In this chapter, I will introduce the concepts of quantum information that are necessary to the understanding of quantum cryptography. I first review the main principles of quantum mechanics. I then introduce concepts that somehow translate information theory into the quantum realm and discuss their unique features. Finally, I conclude this chapter with some elements of quantum optics. Fundamental definitions in quantum mechanics In classical mechanics, a system is described with physical quantities that can take certain values at a given moment in time; we say that is has a state. For instance, the state of an elevator comprises its position and speed at a given time. As dictated by common sense, if the elevator is at a given height, it cannot be simultaneously found at another location. With quantum mechanics, things are fundamentally different and elementary particles can behave against common sense. For instance, a quantum system can simultaneously be in different levels of energy. This is not because our knowledge of the state of the system is incomplete: the behavior of the quantum system is consistent with the fact that it is in simultaneous levels of energy. It would go beyond the scope of this book to describe quantum mechanics in detail. Instead, I propose a synthetic introduction suitable for the understanding of quantum cryptography. For a more detailed introduction, please refer to the books listed at the end of this chapter.
In this chapter, I will introduce the concepts of quantum information that are necessary to the understanding of quantum cryptography. I first review the main principles of quantum mechanics. I then introduce concepts that somehow translate information theory into the quantum realm and discuss their unique features. Finally, I conclude this chapter with some elements of quantum optics. Fundamental definitions in quantum mechanics In classical mechanics, a system is described with physical quantities that can take certain values at a given moment in time; we say that is has a state. For instance, the state of an elevator comprises its position and speed at a given time. As dictated by common sense, if the elevator is at a given height, it cannot be simultaneously found at another location. With quantum mechanics, things are fundamentally different and elementary particles can behave against common sense. For instance, a quantum system can simultaneously be in different levels of energy. This is not because our knowledge of the state of the system is incomplete: the behavior of the quantum system is consistent with the fact that it is in simultaneous levels of energy. It would go beyond the scope of this book to describe quantum mechanics in detail. Instead, I propose a synthetic introduction suitable for the understanding of quantum cryptography. For a more detailed introduction, please refer to the books listed at the end of this chapter.
We may be on the brink of a quantum transfor -mation driven by our increasing ability to realize, detect, control, and manipulate high-purity quantum states. For decades the quantum world has crept into our lives through modern information technology, the most visible side of science that permeates almost every aspect of our daily conduct. Yet quantumness in its full glory remains a promise for the future, as existing mainstream technologies are essentially based on properties of mixed quantum states realized in, e.g., metal–oxide–semiconductor heterostructures. The scientific backbone of current technologies combines quantum theory with information theory and the theory of computation, everything being sprinkled with a healthy dose of mathematics and brought into reality by remarkable engineering achievements. This multidisciplinary approach has enabled us to understand key physical phenomena and material properties, to communicate and process information efficiently, and, ultimately, to build functional devices that provide us with unprecedented convenience and mastery over nature, possibly with the price of being in danger to become the slaves of our computers and gadgets. However, our technological success is built on a combination of essentially distinct and largely independent scientific fields. Harnessing high-purity quantum states, on the other hand, sets these previously separated “continents of knowledge” on a collision path. The main goal of the last three chapters of this book is to provide a bird's eye view of the “collision zone” and prepare the reader to explore, enjoy, and perhaps exploit the newly raising mountains. We start with a brief introduction to basic concepts in quantum information, which also includes a summary of the operational formulation of quantum mechanics (the “natural” language of quantum information theory) and a synopsis of classical information theory ( Chapter 10 ). For a thorough treatment of concepts, problems, and techniques in quantum information theory the reader is referred to the books by Nielsen and Chuang [ 389 ] and Wilde [ 548 ]. The book by Watrous [ 530 ] focuses on the mathematical theory of quantum information. An undergraduate level introduction to this subject can be found, for example, in Schumacher and Westmoreland [ 458 ]. Some of the philosophical implications of the relationship between quantum mechanics and information theory are discussed in Timpson [ 505 ]. Our survey continues with Chapter 11 , where we discuss a few key aspects pertaining to quantum computation, which can be viewed as the quintessential quantum technology involving the controlled realization and manipulation of high-purity quantum states within a large Hilbert space. Finally, Chapter 12 focuses on topological quantum computation, a promising approach to overcoming the fundamental challenge associated with quantum decoherence based on harnessing the underlying robustness of topological quantum states. This also marks our final return to the realm of topological quantum matter.
This book is intended as an introductory textbook at graduate or advanced undergraduate level. Students with diverse backgrounds, in mathematics, physics and computer science, can use it to become acquainted with the quantum theory of information. Its scope is deliberately chosen to be not too broad, while being self-consistent and usable for a semester-long course for beginners. The subject is presented using a balanced mix of intuitive arguments and rigorous formalism. The emphasis is not on the physical interpretation, but rather on the formal and mathematical features of quantum information theory, in order to be accessible from different backgrounds, the only requirement being a basic knowledge of linear algebra. In particular, a background in quantum mechanics is not needed, since quantum theory is used as a mere mathematical framework without entering into physical interpretations. The book introduces all necessary concepts from quantum theory and classical information theory, to cover the elements of quantum Shannon theory: quantum states and channels, data compression and entropy, error correcting codes and channel capacities, as well as the basics of entanglement theory and quantum cryptography. Examples are provided to help the reader in understanding the different concepts, together with exercises at the end of each chapter.
The ability to transfer information securely is essential in the modern world we live in. While the field of cryptography has long researched ways to do this, events over the last couple of years have brought the issues of secure information transfer squarely into the public eye. As a consequence, security and cryptography have quickly become issues of social and political importance, sparking debates about the values of privacy and the role of government. An essential ingredient of modern cryptography is the generation of randomness: cryptographic techniques are built on the premise that one has access to random bits. It is well known, however, that computers cannot produce algorithmically random sequences – that is, sequences with maximal algorithmic information content – but are doomed to produce ‘pseudorandomness’. This lack of randomness can be, and has been, exploited to compromise security, see [4]. The active field of quantum information theory has proposed approaches to provide supposedly ‘unbreakable’ security by exploiting various quantum phenomena. This security unfortunately relies on assumptions about the nature of quantum measurements and their ability to generate random bits. Anton Zeilinger summarises this by postulating that the simplest quantum systems, qubit, can hold only one bit of classical information [6]. This foundational principle is in line with a wider paradigm shift to view quantum information as an extension of classical information, but it is nonetheless unsatisfying to simply postulate this principle, given its importance in determining the practical advantages of quantum information theory. In order to understand better how quantum mechanics can help generate meaningful information, we instead look to relate the outcomes of quantum measurements to formal properties of the system based on more fundamental assumptions. Indeed, we have shown that a) some of the postulated properties of quantum information follow from the formal structure of the theory and b) a purely formal notion of information within a quantum world can generate, via measurement, meaningful and useful information in the macroscopic world (for example, in cryptography). The indeterminism of quantum measurements can be formalised via the notion of value indefiniteness. To explain this concept, let us consider an arbitrary quantum system and ask whether the outcome of a measurement of any observable quantity A (such as the energy of the system, its angular momentum (spin), etc.) is determined prior to the measurement. If this is the case, then we say the observable is value definite with value v(A); otherwise, the observable is value indefinite and v(A) is undefined. Mathematically, one can reduce all observable quantities to so-called projection observables, which can only take the values 0 or 1. Thus, the question of whether the outcome of several measurements can be simultaneously determined in advance can be rephrased in terms of the information ‘carried’ by a particular system in a definite quantum state. Classically, one expects that all quantities are determined in advance, and hence all observables are value definite. Quantum mechanically, however, the belief is that this is not the case, and the information content of quantum systems is limited. Formulating carefully the notion of value indefiniteness allows us to formalise the notion of (quantum) indeterminism; however, this doesn’t help clarify whether quantum systems are indeed value indefinite or not. Staying in this formal framework, the Kochen-Specker theorem [5] provides a first positive result, showing that at least some observables must be value indefinite if one makes the assumption known as non-contextuality, which states that any definite values that exist must be independent of other compatible measurements that may or may not be performed on the system. Under the same assumption, this theorem can be strengthened to show that only one single observable can have the definite value 1 (see [2]). Furthermore, only observables that can be measured simultaneously with this single one can have the definite value 0; the rest must all be value indefinite. Since the preparation of a system involves precisely ensuring that, usually via measurement, the system is in a definite state with respect to some desired observable, this result shows that no other incompatible observable can be value definite. That is, preparing a system in a definite state by making the ‘preparation’ observable value definite specifies completely the information content of the quantum system. This is an example of syntactical quantum information acquiring meaning at the level of the quantum itself. Can the syntactical information at the level of the quantum generate meaning outside the quantum, that is, at the macroscopic level? The results cited above hold in the Hilbert-space framework of quantum mechanics and are formulated only in terms of a syntactical notion of information. Their real importance becomes evident when one interprets them in the context of quantum measurements. Specifically, if we prepare a quantum system in a known state, they allow us to ‘locate’ observables which we can measure, but which are value indefinite; that is, observables whose measurement results are not specified by any pre-existing property of the quantum system. Furthermore, with respect to a mathematical model of unpredictability which we developed in [3], the results of these measurements can be shown to be absolutely unpredictable. In this way we get a mathematical explanation and justification of the largely accepted intuition that quantum mechanics is inherently unpredictable, and that this unpredictability arises from the phenomenon of indefiniteness within the quantum world. Furthermore, if one considers a hypothetical infinite sequence generated by the repeated measurement of a quantum value indefinite observable, one can prove that these sequences must be strongly incomputable, technically ‘bi-immune’ [1]. Such sequences cannot be generated by any Turing machine or classical computer, showing that value indefiniteness leads to a clear classical/quantum split in a purely algorithmic context. Importantly, bi-immunity is a property of observed, macroscopic quantities, quite separate from the quantum framework in which the value indefiniteness is formalised. This form of macroscopic meaning created from the lack of syntactic information is precisely the scenario that quantum random number generators try to create, and which is essential for the development and certification of quantum cryptographic systems. The fact that the macroscopic information created goes beyond anything classically obtainable serves as a valuable practical resource, outside of and removed from the quantum formalism. To conclude, quantum information creates meaning within the quantum and via measure- ment, the lack of information within the quantum, creates meaning and valuable information at the macroscopic level. Acknowledgments The second author thanks Prof. S. Marcus for useful conversations on quantum information theory. This work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. References and Notes Abbott, A.A.; Calude, C.S.; Svozil, K. Strong Kochen-Specker theorem and incomputability of quantum randomness. Physical Review A 2012, 86, 062109. Abbott, A.A.; Calude, C.S.; Svozil, K. Value-indefinite observables are almost everywhere. Physical Review A 2013, 89, 032109. Abbott, A.A.; Calude, C.S.; Svozil, K. On the unpredictability of individual quantum measurement outcomes. CDMTCS Research Report 2014, 458. Calude, C.S. Quantum randomness & cryptology. CyberTalk 2013, 3, 42–43. Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 1967, 17, 59–87. Zeilinger, A. A foundational principle for quantum mechanics. Foundations of Physics 1999, 29(4),631–643.
In the history of cryptography, quantum cryptography is a new and important chapter. It is a recent technique that can be used to ensure the confidentiality of information transmitted between two parties, usually called Alice and Bob, by exploiting the counterintuitive behavior of elementary particles such as photons. The physics of elementary particles is governed by the laws of quantum mechanics, which were discovered in the early twentieth century by talented physicists. Quantum mechanics fundamentally change the way we must see our world. At atomic scales, elementary particles do not have a precise location or speed, as we would intuitively expect. An observer who would want to get information on the particle's location would destroy information on its speed – and vice versa – as captured by the famous Heisenberg uncertainty principle . This is not a limitation due to the observer's technology but rather a fundamental limitation that no one can ever overcome. The uncertainty principle has long been considered as an inconvenient limitation, until recently, when positive applications were found. In the meantime, the mid-twentieth century was marked by the creation of a new discipline called information theory . Information theory is aimed at defining the concept of information and mathematically describing tasks such as communication, coding and encryption. Pioneered by famous scientists like Turing and von Neumann and formally laid down by Shannon, it answers two fundamental questions: what is the fundamental limit of data compression, and what is the highest possible transmission rate over a communication channel?
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
We have seen that there is an intimate relationship between probability and information. The values we assign to probabilities depend on the information available, and information is a function of probabilities. This connection makes it inevitable that information will be an important concept in any statistical theory, including thermodynamics and, of course, quantum physics. The probabilistic interpretation of quantum theory has probability amplitudes rather than probabilities as the fundamental quantities. This feature, together with the associated superposition principle, is responsible for intrinsically quantum phenomena and gives quantum information theory its distinctive flavour. We shall see that the quantum rules for dynamical evolution and measurement, together with the existence of entangled states, have important implications for quantum information. They also make it possible to perform tasks which are either impractical or impossible within the classical domain. In describing these we shall make extensive use of simple but fundamental ideas in quantum theory. This chapter introduces the mathematical description of quantum physics and the concepts which will be employed in our study of quantum information.