help of quantum information theory and quantum Shannon theory, which are the main subjects of this chapter. We will provide an overview of the most important differences between the compressibility of classical bits and quantum bits, and between the capacities of classical and quantum communications channels. To represent classical information with quantum states, we might use pure orthogonal states. In this case there is no difference between the compressibility of classical and quantum bits. But what happens if we use non-orthogonal quantum states?

Similarly, a quantum channel can be used with pure orthogonal states to realize classical information transmission, or it can be used to transmit non-orthogonal states or even quantum entanglement. Information transmission also can be approached using the question of whether the input consists of unentangled or entangled quantum states. This leads us to say that for quantum channels many new capacity definitions exist in comparison to a classical communications channel.

Quantum information theory also has relevance to the discussion of the capacity of quantum channels and to information transmission and storage in quantum systems.

As we will see in this chapter, while the transmission of product states can be described similar to classical information, conversely, the properties of quantum entanglement cannot be handled by the elements of classical information theory. Of course, the elements of classical information theory can be viewed as a subset of the larger and more complex quantum information theory.

The relation between classical and quantum information theory is illustrated in Figure 2.1.

Figure 2.1. The elements of classical information theory can be viewed as a subset of quantum information theory.

Before we can begin our introduction to quantum information theory, we have to make a clear distinction between quantum information theory and quantum information processing. Quantum information theory is a generalization of the elements and functions of classical information theory to describe the properties of quantum systems, storage of information in quantum systems, and the various quantum phenomena of quantum mechanics. While quantum information theory aims to provide a stable theoretical background, quantum information processing is a more general and rather experimental field: it answers what can be achieved in engineering with the help of quantum information. Quantum information processing includes the computing, error-correcting schemes, quantum communication protocols, field of communication complexity, and so on.

This chapter gives an overview of quantum information theory. This field began to grow with exponential speed in the last decade; we summarize the most important results and also discuss the most recent developments.

2.1.1 Brief History

The character of classical information and quantum information is significantly different. There are many phenomena in quantum systems that cannot be described classically, such as entanglement, which makes it possible to store quantum information in the correlation of quantum states. Entangled quantum states are named to EPR states after Einstein, Podolsky and Rosen, or Bell states, after J. Bell.

Quantum entanglement was discovered in the 1930s, and it may still yield many surprises in the future. Currently it is clear that entanglement has many classically indescribable properties and many new communication approaches based on it. Quantum entanglement plays a fundamental role in advanced quantum communications, such as teleportation and quantum cryptography.

The elements of quantum information theory are based on the laws of quantum mechanics. The main results of quantum information processing were laid down during the end of the twentieth century, the most important results being stated by Feynman, Bennett, DiVincenzo, Devetak, Deutsch, Holevo, Lloyd, Schumacher, Shor, and many more. After the basic concepts of quantum information processing had been stated, researchers started to look for efficient quantum error correction schemes and codes, and started to develop the theoretical background of fault-tolerant quantum computation. The main results from this field were presented by Bennett, Schumacher, Gottesman, Calderbank, Preskill, Knill, and Kerckhoff. On the other hand, there are still many open questions about quantum computation. The theoretical limits of quantum computers were discovered by Bennett, Bernstein, Brassard, and Vazirani: quantum computers can provide at best a quadratic reduction in the complexity of search-based problems, hence if we give an NP-complete problem to quantum computer, it still cannot solve it. Recently, the complexity classes of quantum information processing have been investigated, and many new classes and lower bounds have been found.

By the end of the twentieth century, many advanced and interesting properties of quantum information theory had been discovered, and many possible applications of these results in future communication had been developed. One of the most interesting revealed connections was that between quantum information theory and the elements of geometry. The space of quantum states can be modeled as a convex set which contains points with different probability distributions, and the geometrical distance between these probability distributions can be measured by the elementary functions of quantum information theory, such as von Neumann entropy or the quantum relative entropy function. The connection between the elements of quantum information theory and geometry leads us to the application of advanced computational geometrical algorithms to quantum space, to reveal the still undiscovered properties of quantum information processing, such as the open questions on the capacities of the quantum channels or their additivity properties. The connection between the Hilbert space of quantum states and the geometrical distance can help us to reveal the fantastic properties of quantum bits and quantum state space.

Several functions have been defined in quantum information theory to describe the statistical distances between the states in the quantum space: one of the most important is the quantum relative entropy function, which plays a key role in the description of entanglement, too. This function has many different applications, and maybe this function plays the most important role in the questions regarding the capacity of quantum channels. The possible applications of the quantum relative entropy function have been studied by Schumacher and Westmoreland and by Vedral.

Quantum information theory plays fundamental role in the description of the data transmission through quantum communication channels. Most questions concerning quantum channel capacities have already been answered. On the other hand, at the dawn of this millennium new problems have arisen, whose solutions are still not known, and which have opened the door to many new promising results such as the superactivation of zero-capacity quantum channels in 2008, and then the superactivation of the zero-error capacities of the quantum channels in 2009 and 2010.

One of the earliest works on the capacities of quantum communication channels was published in the early 1970s. Along with other researchers, Holevo showed that there are many differences between the properties of classical and quantum communication channels, and illustrated this with the benefits of using entangled input states. Later, he also stated that quantum communication channels can be used to transmit both classical and quantum information. Next, many new quantum protocols were developed, such as teleportation or superdense coding. About thirty years after Holevo published his work, he, with Schumacher and Westmoreland, presented one of the most important results in quantum information theory, called the Holevo-Schumacher-Westmoreland (HSW) theorem. As we will see in Chapter 3, the HSW theorem is a generalization of the classical noisy channel coding theorem from classical information theory to a noisy quantum channel. The HSW theorem is also called the product-state classical channel capacity theorem of a noisy quantum channel. The understanding of the classical capacity of a quantum channel was completed in 1997 by Schumacher and Westmoreland, and in 1998 by Holevo, and it has tremendous relevance in quantum information theory, since it was the first to give a mathematical proof that a noisy quantum channel can be used to transmit classical information in a reliable form. The HSW theorem was a very important result in the history of quantum information theory, though on the other hand it raised a lot of questions regarding the transmission of classical information over general quantum channels. For the complete historical background with references see Section 2.10, Further Reading.

An interesting historical fact is that the HSW theorem from 1997 used the concept of Holevo information, which was discovered about twenty-five years before, in 1973. Holevo in 1998 used the same definition of subspaces as Bennett in 1999; however, he applied it to a completely different problem.

2.2 BASIC DEFINITIONS AND FORMULAS

In this section, we summarize the basic definitions and formulas of quantum information theory. Those readers who are familiar with density matrices, entropies, and the like may run through this chapter and focus only on notations (the collection of the notations used in this book can be found in Notations and Abbreviations) and can return later if it is required when processing another chapter.

2.2.1 Density Matrices and Trace Operator

We introduce a basic concept of quantum information theory, called the density matrix.

Before we start to discuss the density operator, we introduce some terms. An n × n square matrix A is called positive-semidefinite if 〈ψ|A|ψ〉 is a non-negative real number for every vector |ψ〉. If A = A†, that is, A has Hermitian matrix and the {λ1, λ2, … λn} eigenvalues of A are all non-negative real numbers then it is positive-semidefinite. This definition has an important role in quantum information theory, since every density matrix is positive-semidefinite. It means, for any vector |ψ〉 the positive-semidefinite property says that

(2.1) 

In (2.1) the density matrix is denoted by ρ, and it describes the system by the classical probability weighted sum of possible states

(2.2) 

where |ψi〉 is the ith system state occurring with classical probability pi. As can be seen, this density matrix describes the system as a probabilistic mixture of the possible known states, the so-called pure states. For pure state |ψ〉 the density matrix is ρ = |ψ〉〈ψ| and the rank of the matrix is equal to one. Trivially, classical states, for example |0〉 and |1〉 are pure, however, if we know that our system is prepared to the superposition then this state is pure, too. Clearly speaking, while superposition is a quantum linear combination of orthonormal basis states weighted by probability amplitudes, mixed states are a classical linear combination of pure superpositions (quantum states) weighted by classical probabilities.

The density matrix contains all the possible information that can be extracted from the quantum system. It is possible that two quantum systems possess the same density matrices: in this case, these quantum systems are called indistinguishable, since it is not possible to construct a measurement setting, which can distinguish between the two systems.

The density matrix ρ of a simple pure quantum system which can be given in the state vector representation |ψ〉 = α|0〉 + β|1〉 can be expressed as the outer product of the ket and bra vectors, where bra is the transposed complex conjugate of ket, hence for the density matrix is

(2.3) 

The density matrix contains the probabilistic mixture of different pure states, representation of which is based on the fact that the mixed states can be decomposed into the weighted sum of pure states.

To reveal important properties of the density matrix, we introduce the concept of the trace operation. The trace of a density matrix is equal to the sum of its diagonal entries. For an n × n square matrix A, the Tr trace operator is defined as

(2.4) 

where aii are the elements of the main diagonal. The trace of the matrix A is also equal to the sum of the eigenvalues of its matrix.

The eigenvalue is the factor by which the eigenvector changes if it is multiplied by the matrix A, for each eigenvector. The eigenvectors of the square matrix A are those non-zero vectors whose direction remains the same to the original vector after being multiplied by the matrix A. It means the eigenvectors remain proportional to the original vector. For square matrix A, the non-zero vector v is called eigenvector of A, if there is a scalar λ for which

(2.5) 

where λ is the eigenvalue of A corresponding to the eigenvector v.

The trace operation gives us the sum of the eigenvalues of positive-semidefinite A, for each eigenvector, hence

(2.6) 

Using the eigenvalues, the spectral decomposition of density matrix ρ can be expressed as

(2.7) 

where |φi〉 are orthonormal vectors.

The trace is a linear map, hence for square matrices A and B

(2.8) 

and

(2.9) 

where s is a scalar. Another useful formula is that for m × n matrix A and n × m matrix B,

(2.10) 

which holds for any matrices A and B for which the product matrix AB is a square matrix, since

(2.11) 

Finally, we mention that the trace of a matrix A and the trace of its transpose AT are equal, hence

(2.12) 

If we take the conjugate transpose A* of the m × n matrix A, then we will find that

(2.13) 

which will be denoted by 〈A, A〉 and is called the inner product. For matrices A and B, the inner product is 〈A, B〉 = Tr(B*A), which can be used to define the angle between the two vectors. The inner product of two vectors will be zero if and only if the vectors are orthogonal.

As we have seen, the trace operation gives the sum of the eigenvalues of matrix A, and this property can be extended to the density matrix, hence for each eigenvector λi of density matrix ρ

(2.14) 

Now, having introduced the trace operation, we apply it to a density matrix. If we have an n-qubit system being in the state then

(2.15) 

where we exploited the relation for unit-length vectors |ψi〉

(2.16) 

Thus the trace of any density matrix is equal to one

(2.17) 

The trace operation can help to distinguish pure and mixed states since for a given pure state ρ

(2.18) 

while for a mixed state σ,

(2.19) 

where and , where ωi are the eigenvalues of density matrix σ.

Similarly, for a pure entangled system ρEPR

(2.20) 

while for any mixed subsystem σEPR of the entangled state (i.e., for a half-pair of the entangled state), we will have

(2.21) 

The density matrix also can be used to describe the effect of a unitary transform on the probability distribution of the system. The probability that the whole quantum system is in |ψi〉 can be calculated by the trace operation.

If we apply unitary transform U to the state , the effect can be expressed according to the Second Postulate of Quantum Mechanics as follows

(2.22) 

What will happen with the density matrix if the applied transformation is not unitary? To describe this case, we introduce a more general operator denoted by G, and with the help of this operator the transform can be written as

(2.23) 

where for every matrix Ai. In this sense, operator G describes the physically admissible or completely positive trace preserving (CPTP) operations. The application of a CPTP operator G on density matrix ρ will result in a matrix G(ρ), which in this case is still a density matrix.

Now we can summarize the two most important properties of density matrices:

1. The density matrix ρ is a positive-semidefinite matrix; see (2.1).

2. The trace of any density matrix ρ is equal to 1; see (2.15).

2.2.2 Quantum Measurement

Now, let us turn to measurements and their relation to density matrices. Assuming a projective measurement device (we will give the formulas for general measurement in Section 2.2.4) defined by measurement operators—that is, projectors {Pj}. The projector Pj is a Hermitian matrix, for which and . According to the Third Postulate of Quantum Mechanics the trace operator can be used to give the probability of outcome j belonging to the operator Pj in the following way:

(2.24) 

After the measurement, the measurement operator Pj leaves the system in a post-measurement state

(2.25) 

If we have a pure quantum state |ψ〉 = α|0〉 + β|1〉, where α = 〈0|ψ〉 and β = 〈1|ψ〉. Using the trace operator, the measurement probabilities of |0〉 and |1〉 can be expressed as

(2.26) 

and

(2.27) 

in accordance with our expectations.

Let us assume we have an orthonormal basis M = {|x1〉〈x1|, … , |xn〉〈xn|} and an arbitrary (i.e., non-diagonal) density matrix ρ. The set of Hermitian operators Pi = {|xi〉〈xi|} satisfies the completeness relation, where Pi = |xi〉〈xi| is the projector over |xi〉, that is, quantum measurement operator Mi = |xi〉〈xi| is a valid measurement operator. The measurement operator Mi projects the input quantum system |ψ〉 to the pure state |xi〉 from the orthonormal basis M = {|x1〉〈x1|, … , |xn〉〈xn|}. Now, the probability that the quantum state |ψ〉 is after the measurement in basis state |xi〉 can be expressed as

(2.28) 

In the computational basis {|x1〉, … , |xn〉}, the state of the quantum system after the measurement can be expressed as

(2.29) 

and the matrix of the quantum state ρ′ will be diagonal in the computational basis {|xi〉}, and can be given by

(2.30) 

To illustrate it, let assume we have an initial (not diagonal) density matrix in the computational basis {|0〉, |1〉} for example |ψ〉 = α|0〉 + β|1〉 with p = |α|2 and 1 − p = |β|2 as

(2.31) 

and we have orthonormal basis M = {|0〉〈0|, |1〉〈1|}. In this case, the after-measurement state can be expressed as

(2.32) 

As can be seen, the matrix of ρ′ is a diagonal matrix in the computational basis {|0〉, |1〉}. Eq. (2.31) and (2.32) highlight the difference between quantum superpositions (probability amplitude weighted sum) and classical probabilistic mixtures of quantum states.

Now, let us see the result of the measurement on the input quantum system ρ

(2.33) 

For the measurement operators M0 = |0〉〈0| and M1 = |1〉〈1| the completeness relation holds

(2.34) 

Using input system ρ = |ψ〉〈ψ|, where |ψ〉 = α|0〉 + β|1〉, the state after the measurement operation (see later, in (2.64)) is

(2.35) 

As we have found, after the measurement operation M(ρ), the off-diagonal entries will have zero values, and they have no relevance. As follows, the initial input system ρ = |ψ〉〈ψ| after operation M becomes

(2.36) 

2.2.2.1 Orthonormal Basis Decomposition 

Let assume we have orthonormal basis {|b1〉, |b2〉, … , |bn〉}, which basis can be used to rewrite the quantum system |ψ〉 in a unique decomposition

(2.37) 

with complex bi. Since 〈ψ|ψ〉 = 1, we can express it in the form

(2.38) 

where is the complex conjugate of probability amplitude bi, thus |bi|2 is the probability pi of measuring the quantum system |ψ〉 in the given basis state |bi〉, that is,

(2.39) 

Using (2.2), (2.37), and (2.38), the density matrix of quantum system |ψ〉 can be expressed as

(2.40) 

This density matrix is a diagonal matrix with the probabilities in the diagonal entries

(2.41) 

The diagonal property of density matrix (2.40) in (2.41) can be checked, since the elements of the matrix can be expressed as

(2.42) 

where .

2.2.2.2 The Projective and POVM Measurement 

The projective measurement is also known as the von Neumann measurement and formally can be described by the Hermitian operator , which has the spectral decomposition

(2.43) 

where Pm is a projector to the eigenspace of with eigenvalue λm. For the projectors

(2.44) 

and they are pairwise orthogonal. The measurement outcome m corresponds to the eigenvalue λm, with measurement probability

(2.45) 

When a quantum system is measured in an orthonormal basis |m〉, then we make a projective measurement with projector Pm = |m〉〈m|, thus (2.43) can be rewritten as

(2.46) 

The POVM (positive operator valued measurement) is intended to select among the non-orthogonal states and is defined by a set of POVM operators , where

(2.47) 

and since we are not interested in the post-measurement state the exact knowledge about measurement operator i is not required. For POVM operators i the completeness relation holds,

(2.48) 

For the POVM the probability of a given outcome n for the state |ψ〉 can be expressed as

(2.49) 

The POVM also can be imagined as a “black box,” which outputs a number from 1 to m for the given input quantum state ψ, using the set of operators

(2.50) 

where {1, … , m} are responsible for distinguishing m different typically non-orthogonal states, that is, if we observe i ∈ [1, m] on the display of the measurement device we can be sure that the result is correct. However, because unknown non-orthogonal states can not be distinguished with probability 1, we have to introduce an extra measurement operator, m+1, as the price of the distinguishability of the m different states and if we obtain m+1 as measurement results we can say nothing about |ψ〉. This operator can be expressed as

(2.51) 

Such m+1 can be always constructed if the states in are linearly independent. We note, we will omit listing operator m+1 in further parts of the book.

The POVM measurement apparatus will be a key ingredient to distinguish quantum codewords with zero-error (see Chapter 3), and to reach the zero-error capacity of quantum channels.

The POVM can be viewed as the most general formula from among any possible measurements in quantum mechanics. Therefore the effect of a projective measurement can be described by POVM operators, too. Or, in other words, the projective measurements are the special case POVM measurement [Imre05]. The elements of the POVM are not necessarily orthogonal, and the number of the elements can be larger than the dimension of the Hilbert space which they are originally used in.

In our book we do not discuss the various measurement schemes in detail; for further information see the work of Nielsen [Nielsen00] or the previous part of the book [Imre05].

2.2.3 Partial Trace

If we have a density matrix which describes only a subset of a larger quantum space, then we talk about the reduced density matrix. The larger quantum system can be expressed as the tensor product of the reduced density matrices of the subsystems, if there is no correlation (entanglement) between the subsystems.

On the other hand, if we have two subsystems with reduced density matrices ρA and ρB, then from the overall density matrix denoted by ρAB the subsystems can be expressed as

(2.52) 

where TrB and TrA refers to the partial trace operators. So, this partial trace operator can be used to generate one of the subsystems from the joint state ρAB = |ψA〉〈ψA| ⊗ |ψB〉〈ψB|, then

(2.53) 

Since the inner product is trivially 〈ψB|ψB〉 = 1, therefore

(2.54) 

In the calculation, we used the fact that Tr(|ψ1〉〈ψ2|) = 〈ψ2|ψ1〉. In general, if we have two systems, A = |i〉〈k| and B = |j〉〈l|, then the partial trace can be calculated as

(2.55) 

since

(2.56) 

where |i〉〈k| ⊗ |j〉〈l| = |i〉|j〉(|k〉|l〉)T