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In information processing, as in physics, our classical world view provides an incomplete approximation to an underlying quantum reality. Quantum effects like interference and entanglement play no direct role in conventional information processing, but they can—in principle now, but probably eventually in practice—be harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations.

Charles H. Bennett, David Peter Divincenzo

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Information and computation theory have undergone a spurt of new growth, and a renewal of their historic connection to basic physics, as they have expanded to treat the intact transmission and processing of quantum states, and the interaction of such 'quantum information' with traditional forms of information. We may wonder why this did not happen earlier, as quantum principles have long been accepted as fundamental to all of physics. Perhaps the founders of information and computation theory, such as Shannon, Turing and von Neumann, were too accustomed to thinking of information processing in macroscopic terms, not yet having before them the powerful examples of the genetic code and ever-shrinking microelectronics. Be that as it may, information until recently has largely been thought of in classical terms, with quantum mechanics playing a supporting role in the design of the equipment to process it, and setting limits on the rate at which it could be sent through certain channels. Now we know that a fully quantum theory of information and information processing offers, among other benefits, a brand of cryptography whose security rests on fundamental physics, and a reasonable hope of constructing quan- tum computers that could dramatically speed up the solution of certain mathematical problems. These benefits depend on distinc- tively quantum properties such as uncertainty, interference and entanglement. At a more fundamental level, it has become clear that an information theory based on quantum principles extends and completes classical information theory, just as complex numbers extend and complete the reals. Besides quantum generalizations of classical notions such as sources, channels and codes, the new theory includes two complementary, quantifiable kinds of information— classical information and quantum entanglement. Classical infor- mation can be copied at will, but can only be transmitted forward in time, to a receiver in the sender's forward light cone. Entanglement in contrast, cannot be copied, but can connect any two points in space-time. Conventional data processing operations destroy entanglement, but quantum operations can create it and use it for various purposes, such as speeding up certain classical computa- tions and assisting in the transmission of classical information or intact quantum states. Part of the new quantum information theory is the qualitative and quantitative study of entanglement, and its interactions with classical information. Any means, such as an optical fibre, of delivering quantum systems more or less intact from one place to another, may be viewed as a quantum channel. Unlike classical channels, which are well characterized by a single capacity, quantum channels have several distinct capacities, depending on what one is trying to use them for, and what auxiliary resources are brought into play. New effects involving quantum information continue to be discovered, not only in the traditional areas of computation, channel capacity, and cryptography, but in areas such as commu- nication complexity and game theory.

Related papers

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Charles H. Bennett

The new theory of coherent transmission and transformation of information in the form of intact quantum states represents a major extension and generalization of classical information and computation theory, and has a number of distinctive features, including dramatically faster algorithms for certain problems, more complex kinds of channel capacity and cryptography, and a second, quantifiable kind of information—entanglement—which interacts with classical information in phenomena such as quantum teleportation.

CRC Press eBooks•Nov 16, 2018•218 citations

Quantum computing

Andrew M. Steane

The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high- Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.

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Introduction

Mark M. Wilde

Before quantum information became an established discipline, John R. Pierce issued the following quip at the end of his 1973 retrospective article on the history of information theory (Pierce, 1973): “I think that I have never met a physicist who understood information theory. I wish that physicists would stop talking about reformulating information theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases.” Since the publication of Pierce's article, we have learned much more about quantum mechanics and information theory than he might have imagined at the time, but we have also realized that there is much more to discover. In spite of all that we have learned, we still unfortunately have not been able to address Pierce's concern in the above quote in full generality. The most basic question that we could ask in quantum Shannon theory (and the one with which Pierce was concerned) is how much classical information a sender can transmit to a receiver by exploiting a quantum channel. We have determined many special cases of quantum channels for which we do know their classical capacities, but we also now know that this most basic question is still wide open in the general case. What Pierce might not have imagined at the time is that a quantum channel has a much larger variety of capacities than does a classical channel. For example, we might wish to determine the classical capacity of a quantum channel assisted by entanglement shared between the sender and receiver. We have seen that in the simplest of cases, such as the noiseless qubit channel, shared entanglement boosts the classical capacity up to two bits, and we now refer to this phenomenon as the super-dense coding effect (see Chapter 6). Interestingly, the entanglementassisted capacity of a quantum channel is one of the few scenarios where we can claim to have a complete understanding of the channel's transmission capabilities. From the results regarding the entanglement-assisted capacity, we have learned that shared entanglement is often a “friend” because it tends to simplify results in both quantum Shannon theory and other subfields of quantum information science.

Cambridge University Press eBooks•Nov 2, 2016•0 citations

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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?

Cambridge University Press eBooks•Jun 29, 2006•0 citations

Introduction to Quantum Information Processing

Elham Kashefi

The topic of these lectures lies in the new and rapidly growing field of quantum computing,which explores connections between physics and computing in general. Quantum informa-tion processing is a cross-disciplinary field and is of great importance from both a funda-mental, as well as technological perspective [27]. From the fundamental perspective we havedeepened our understanding of the relationship between physics, information and computa-tion in general, and have also gained a deeper understanding of the fundamental aspects ofquantum theory - non-locality and entanglement in particular [31]. From the technologicalperspective we have manipulated larger and larger quantum systems and obtained power-ful practical applications in the domain of communication and cryptography such as theunconditionally secure quantum cryptography (key exchange) and quantum teleportation[7, 14, 5].Historically, the greater potential of the quantum computer was first realised by Feyn-man, who noted that quantum systems appear to be exponentially hard to simulate withclassical computers [15]. He speculated that, therefore, quantum computers could poten-tially be much more powerful than their classical counterparts. This intuition has beenproven to be correct for some tasks, such as factoring large numbers and searching unstruc-tured databases. Every computer is fundamentally a physical system, and any computationis just a physical process undergone by this system. Quantum physics is the most accurateway of describing physical systems and their behaviour in general. Encoding informationinto quantum systems and processing it according to the laws of quantum physics results innew features which do not exist in the classical computation.Large scale quantum computation is still hypothetical. However, Moore’s law

•Jan 1, 2004•0 citations

Quantum Information

Charles H. Bennett

An expanded theory of information transmission and processing has emerged over the past few years, encompassing the processing and transmission of intact quantum states, the interaction of quantum and classical information, the quantitative theory of entanglement, and the use of quantum information processing to speed up certain classical computations. We review this field, concentrating on the parallels with and differences from classical information theory, which is now best seen as a part of the new theory.

Physica Scripta•Jan 1, 1998•66 citations
Information and computation theory have undergone a spurt of new growth, and a renewal of their historic connection to basic physics, as they have expanded to treat the intact transmission and processing of quantum states, and the interaction of such 'quantum information' with traditional forms of information. We may wonder why this did not happen earlier, as quantum principles have long been accepted as fundamental to all of physics. Perhaps the founders of information and computation theory, such as Shannon, Turing and von Neumann, were too accustomed to thinking of information processing in macroscopic terms, not yet having before them the powerful examples of the genetic code and ever-shrinking microelectronics. Be that as it may, information until recently has largely been thought of in classical terms, with quantum mechanics playing a supporting role in the design of the equipment to process it, and setting limits on the rate at which it could be sent through certain channels. Now we know that a fully quantum theory of information and information processing offers, among other benefits, a brand of cryptography whose security rests on fundamental physics, and a reasonable hope of constructing quan- tum computers that could dramatically speed up the solution of certain mathematical problems. These benefits depend on distinc- tively quantum properties such as uncertainty, interference and entanglement. At a more fundamental level, it has become clear that an information theory based on quantum principles extends and completes classical information theory, just as complex numbers extend and complete the reals. Besides quantum generalizations of classical notions such as sources, channels and codes, the new theory includes two complementary, quantifiable kinds of information— classical information and quantum entanglement. Classical infor- mation can be copied at will, but can only be transmitted forward in time, to a receiver in the sender's forward light cone. Entanglement in contrast, cannot be copied, but can connect any two points in space-time. Conventional data processing operations destroy entanglement, but quantum operations can create it and use it for various purposes, such as speeding up certain classical computa- tions and assisting in the transmission of classical information or intact quantum states. Part of the new quantum information theory is the qualitative and quantitative study of entanglement, and its interactions with classical information. Any means, such as an optical fibre, of delivering quantum systems more or less intact from one place to another, may be viewed as a quantum channel. Unlike classical channels, which are well characterized by a single capacity, quantum channels have several distinct capacities, depending on what one is trying to use them for, and what auxiliary resources are brought into play. New effects involving quantum information continue to be discovered, not only in the traditional areas of computation, channel capacity, and cryptography, but in areas such as commu- nication complexity and game theory.
The new theory of coherent transmission and transformation of information in the form of intact quantum states represents a major extension and generalization of classical information and computation theory, and has a number of distinctive features, including dramatically faster algorithms for certain problems, more complex kinds of channel capacity and cryptography, and a second, quantifiable kind of information—entanglement—which interacts with classical information in phenomena such as quantum teleportation.
The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high- Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.
Before quantum information became an established discipline, John R. Pierce issued the following quip at the end of his 1973 retrospective article on the history of information theory (Pierce, 1973): “I think that I have never met a physicist who understood information theory. I wish that physicists would stop talking about reformulating information theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases.” Since the publication of Pierce's article, we have learned much more about quantum mechanics and information theory than he might have imagined at the time, but we have also realized that there is much more to discover. In spite of all that we have learned, we still unfortunately have not been able to address Pierce's concern in the above quote in full generality. The most basic question that we could ask in quantum Shannon theory (and the one with which Pierce was concerned) is how much classical information a sender can transmit to a receiver by exploiting a quantum channel. We have determined many special cases of quantum channels for which we do know their classical capacities, but we also now know that this most basic question is still wide open in the general case. What Pierce might not have imagined at the time is that a quantum channel has a much larger variety of capacities than does a classical channel. For example, we might wish to determine the classical capacity of a quantum channel assisted by entanglement shared between the sender and receiver. We have seen that in the simplest of cases, such as the noiseless qubit channel, shared entanglement boosts the classical capacity up to two bits, and we now refer to this phenomenon as the super-dense coding effect (see Chapter 6). Interestingly, the entanglementassisted capacity of a quantum channel is one of the few scenarios where we can claim to have a complete understanding of the channel's transmission capabilities. From the results regarding the entanglement-assisted capacity, we have learned that shared entanglement is often a “friend” because it tends to simplify results in both quantum Shannon theory and other subfields of quantum information science.
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 topic of these lectures lies in the new and rapidly growing field of quantum computing,which explores connections between physics and computing in general. Quantum informa-tion processing is a cross-disciplinary field and is of great importance from both a funda-mental, as well as technological perspective [27]. From the fundamental perspective we havedeepened our understanding of the relationship between physics, information and computa-tion in general, and have also gained a deeper understanding of the fundamental aspects ofquantum theory - non-locality and entanglement in particular [31]. From the technologicalperspective we have manipulated larger and larger quantum systems and obtained power-ful practical applications in the domain of communication and cryptography such as theunconditionally secure quantum cryptography (key exchange) and quantum teleportation[7, 14, 5].Historically, the greater potential of the quantum computer was first realised by Feyn-man, who noted that quantum systems appear to be exponentially hard to simulate withclassical computers [15]. He speculated that, therefore, quantum computers could poten-tially be much more powerful than their classical counterparts. This intuition has beenproven to be correct for some tasks, such as factoring large numbers and searching unstruc-tured databases. Every computer is fundamentally a physical system, and any computationis just a physical process undergone by this system. Quantum physics is the most accurateway of describing physical systems and their behaviour in general. Encoding informationinto quantum systems and processing it according to the laws of quantum physics results innew features which do not exist in the classical computation.Large scale quantum computation is still hypothetical. However, Moore’s law
An expanded theory of information transmission and processing has emerged over the past few years, encompassing the processing and transmission of intact quantum states, the interaction of quantum and classical information, the quantitative theory of entanglement, and the use of quantum information processing to speed up certain classical computations. We review this field, concentrating on the parallels with and differences from classical information theory, which is now best seen as a part of the new theory.