The main view promoted by this book is that underlying many different aspects of reality is some form of information processing. The theory of information started rather innocently, as the result of a very specific question that Shannon considered, which was how to maximize the capacity of communication between two users. Shannon showed that all we need is to associate a probability to an event, and defined a metric that allowed you to quantify the information content of that event. Interestingly, because of its simplicity and intuitiveness, Shannon’s views have been successfully applied to many other problems. We can view biological information through Shannon’s theory as a communication in time (where the objective of natural selection is to propagate the gene pool into the future). But it is not only that communications and biology are trying to optimize information. In physics, systems arrange themselves so that entropy is maximized, and this entropy is quantified in the same way as Shannon’s information. We encounter the same form of information in other phenomena. Financial speculation is also governed by the same concept of entropy, and optimizing your profit is the same problem as optimizing your channel capacity. In social theory, society is governed by its interconnectedness or correlation and this correlation itself is quantified by Shannon’s entropy. Underlying all these phenomena was the classical Boolean logic where events had clear outcomes, either yes or no, on or off, and so on. In our most accurate description of reality, given by quantum theory, we know that bits of information are an approximation to a much more precise concept of qubits. Qubits, unlike bits, can exist in a multitude of states, any combination of yes and no, on and off. Shannon’s information theory has been extended to account for quantum theory and the resulting framework, quantum information theory, has already shown a number of advantages. The greater power of quantum information theory is manifested in more secure cryptographic protocols, a completely new order of computing, quantum teleportation, and a number of other applications that were simply not possible according to Shannon’s view.
The main view promoted by this book is that underlying many different aspects of reality is some form of information processing. The theory of information started rather innocently, as the result of a very specific question that Shannon considered, which was how to maximize the capacity of communication between two users. Shannon showed that all we need is to associate a probability to an event, and defined a metric that allowed you to quantify the information content of that event. Interestingly, because of its simplicity and intuitiveness, Shannon’s views have been successfully applied to many other problems. We can view biological information through Shannon’s theory as a communication in time (where the objective of natural selection is to propagate the gene pool into the future). But it is not only that communications and biology are trying to optimize information. In physics, systems arrange themselves so that entropy is maximized, and this entropy is quantified in the same way as Shannon’s information. We encounter the same form of information in other phenomena. Financial speculation is also governed by the same concept of entropy, and optimizing your profit is the same problem as optimizing your channel capacity. In social theory, society is governed by its interconnectedness or correlation and this correlation itself is quantified by Shannon’s entropy. Underlying all these phenomena was the classical Boolean logic where events had clear outcomes, either yes or no, on or off, and so on. In our most accurate description of reality, given by quantum theory, we know that bits of information are an approximation to a much more precise concept of qubits. Qubits, unlike bits, can exist in a multitude of states, any combination of yes and no, on and off. Shannon’s information theory has been extended to account for quantum theory and the resulting framework, quantum information theory, has already shown a number of advantages. The greater power of quantum information theory is manifested in more secure cryptographic protocols, a completely new order of computing, quantum teleportation, and a number of other applications that were simply not possible according to Shannon’s view.
The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
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.
The success of the abstract model of computation, in terms of bits, logical operations, programming language constructs, and the like, makes it easy to forget that computation is a physical process. Our cherished notions of computation and information are grounded in classical mechanics, but the physics underlying our world is quantum. In the early 80s researchers began to ask how computation would change if we adopted a quantum mechanical, instead of a classical mechanical, view of computation. Slowly, a new picture of computation arose, one that gave rise to a variety of faster algorithms, novel cryptographic mechanisms, and alternative methods of communication. Small quantum information processing devices have been built, and efforts are underway to build larger ones. Even apart from the existence of these devices, the quantum view on information processing has provided significant insight into the nature of computation and information, and a deeper understanding of the physics of our universe and its connections with computation. We start by describing aspects of quantum mechanics that are at the heart of a quantum view of information processing. We give our own idiosyncratic view of a number of these topics in the hopes of correcting common misconceptions and highlighting aspects that are often overlooked. A number of the phenomena described were initially viewed as oddities of quantum mechanics. It was quantum information processing, first quantum cryptography and then, more dramatically, quantum computing, that turned the tables and showed that these oddities could be put to practical effect. It is these application we describe next. We conclude with a section describing some of the many questions left for future work, especially the mysteries surrounding where the power of quantum information ultimately comes from.
What is information? A short answer could be: Anything that changes our minds and states of thinking! Information is virtual. It can be carried by speech, an image, scratches on a stone, patterns in a photon field, the connections of neurons in our brains or even by the wavefunction of an electron. Information does not depend on the actual means of transmission nor does its identity change when changing the carrier. Information is physical. It needs to be sustained by a physical substrate and rearranging that substrate to contain the wanted information requires to spend energy or work. Without a physical world, information can not be stored, processed, or transmitted. Our world is permeated with different representations of information. Actually Information about the physical laws governing our Universe can be found everywhere and in everything. Could it then be that the real fundamental elements of this world are tiny bits of information? Could we get it from bit?1 A number of researcher investigate this possibility, and several of their attempts to recover physical laws from the principles governing information are reported about in this special issue on physics of information. The articles by Gregg Jaeger,2 by Ariel Caticha,3 and by John Skilling and Kevin Knuth4 address the reconstruction of quantum mechanics from information theory. Furthermore, Kevin Knuth and James Walsh5 show how to recover space-time from particles sending each others signals in an initially geometry-free setting. A central element of most works on information is the concept of a probability. Probabilities can be defined via various routes. Most familiar might be their definitions by the frequentists school as relative frequencies of events (in the limit of infinite amounts of such). A more general definition of probabilities is the Bayesians' one, which regards probabilities as quantifiers of statements, expressing how much those should be expected to be true. Although this definition looks subjective on the first glance, it rests on a solid mathematical theorem by Richard Cox6 that any extension of binary logic to a continuum of single valued truthfulness quantifier that respects a minimum of requirements must be isomorphic to probability theory. And this definition turns out to embrace the frequentists' definition in case that infinite numbers of comparable events are actually available and can be counted. The majority of authors of this special issues seem to favor the Bayesian perspective, which is probably a result of our own, the guest editors', preference and not necessarily a reflection of the prevailing position of contemporary scientists. The different views on probabilities of frequentists and Bayesians are addressed by the article of Allen Caldwell.7 He builds a bridge between these antagonistic camps by explaining frequentists' constructions in a Bayesian language. In particular, his argumentation why the omnipresent p-value in statistical and experimental literature might often capture a relevant signal of model misfit is reassuring, given that a large fraction of our medical system is based on decisions backed up by p-values. A Bayesian might add that it would be even more reassuring, if a system rigorously build on logic and its continuous extension would be used for decisions affecting live-or-death questions. The laws governing probabilities themselves seem to reflect that of more elementary mathematical structures. Kevin Knuth shows how these emerge from lattice theory, which deals with relations in partially ordered sets like that of collections of possibilities.8 The sum rule of probability theory is a direct consequence of the underlying lattice structure. Interestingly, a number of other theories are resting also on such lattices and therefore exhibit their own versions of sum rules: measures of sets, mutual information, the Euler characteristics of topological spaces, as well as quantum mechanical interactions that are quantified by pairs of numbers, the real and imaginary parts of complex probability amplitudes (see also4). The theory of information has very practical application in physics and elsewhere. Manfred Opper discusses how the solutions of stochastic differential equations can be inferred via variational methods, which have their origin in thermodynamics.9 Torsten Enßlin introduces information field theory (IFT), the information theory for fields, as a means to infer the configuration of a physical field from incomplete and noisy data.10 The mathematical apparatus for this is that of quantum field theory (QFT), just with ordinary probabilities instead of complex probability amplitudes as in the latter. Could it be that the difference between IFT and QFT is just be the same as between classical information theory and quantum mechanics, the transition from real valued to complex valued quantifier for statements? Anyhow, any practical inference machinery, be it for dynamical systems or for fields, requires sophisticated numerical methods. Michael Betancourt reviews Markov Chain Monte Carlo Methods to calculate expectation values over high dimensional probabilities.11 Theo Steininger and coworkers present the NIFTy3 software package for numerical information field theory calculations, which can handle very high dimensional field inference problems.12 Finally, in an article focused on the subtleties of the Monty Hall problem and mostly aiming at information theoretical entertainment, Torsten Enßlin and Margret Westercamp investigate the possibility that human intuition might be more rational than contemporary psychology thinks.13 A number of possible topics for an issue on physics of information are not addressed here. We would have loved to have articles on Landauer's principle, which states that erasing information has to cost energy, on QBism, a Bayesian version of quantum mechanics, on artificial and biological neural networks, as well as any other aspect of the information-soaked biological world. What has a reader of this special issue to be equipped with in order to follow the lines of thoughts? In addition to the usual physicist's education, the minimal additional equipment are the basics of probability theory and the concept of entropy. Most of the other concepts used in the articles rest on those and are introduced there. Let A, B, and C be propositions (or possibilities). The probability P ( A | B ) of A being true assuming that B is true is a number between 0 and 1, with P ( A | B ) = 1 expressing certainty of A under B ( B ⇒ A ) and P ( A | B ) = 0 expressing certainty of not A being the case under B ( B ⇒ ¬ A ).
Information as accessed by the human mind, and the myriad ways of processing it, is what has set humankind apart in the animal world. The ability to use the physical systems around us to encode and then to process information has been evolving in leaps and bounds ever since the dawn of our race! The earliest form of computation was probably in the form of account-keeping by counting pebbles or on the fingers. This developed into the abacus, writing symbols, the computing machine, and now, a laptop or a supercomputer: everywhere we represent information by means of a physical system, and perform manipulations on that system to process the information in many desired ways, including communication. The stress here is on the realization that the basis of information is a physical system. The more advanced that system is and the set of rules it functions on, the more capable our means of information processing and communication. When the underlying physical system used for encoding and processing information is a quantum system, we have quantum information processing.
Introduction The quantum information theory began to rise in the end of 20th century. The EPR (relation)really is quantum entanglement, Consequently, it makes the concept of information in science expand from classical information theory to quantum information theory, and a new concept of quantum information formed. The primary concept of information faced to new challenges and produced some important philosophical questions. For example, can we set up the concept of quantum information? What is the nature of quantum information? The past common concept of information (the information is the elimination of uncertainty) could be used in quantum information? Whether does the emergence of quantum computation mean to eliminate no-boundaries between the quantum and the classical? My aim in this paper is to discuss these questions. My philosophical analysis focuses, however, on the basic quantum information theory. Methods The research method is analytical philosophy and logic that make quantum conception more clear and accurate. Results and Discussion The concept of information could be set up. fundamentally speaking, it did be supported by the Shannon information theory and further philosophical reflections. According of information theory, (classical) information concept could be set up, and there are two necessary preconditions: (a) the premise is a set of possible events, and they can be described by the probability. (b) the information is the elimination of uncertainty. Fundamentally, quantum information is also satisfied with the family-resemblance two preconditions: (1) Quantum information describes the possibility of microscopic things, and quantum information is the expression of wave function. The wave function describes the state of microscopic particles, and presentation of the possibility of microscopic objects. (2) On the aspect of eliminating the uncertainty, the quantum communication is same as classical communication. The signal receiver once received quantum signals transmitted, and the receiver eliminates the corresponding uncertainty of quantum. Therefore, the quantum information is the elimination of quantum uncertainty, and which is called quantum information I. From two point of views, the possible events and eliminate uncertainty, the quantum information and classical information have family resemblance. Quantum information like Classical information way of transmission, it is delivered by the information source, and is received by recipient through the information channel. Quantum Information is similar to classical information, but more essential difference between them, specifically in the following three aspects: (1) The probability directly describe the possibility of classic events, while wave function (or probability amplitude) describes quantum events in the quantum world, and the absolute square of wave function is corresponded to classical probability. (2)There have the essential difference between qubits (quantum qits) and classical bits. (3) Classical information can be completely cloned or deleted. nevertheless, quantum information is neither cloned (no-cloning), nor removed completely. To my point of view, through the analysis of the process of quantum teleportation, a quantum uncertainty can be divided into three categories: the first is the external uncertainties of microscopic particles; the second the uncertainty of the superposition state; the third the inherent uncertainty of microscopic particles. These three uncertainties can be eliminated by three different ways that are transmission of quantum information, accurate quantum transformations and the creations of quantum entanglement. Therefore we can give the general definition of certainty: a situation (the state of events, things) is of certainty, if they are satisfied with the following conditions: (a) a situation is entirely determined by n events parameters; (b) n parameters of events has an accurate values simultaneously. The existences of quantum states form quantum facts. Quantum states show themselves, and therefore it releases the information to the outside world. For this reason, there have two aspects of quantum state which need to express: existence of quantum state is the quantum fact, here we emphasize "existence"; and quantum state can show to the external world, which is the quantum information, here we emphasize "Show". Hence, quantum information is the show(Zeigt) of quantum state, which is called quantum information II. The nature of quantum information is the show of quantum events and their correlated ways. Quantum information is a kind of certainty and order. Quantum information can be divided into quantum informationⅠand quantum informationⅡ.. Quantum InformationⅠjust only a phenomenonal definition, it does not go deeply into the nature of the information. Therefore, the elimination of uncertainty can not be seen as the nature of quantum information itself . In the process of quantum teleportation, the source of quantum information (such as quantum state , a and b are coefficients) is delivered to the receiver, what is the essence of quantum information? From the point of view of quantum information theory, quantum state represents quantum information, quantum state is the state of quantum system (or microscopic particles) or quantum events. in the quantum state , what quantum states disclose the quantum information does not be the absolute size of quantum information, but the relationship, since the coefficients a and b satisfy the condition , and it do not be important numerical a and b, but between their relative size and relationship, which reflects the correlated ways between a and b. The connotation of quantum InformationⅠand quantum information Ⅱ offers support for the common view of classical information concept. Which is more deeply to disclose the nature of quantum information for both of them? Obviously, quantum information Ⅱ reflects the nature of quantum information. Quantum reality and quantum information are unified. What the wave function expresses is quantum reality. The wave function is unified of quantum information and quantum reality. There is no quantum reality, there is no quantum information. Because of the information and reality are unified, and quantum information different from classical information, so we can deduce that quantum reality is different from classical reality, and quantum computing differs from classical computing. In the classical parallel computing, the value calculated in different circuits respectively, but in parallel quantum computing the value calculated in one circuit, which is the essential difference between classical and quantum parallel computing. Many worlds interpretation of quantum mechanics does not mean that the difference between quantum and classical disappears: (1) Superposition in the microscopic world is very different from classical macroscopic physical world, so it is impossible that the superposition principle is fit for quantum world used for the classical world; (2) "To perform a quantum measurement, quantum world can be split up numerous classical world" is not supported by any experience and evidence; (3) Each branches of quantum computing are in quantum states, and quantum states are coherent, holistic, and some even entangled. Nevertheless, it is impossible for classical computing to achieve the quantum computing’s specific features; (4) The essential differences between quantum and the classical are decided by the development history of quantum mechanics and quantum information theory. The differences between quantum and the classical is really established by the quantum theory, concepts and their experiments that include Heisenberg’s matrix mechanics of Schrodinger’s wave equation, uncertainty principle of quantum mechanics and quantum entanglement phenomena and so on. They are completely different from the classical physical theories, concepts and experiments. Figure 1. Quantum information processing (see PDF version for the Figure). Conclusions This fruitful work gives an discussion of quantum concept that is the quantum information is the elimination of quantum uncertainty, which is called quantum information I and quantum information is the show (Zeigt) of quantum state, which is called quantum information II. Quantum information II reflects the nature of quantum information. The nature of quantum information is the show of quantum events and their correlated ways. Acknowledgments This research work of this paper is supported by 2011 the Philosophy and Social Sciences Foundation of the Ministry of Education of P. R. China (Project Number: 11JZD007) " Studies of the development trend of contemporary technological philosophy" . References and Notes Weiner, N., Cybernetics and Society. Houghton Mifflin Company, 1950. Shannon, C. E., The Mathematical Theory of Communication. The Bell System Technical Journal. 1948, Vol.27, pp.379-423, 623-656. Kornwachs K., Jacoby K., Introduction: What’s new about information? In Kornwachs K., Jacoby K.eds. Information. Berlin: Akademic Verlag GmbH. 1996. Bouwmeester, D. Pan, J.-W.，et al. Experimental Quantum Teleportation. Nature. 1997. 390. Bennett C. H. and Di Vinecenzo D. P. quantum information and computation, Nature, 2000, 404. Einstein, A. Podolsky B., and Rosen, N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review. 1935,47. Nielsen, M. A. Chuang, I. 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