One of the founders of the quantum theory of information Corresponding Member of the Russian Academy of Sciences Alexander Holowo believes that we may be approached by the boundaries of knowledge

TO The cable computer is one of the most discussed science. Unfortunately, so far the further experiments that are conducted in many countries of the world, including Russia, did not matter, although the results of their promising.

In parallel, but with significantly great success, there is a quantum cryptography systems. Such systems are already at the stage of experienced implementation.

The very idea of \u200b\u200bthe possibility of creating a quantum computer and quantum cryptography systems is a quantum information theory. One of her founders - Alexander Holedo, russian mathematician, Corresponding Member of the Russian Academy of Sciences, Head of Probability Theory Department and Mathematical Statistics of the Mathematical Institute. V. A. Steklov RAS. In 2016, he received the Shannon Prize, the most prestigious information on the theory of information that the Institute of Electrical Engineering Engineers and Electronics was awarded - IEEE. Back in 1973, the cholevo formulated and proved the theorem that obtained his name and the most important quantum cryptography: it establishes the upper limit of the amount of information that can be extracted from quantum states.

You formulated our most famous theorem in 1973. As far as I remember, there was no such words in the public space as a quantum theory of information. Why did you get interested?

Indeed, then, yes, and then some time, in the public space she did not sound, but in scientific literature It was then, in the 1960s - early 1970s, publications were emerging on the issue of which fundamental restrictions impose a quantum nature of the media carrier (for example, a laser radiation fields) on its transfer. The question of fundamental restrictions arose no, almost immediately after the creation of Crate Shennon the foundations of the theory of information. By the way, in 2016 it was one hundred years from the day of birth, and his famous work on the theory of information appeared in 1948. And already in the 1950s, experts began to think about quantum restrictions. One of the first was the article by Denis Gabor (who received the Nobel Prize for the invention of holography). He set such a question: what fundamental limitations the quantum nature of the electromagnetic field imposes to the transmission and playback of information? After all, the electromagnetic field is the main storage medium: in the form of light, radio waves or other frequencies.

If there is a communication channel, which is considered as a quantum, then the Shannon number of classical information, which can be transmitted by such a channel, is limited on top of some kind of completely specific magnitude

After that began to appear physical work about this theme. Then it was called non-quantum information theory, but Quantum Communication, that is, a quantum transmission theory of messages. Of the domestic scientists, already interested in this problem, I would call Ruslan Leontyevich Stratonovich. It was a major specialist in statistical thermodynamics, which wrote on these topics.

In the late 1960s, I defended the PhD dissertation for mathematical statistics Random processes, began to think what to do next, and came across work on this issue. I saw that this is a huge field of activity, if, on the one hand, approach these tasks from the point of view of the mathematical foundations of quantum theory, and on the other - to use what I know about mathematical statistics. This synthesis turned out to be very fruitful.

The essence of the theorem proved by me in 1973 is as follows: if there is a communication channel, which is considered as a quantum, then the Shannon number of classical information, which can be transmitted by such a channel, is limited on top of a certain specific value - it was later called χ-quantity (Hee quiet). Essentially, all channels of communication are quantum, only in most cases their "quantum" can be neglected. But if the noise temperature in the channel is very low or the signal is very weak (for example, a signal from a remote star or a gravitational wave), then there is a need to take into account quantum-mechanical errors arising due to the presence of quantum noise.

- Limited from above, so we are talking about the maximum amount of information transmitted?

Yes, about the maximum number of information. I took up this issue because it was essentially a mathematical task. The existence of such inequality of physics guess, it was formulated as an assumption and appeared in such a capacity at least a decade, and maybe more. It was not possible to find contradictory examples, but the proof did not work, I decided to do it. First of all, the assumption was necessary to formulate mathematically to really prove it as theorem. After that, a couple more years have passed, while someone in the subway did not come insight. As a result, it turned out this inequality. And in 1996 I managed to show that this upper border is reaches the limit of very long messages, that is, it gives the bandwidth of the channel.

It is important that this upper limit for information does not depend on how the output is measured. This border, in particular, has found important applications in quantum cryptography. If there is a secret channel of communication and a certain attacker is trying to overhear it (such an attacker is usually called Eve from the English. Eavesdropper - listening), it is unknown what kind of Eva is overhears. But then the amount of information that it will still be dehydrated to steal, limited from above this absolute value that does not depend on the measurement method. Knowledge of this value is used to enhance the transmission secrecy.

- information can be understood as with mathematical and from a physical point of view. What is the difference?

In mathematical theory of information, this is not about its content, but about quantity. And from this point of view, the method of physical implementation of information is indifferent. Whether it is about image, music, text. It is essential only how many memory this information is in digital form. And how it can be encoded well, usually in binary form, because for classical information it is the most convenient way digital presentation. The number of such information is measured in binary units - bits. If the information is uniform in this way, it opens up the possibilities for a single approach that does not depend on the nature of the information carrier, while we only consider the "classic" carriers.

Distinctive property Quantum information is the impossibility of its "cloning". In other words, the laws of quantum mechanics prohibit the "quantum xerox". This, in particular, makes quantum information by a suitable means for transmitting secret data.

However, the transition to quantum media - photons, electrons, atoms - opens up fundamentally new opportunities, and this consists of one of the main sections of the quantum theory of information. Arises the new kind information - quantum information, a unit of measurement of which is a quantum bit - qubit. In this sense, "information is physical", as one of the founding fathers of the quantum theory of information Rolf Landauer said. The distinctive feature of quantum information is the uncealty of its "cloning". In other words, the laws of quantum mechanics prohibit the "quantum xerox". This, in particular, makes quantum information with a suitable means for transmitting secret data.

I must say that our compatriot Vladimir Aleksandrovich Kotelnikov said his word in the theory of information before Shannon. In 1933, in 1933, "Materials for the First All-Union Congress on Communication Reconstruction issues" published the famous "Directory Theorem". The value of this theorem is that it allows continuous information, analog signal Translate to discrete shape (count). Our work in this area was furnished with great secrecy, so such a resonance, as the work of Shannon, the work of Kotelnikov did not receive, and in the West in general, until some moments were unknown. But in the late 1990s, the Institute of Electrical Engineering Engineers and Electronics, IEEE, awarded Kotelnikov the highest award - the medal named after A. G. Bella, and the German Foundation Eduard Reina - a prize for fundamental research, namely, for the countdown theorem.

- And for some reason about Kotelnikov, I even remembered so little ...

His works were classified. In particular, the kotelnikov made a lot in the field of government relations, far cosmic communication. By the way, Vladimir Alexandrovich was interested in the interpretation of quantum mechanics, he has work on this topic.

Shannon became famous for its 1948 article on the theory of information. But the first of its famous work dedicated to the use of logic algebra and boolean functions, that is, the functions of binary variables for analysis and synthesis electrical schemes (relay, switching schemes), was written back in 1937, when he was a student of the Massachusetts Institute of Technology. Sometimes it is called the most outstanding the graduation work Twentieth century.

It was a revolutionary idea, which, however, at that time Vitala in the air. And in this Shannon had a predecessor, Soviet physicist Viktor Shestakov. He worked on Physician MSU and proposed the use of binary and more general multi-valued logic for analyzing and synthesizing electrical circuits in 1934. He then defended himself, but did not immediately publish his research, as it was believed that it was important to get the result, and the publication could wait. In general, he published his work only in 1941, after Shannon.

Interestingly, at that time, in the 1940-1950s, it turned out so well: everything that has allowed to develop the theory of information and ensure its technical implementation, it appeared almost simultaneously.

Indeed, at the end of the war there were electronic computing machines. Then almost simultaneously with the publication of Shannon's article invented the transistor. If it were not for this discovery and if technological progress was slow down in this regard, the ideas of the information theory would not have found a long time, because it was difficult to realize them on huge cabinets with radiologists that were heated and demanded a niagara for their cooling was difficult. All coincided. It can be said that these ideas arose very timely.

Photo: Dmitry Lykov

Shannon received a diploma mathematics and at the same time a diploma engineer-electrician. He knew mathematics so much as an engineer needed, and at the same time he had a stunning engineering and mathematical intuition. The importance of Shannon's work for mathematics was realized in the Soviet Union by Andrey Kolmogorov and his school, while some Western mathematicians treated Shannon's work quite arrogant. It was criticized for the fact that he would not write that he had some mathematical flaws, although by and large he had no serious flaws, but intuition was completely unmistakable. If he claimed something, it was usually not discharged with the general conditions under which this is true, but a professional mathematician, having bothering, could always find accurate wording and evidence in which the corresponding result will be strict. As a rule, it was very new and deep ideas that had global consequences. In this regard, it is even compared with Newton and Einstein. So were laid theoretical basis For an information era, which began in the middle of the twentieth century.

In their works, you write about the connection of such properties of the quantum world, as "complementary" and "clutch" with information. Explain this, please.

These are two main, fundamental properties that distinguish a quantum world from classic. An additional amount in quantum mechanics is that there are some aspects of a quantum-mechanical phenomenon or an object that both belong to this object, but cannot be exactly fixed at the same time. For example, if the position of the quantum particle focuses, the pulse is blurred, and vice versa. And this is not only coordinates and impulse. As Nils Bor pointed out, additional value is the property of not only quantum-mechanical systems, it is also manifested in biological and in social systems. In 1961, a remarkable collection of articles of Bora "Nuclear Physics and Human Cognition" came out in the Russian language. It says, for example, about the complementarity between reflection and action, while reflection is an analogue of the situation, and the effect is an analogue of the impulse. We know perfectly well that there are people actions, there are people thinking, and it is difficult to combine in one person. There are some fundamental limits that do not allow you to combine these properties. Mathematically, additionalness is expressed in the fact that unbeatable objects, matrices or operators are used to describe quantum values. The result of their multiplication depends on the order of the factory. If we measure one size first, then another, and then do it in the opposite order, then we get different results. This is a consequence of additionalness, and nothing of the kind in the classic description of the world does not exist, if you understand this, let's say, the theory of probabilities Kolmogorov. In it, in what order would be measured by random variables, they will have the same joint distribution. Mathematically, this is a consequence of the fact that random variables are not matrices, but functions that are permutable in the sense of multiplication.

Shannon received a diploma mathematics and at the same time a diploma engineer-electrician. He knew mathematics as much as the engineer needed, and at the same time he had a stunning engineering and mathematical intuition

- How does this affect the theory of information?

The most important consequence of complementarity is that if you measure one value, then indignant to it. It works, for example, in quantum cryptography. If unauthorized intervention was unauthorized in the communication channel, it must manifest itself. In this principle ...

- Built the security of information?

Yes, one of the "quantum" ways to protect information based on the complementary property.

The second method uses "clutch" (intricate). Coupling is another fundamental property of quantum systems that has no classic analogues. It refers to composite systems. If additional is also manifested for a single system, the adhesion property speaks about the connection between parts of the composite system. These parts can be spatially separated, but if they are in a linked quantum condition, a certain mysterious connection arises between their internal properties, the so-called quantum pseudo-tracking. Measuring one subsystem, one can somehow affect the other, and instantly, but influenced in a very thin way. The measure of such a coupling is determined by the correlation of Einstein-Podolsky-Rosen. It is stronger than any classical correlation, but does not contradict the theory of relativity, which prohibits the transmission of information at speeds, greater light speed. It is impossible to transfer information, and you can catch this correlation, and you can use it. The second class of cryptographic protocols is just based on the creation and use of units between the participants of this protocol.

- If someone interferes, then because of the coupling, you can learn about it?

If they interfere with one thing, another inevitably feels.

Coupling is probably the transfer of something. Any transmission occurs through something. What is the coupling mechanism?

I would not talk about the coupling mechanism. This property of a quantum-mechanical description. If you accept this description, the clutch flows out of it. How is the interaction been transmitted? With some particles. In this case, there are no such particles.

But there are experiments that confirm the existence of this property. In the 1960s, the Irish physicist John Bell brought an important inequality that allows you to experimentally determine whether there is a quantum clutch at long distances. Such experiments were carried out, and the presence of adhesion was confirmed experimentally.

If you want to create a consistent axiom system for a sufficiently substantive mathematical theory, it will always be incomplete in the sense that it will have a proposal, the truth or the falsity of which is not applicable

The coupling phenomenon is indeed very contrintuitive. His quantum-mechanical explanation was not made by some outstanding physicists, such as Einstein, de Broglel, Schrödinger ... They did not take a probabilistic interpretation of quantum mechanics with which the coupling phenomenon is connected, and it was believed that there should be a certain "deeper" theory that would allow to describe The results of quantum-mechanical experiments, in particular, the presence of a "realistic" adhesion, as, say, the classical field theory describes electromagnetic phenomena.

Then it would be harmonious to combine this property with the theory of relativity and even with the general theory of relativity. Currently, this is perhaps the most profound problem of theoretical physics: as a quantum mechanic to coordinate with the requirements of the general theory of relativity. Quantum field theory consistent with the special theory of relativity by the price of what amendments (renormal) are being made to the subtraction of the "endless constant". A fully mathematically consistent unified theory still does not exist, attempts to build it so far be resting in a dead end. Two fundamental theories that arose at the beginning of the twentieth century: a quantum theory and theory of relativity, - still not fully reduced together.

- Thinking also form processing information. What is the connection of thinking and theory of information?

In 2015, the two hundredth anniversary of George Bul was noted. This is an Irish mathematician, which opened the calculation of the functions of binary variables, as well as the logic algebra. He proposed to give the importance of "0" with a false statement, the value of "1" to the true statement and showed that the laws of logic are perfectly described by the corresponding algebra logic. It must be said that the impulse for this discovery was his desire to figure out the laws of human thinking. As they write in his biographies, when he was a young man, his mystical revelation was visited and he felt that he had to deal with the disclosure of the laws of human thinking. He wrote two important books that were not truly in demand at that time. His discoveries found wide applications Only in the twentieth century.

- In the famous sense, logic algebra, in fact, demonstrates the connection of thinking and mathematics?

You can say so. But, if we talk about the connection of thinking and mathematics, then in the twentieth century the most impressive achievement speaking about some deep internal contradictions or paradoxes, which are laid in the laws of human thinking, were the works of Kurt Gödel, who put the cross on the utopian and too optimistic idea David Hilbert axiomatize all mathematics. From the results of Gödel, in particular, it follows that such a goal is in principle unattainable. If you want to create a consistent axiom system for some sufficiently substantial mathematical theory, it will always be incomplete in the sense that it will have a proposal, the truth or the felt of which is not applicable. This sends some distant parallel with the principle of complementary in quantum theory, which also indicates the incompatibility of certain properties. Completeness and consistency turn out to be mutually additional properties. If this parallel is further done, then you can come to the thought, which may be for modern science It will seem confusion: knowledge has borders. "Commission, proud person," - as Fedor Mikhailovich Dostoevsky said. Of course, the electron, of course, is inexhaustible, but knowledge has boundaries due to the limb of that thought apparatus, which man has. Yes, we still do not fully know all the possibilities, but already somewhere, in some aspects, apparently, approach the borders. It is possible, so the problem of creating a scalable quantum computer is also so complicated.

Of course, the electron, of course, is inexhaustible, but knowledge has boundaries due to the limb of that thought apparatus, which man has. Yes, we still do not fully know all the possibilities, but already somewhere, in some aspects, apparently approach the boundaries

Maybe the fact is that it is not easy missing the possibilities of human thinking, and that the world as such is so internally contradictory that it is impossible to know it?

It can only show the future. In some sense, it is so clearly visible on the example of public life: how many attempts to build a harmonious society, and, although they led to new development - unfortunately, with great efforts and victims, the harmonious society has not been created. This internal contradiction, of course, is present in our world. However, as a dialectic teaches, contradiction, denial of denial is a source of development. By the way, certain dialectivity is present in quantum theory.

Of course, the fact that I am now saying is contrary to the existing historical optimism, roughly speaking, that you can build the "theory of everything" and explain everything.

Ludwig Faddeev, as he spoke in an interview to me, a supporter of the point of view that sooner or later such a theory will arise.

Such a point of view is probably based on the extrapolation of the ideas of the eyelid of enlightenment, the culmination of which was the unprecedented scientific and technical jerk of the twentieth century. But the reality puts us all the time in the face of the fact that science can a lot, but still not Almighty. The situation when different fragments of reality are successfully described by various mathematical models, only in principle consistent with the border regimens, may be laid in the very nature of things.

- You mentioned about a quantum computer. But his idea was born based on quantum theory of information ...

The idea of \u200b\u200beffective quantum computing was expressed by Yuri Ivanovich Manin in 1980. Richard Feynman wrote an article in 1984, in which he wondered: since the modeling of complex quantum systems, such as large molecules, takes everything more places and time to ordinary computerswhether it is impossible to use quantum systems for modeling quantum systems?

- Based on the fact that the complexity of the quantum system is adequate to the complexity of the task?

Approximately so. The ideas of quantum cryptography appeared, and the idea of \u200b\u200ba quantum computer sounded most loudly after Peter Shor suggested an expansion algorithm for multiple components natural Numberbased on the idea of \u200b\u200bquantum parallelism. Why did it cause such a resonance? The assumption of the complexity of solving such a task is the basis of modern open-key encryption systems, which are widely used, in particular on the Internet. Such complexity does not allow, even having a supercomputer, hack the cipher for any weather time. At the same time, the shore algorithm allows you to solve this task for an acceptable time (about a few days). This, as it were created a potential threat to the entire Internet system and everything that uses such encryption systems. On the other hand, it was shown that the methods of quantum cryptography are not amenable to hacking even with the help of a quantum computer, that is, they are physically protected.

Another important discovery was that it is possible to offer quantum codes that correct errors as in classical theory information. Why is the digital information so highly stored so high? Because there are codes that correct errors. You can scratch a CD, and still it will play recording correctly, without distortion, thanks to such corrective codes.

A similar, but significantly more sophisticated design was also proposed for quantum devices. Moreover, it is theoretically proved that if the probability of failures does not exceed some threshold, you can almost any scheme that performs quantum calculations, to make errors resistant by adding special blocks that are engaged not only to correction, but also internal security.

It is possible that the most promising path is the creation of a non-large quantum processor, but a hybrid device in which several qubians interact with a classic computer

When the experimenters began to work on the embodiment of the ideas of quantum informatics, difficulties were clear on the ways of their implementation. The quantum computer should consist of a large number of qubits - quantum memory cells and quantum logical processors that carry out operations on them. Our physicist Alexey Ustinov in 2015 realized a superconducting quantum qubit. Now there are schemes from dozens of qubits. Google promises in 2017 to build a computing device out of 50 qubs. At this stage, it is important that physicists successfully master innovative experimental methods that allow "to measure and target individualizing individual quantum systems" (Nobel Prize in Physics 2012). Chemists create molecular machines (Nobel Prize in Chemistry 2016) are moving in the same direction.

The practical implementation of quantum computing and other ideas of quantum informatics are a promising task. There is a constant resistant work of physicists, experimenters. But so far there has not been a technological breakthrough like the invention of the transistor, there are no quantum technologies that would be reproduced massively and relatively cheap, like the production of integrated circuits. If you could buy parts in the store and solder electronic circuits in the garage for the manufacture of a classic personal computer in the store, then it will not work with quantum.

It is possible that the most promising path is the creation of a non-large quantum processor, but a hybrid device in which several qubians interact with a classic computer.

Perhaps the human brain is a similar hybrid computer. In the popular book of English Physics, Roger Penrose "New King's Mural", the author expresses the opinion that there are certain biophysical mechanisms that are capable of performing quantum calculations, although this opinion is not divided by everything. Famous Swiss theorist Claus Hepp says that it cannot imagine the wet and warm brain to carry out quantum operations. On the other hand, Yuri Mannin, which has already been mentioned, allows that the brain is a large classic computer in which a quantum chip is present responsible for intuition and other creative tasks. And also probably for "freedom of will", since in quantum mechanics the accident is laid fundamentally, in the very nature of things.

Unlike conventional systems (with a secret key), systems that allow open transmission (open) part of the key along an unprotected communication channel are called public key systems. In such systems, the public key (encryption key) differs from the personal key (key of the decryption), so they are sometimes called asymmetric systems or two-on systems.

Alexander Holedo

Modern teenagers are difficult to imagine the world without mobile phones, computers, digital cameras, MP3 players and other attributes of the information technologies. Meanwhile, the historical moment, a predetermined principal transition to the "digit", is determined rather accurately

The digital revolution began in 1948, when the transistor who opened the miniaturization road was invented electronic devices and a radical decrease in material and energy costs to create information processing systems (Hardware). In the same year, the fundamental work of the American Mathematics engineer Claude Shannon, the Father of the Theory of Information, which has substantiated transition to digital presentation and digital data processing (Software). Even earlier, the works of our scientist V.A. Kotelnikova on the basics of noise-resistant communication, which anticipated some ideas of Shannon.

Strong and at the same time weak Party The classical theory of information providing its versatility has become abstraction from the content and nature of the transmitted data. Such a theory is interested only in two aspects: the number of information transmitted and the transfer quality. These characteristics are associated with the inverse dependence: the more accurate we want to transfer the message if there is no interference in the communication channel, the more slows down the transmission. Special attention in the theory of information is paid to optimal characteristics, such as channel bandwidth, i.e. The maximum possible transmission rate when using coding-decoding, providing correction of errors caused by interference.

Information Physical

One of the pioneers of the physical theory of information of Rolf Landauer, for many years worked in IBM, argued that information is physical, and distracted by her physical natureThe researcher does not always justify the assumption. The fundamental storage medium is an electromagnetic field, for example in the form visible lightor radio waves. Under normal interference conditions during signal transmission due to the chaotic behavior of the field quanta (photons), which has a thermal nature. It turns out, a decrease in temperature to absolute zero It does not lead to the complete disappearance of noise: the so-called vacuum fluctuations caused by the quantum nature of radiation. The quantum properties of light are particularly pronounced in the coherent radiation of the laser, which differs from the radiation of a natural heat source in the same way as the ordered column of soldiers differs from a motley fair crowd. Already in the 1950s. Scientists thought about the fundamental quantum mechanical limits of accuracy and the rate of information transfer. Further development of information technologies, the achievement of quantum optics, electronics and supramolecular chemistry, exploring the cybernetic properties of high molecular compounds, makes it suggest that in the near future such restrictions will become the main obstacle to further extrapolating existing technologies and the principles of information processing.

New questions for old theory

To explore the qualitative conclusions of physicists in the exact form, the synthesis of mathematical ideas of the theory of information and quantum mechanics took place. In the 1960s There were already quantum statistical mechanics and quantum field theory, however, these disciplines are aimed at a different range of tasks associated with the dynamics of quantum systems. So, in the statistical mechanics, the nearest relative of the information - entropy occurs and is widely used, but it acts there only as a thermodynamic characteristic. The information meaning of quantum entropy was clarified in the work of Ben Schumacher dedicated to quantum compression of data and published in Physical Reviews in 1995. The theory of quantum measurement was still close to the needs of a quantum measurement that was not yet born. However, she needed substantial improvement and development.

Any information transmission circuit consists of a transmitter (perhaps that includes a device encoding), a communication channel and, finally, a receiver (along with a possible decoding device). Usually all three mentioned components are described in the language of classical physics and statistics. The signal sent by the transmitter (for simplicity 0 or 1) is subjected to random interference and can be distorted. Therefore, the signal at the output of the receiver does not necessarily coincide with the sent signal, and the quality of communication is characterized by a loss of an error. It is usually necessary to develop a receiver design that would provide optimal detection or evaluation of the sent signal for a given channel and the transmission method. Such tasks are solved by methods of the theory of statistical solutions. The theory of information has a more ambitious goal: for a given channel with interference, develop such coding methods and decoding a signal that would allow for a unit of time as many messages, almost invulnerable to interference. The maximum maximum speed of such a transmission is called channel bandwidth. Invented hitrophic error correction methods that are suitable for transmission and reliable storage of information. Capture plays the role of a "catalyst" that detects the hidden information resources of the quantum system, but the in itself does not allow to transmit information: it would mean instantaneous transmission to the final distance

Study quantum communication channels are necessary, because Every physical channel is ultimately quantum. In the quantum world, the transmitter prepares the quantum state of the media of the information depending on the incoming message. For example, a transmitter can be a laser that emits either vertically or horizontally polarized photons. The sending binary signal is encoded by the corresponding state of the radiation field. However, in the communication channel, it is usually distorted, and states other than sent by the transmitter come to the receiver. The receiver carries out a quantum measurement of one or another physical value, possibly, followed by the processing of the obtained classical information. The final result of such a measurement is the output signal 0 or 1, which gives a more or less reliable estimate of the sent source signal, and the quality of the communication line is again characterized by an error probability. Analogy with the classic line of communication is obvious. Thus, the need arises in the quantum theory of statistical solutions and methods of optimal estimation of the parameters of quantum states based on the measurement results. The prospect of creating coding-decoding methods, taking into account the quantum mechanical nature of the media of the information, which would allow to compensate negative influence Quantum noise. Returning to the statistical mechanics, we note that such procedures cause the association with the famous "Maxwell Demon", creating order from disorder, but they are more modest to be put in front of them, but to achieve a goal: maintaining the island of order in the sea of \u200b\u200bchaos. The magnitude of this island and determines the bandwidth of the communication channel.

An close consideration of the concept of quantum measurement from an information and statistical point of view has led to a new paradoxical conclusion: adding an independent quantum noise in observation allows you to increase the amount of information obtained. The paradox is that this is never happening in classical statistics: the addition of noise (randomization) only spoils the quality of observations. In quantum optics there is an example of a real measuring procedure that uses an independent source of quantum noise (a kind of quantum roulette). We are talking about optical heterodinations, in which the radiation, carrying information, consists of reference radiation from an independent source. This kind of procedure allows you to carry out an approximate joint measurement of both components of the signal, electrical and magnetic, despite the fact that the quantum theory prohibits their accurate joint measurability. From a mathematical point of view, such measurements are described by overcrowded vectors other than full orthonormal systems (bases) of the standard measurement theory of neimane. In particular, optical heterodinization statistics are described by the overcrowded system of coherent vectors, such an effectively used in the works of the Nobel Wine Waubeca laureate. Any overflowing system of vectors in the space H can be described as a projection on H basis in some comprehensive space K, resulting from H adding independent (randomizing) degrees of freedom. It turned out that overflowing systems are only a special case more general concept Probabilistic operator-valued measure studied by Soviet Mathematics MA Namerka back in 1940. and found a natural place in the quantum theory of statistical solutions created in 1970-1980.

Natural quantum computer

It is possible that in nature a quantum computer has long existed. It is expressed that the elements of quantum computer are present in human thinking, and then quantum informatics opens up new perspectives for a fundamental explanation of possible thinking algorithms. Let us dwell on those features of human thinking that truly cause associations with quantum natural stances

1. The ability of the holistic perception of information in contrast to the decomposition into the components of the properties; Perhaps the eye is able to take not only the classic states of incoming light, but also directly quantum states of photons than and the special power and bandwidth of visual communications are explained, as well as their organic communication with the recognition of images.
2. The similarity of additionalness between the action and reflection and quantum supplement between the position and speed, which is also paid to Nils Bohr in its physical and philosophical essays. It is noteworthy that when developing the concept of quantum supplity, Bor proceeded from the already existing similar concept of vitalism in biology
3. Trafficking features (or nonlocality) when the information contained in the association of the subsystems of some complex system, surpasses the arithmetic amount of the amount of information obtained from subsystems.
4. The phenomenon of consciousness-subconscious. It is difficult to resist such (of course, extremely simplified) analogy: a noncommutative algebra of quantum mechanical observable, in which at any time "scans" some available observation commutative (classical) subalgebra
5. Organic combination of analog and digital methods, effective parallelism information processing

Of course, these and other considerations, such as the presence of intuition and free will, are indirect in nature and are not inevitable with the inevitability of the human brain or in the nervous system of other living beings there are "quantum microchips" or other quantum-physical mechanisms responsible for non-classical calculations and appropriate behavior. But they may indicate that the work of the brain is fundamentally incorporated by the functions of an arbitrarily perfect and complex classic supercomputer, and then theoretical models of such systems should take into account this non-classicity.

The effectiveness of mathematics

The quantum theory of statistical solutions and information is based on a far-reaching logical development of the mathematical apparatus of quantum physics, supplemented by statistical interpretation. There are other interpretations, such as multi-volume, but all of them are too exotic to seriously compete with the statistical, which is also called "minimal", since it relies only on the possible statistics of quantum measurements and does not attract special assumptions about the mechanism of the emergence of this statistics. Statistical interpretation is so organically fused with the mathematical structure of the quantum theory, which arises as if by itself. Those objects of Hilbert spaces that previously seemed purely mathematical abstractionsThanks to the statistical interpretation become the twins of physical ideas and concepts. This happened with the above-mentioned overflowing systems and probabilistic operator-valued measures, it also happened with an abstract concept of a completely positive display from the theory of operator algebras, which turned out to be an adequate mathematical model of the quantum channel with noise.

Historically, the quantum information theory originated when considering fundamental quantum mechanical restrictions. The simplest of them is known since the 1920s. The ratio of the uncertainty of Heisenberg. In the 1970s. Thinner mathematical facts were established, such as entropy inequality, which limits the amount of information from above, which can be transmitted by the carrier subject to the laws of quantum mechanics (for example, the radiation of the laser). However, in 1980-1990. Scientists have concluded that quantum theory not only introduces its limitations, but also opens up fundamentally new opportunities, such as quantum teleportation and other effective communication protocols, physically resistant quantum cryptography protocols, effective algorithms for solving difficult computing tasks, etc. As a result logical development The apparatus of a quantum theory, equipped with statistical interpretation, and if we assume that quantum theory and its minimum interpretation have unlimited applicability, then there is no reason to doubt the principal possibility of new effective applications quantum theory. However, everything is not so simple. A quantum computer is a hypothetical computing device using specific quantum effects and a much superior in its capabilities of any classical computing machine.

The conferences on the quantum theory of information still retain a pleasant and rather rare feature: they are united by both theoretical specialists, up to specialists in the very abstract sections of mathematics and physicists directly involved in the experiment. At one such conference, the Experimentator's scientist began a report with an illustration on which a luxurious "Cadillac" was depicted with the inscription "Theory" and a modest "trabant" - "Experiment". The separation of the theory from experimental implementations is really great. Any experiment, imposing manipulation of individual microparticles, is extremely complicated due to their supersensitiveness to any external influences. Moreover, the difficulties of implementing the prescriptions of quantum theory are also included in its foundation: it provides a mathematical model for any actually observed phenomenon of the microworld, but it gives only the most common hints on how to move in the opposite direction - from the element of the mathematical model to its material prototype . In the unsurpassed treatise of the Dirac field "Principles of Quantum Mechanics", this problem is described as follows: "There is a natural question: can anyone be measured? Theoretically, this question can be answered - yes. It can practically be that it can be very difficult to build such a device that could It would measure some specific observed. It is possible that the experimenter cannot say how to build such a device, but the theorist can always imagine what measurement can be produced. " In other words, there is no regular way to give a constructive description of the corresponding measuring procedure, neither guarantee that such a description is possible in principle. It remains only to believe that it will be wound sooner or later. We give an example from quantum optics. The theory is well known for the state of radiation with a certain number of photons (they are called the states of the Fock). Today, no one doubts the existence of photons, but there is still no way to generate such states. There were theoretical proposals, in particular, based on the use of optical feedback, and only recently, Japanese scientists managed to implement it in the experiment. But, in particular, the reliability of the quantum cryptography protocol is based on the assumption that the secret key is distributed using single photons. As a real source, a weak coherent laser signal is used, for which the probability of the appearance of more than one photon is small. But it leaves a loophole for a potential interceptor "extra" photons.

To date, a number of fundamental experiments on quantum information processing have already been implemented. We mention only the well-known experiments A. Tsaylinger and J. Kimbla in teleportation of photon states, as well as active quantum cryptographic lines implemented by N. Jizen Group in Switzerland and S.N. Molotkova in Russia. Studies of theoretical and experimental aspects of quantum informatics are conducted in all developed countries, Including in Russia.

Two differences

The fundamental differences between the classic and quantum worlds can be expressed in a nutshell: extraness and clutch. An additional value means the presence of such properties of the same object that are fundamentally not available to joint observation. Various physical measurements of microjects are carried out by different macroscopic experimental settings, each of which implies a complex and specific organization of the spatial-time medium. Methods of such an organization corresponding to different observed properties can be mutually exclusive, i.e. Additional. In the language of mathematics, additional values, such as coordinate and impulse, electrical and magnetic fields, spin components are depicted by non-converted (non-compromising) operators. They have a ratio of uncertainties prohibiting accurate joint measurability, so that precisely is responsible for specific information constraints.

Extra value also leads to the fact that the states of the quantum system cannot be specified by simply enumeration of properties, i.e. Point in any phase space. Instead of this state, are described by vectors in some linear (Hilbert) space H, and any superposition (linear combination) of vectors also sets the state.

The new unusual possibilities of quantum systems are usually associated with adhesion (Entanglement; In Russian literature, the translation "intricacy", "confusion") is also used. It is based on the unusual properties of compound quantum systems, which are described by tensor (and not Cartesome, as in classical mechanics) by the product of HA Z Hb spaces with subsystems. By virtue of the superposition principle, the space of the AB composite system along with the vectors of the works A Z B should contain all sorts of their linear combinations. The states of the composite system defined by vectors are called non-depleted, and all other connected. Coupliness is quantum Property, partly a kindred classical correlation, but it is not reduced to it (in physics they talk about the correlations of Einstein-Podolsky-Rosen). Captured states are not uncommon in quantum physics: usually they arise as a result of the interaction or decay of quantum systems. However, the quantum theory does not exclude the possibility of a detailed state for a pair of particles, which, once provocating, scattered on the macroscopic distance. On the unusual "telepathic" properties of such a pair and pointed out at one time Einstein, Podolsky and Rosen. Recent experiments confirm the possibility of artificially creating the inner coupling of photons and even massive microparticles at distances of about several meters, although such a phenomenon is never observed in natural conditions and disgusting the very nature of the classical macroscopic world. The method of describing the surrounding world, which underlies the reports of the space-time report, was called "Local Realism". Whatever the combination of quantum mechanics and the general theory of relativity - on noncommutative geometry, strings theory, nonlinear quantum mechanics, trajectory or other approaches - it will have to resolve a contradiction between quantum coupling and local realism.

Quantum Channels and Information

The large section of the quantum theory of information is devoted to the quantitative coupling theory. It turns out that clutch can be measured quantitatively as the temperature or other physical characteristics of the state. Moreover, it can be concentrated, "dilute", forward; It can exist in the latent "bound" form and manifest only in special circumstances.

In the case of composite quantum systems, it makes sense to speak not only about adhesive and non-delicate states, but also on the corresponding measurements. At the same time, if the quantum systems A and B are in a non-depleted state, then the maximum amount of state information obtained from the measurements of the composite AB system may be larger than the amount of the amounts of information obtained from measurements of the systems A and B. Such non-classical strict superadditivity of information is manifested in the study The bandwidth of the quantum communication channel.

In the quantum case, the concept of bandwidth is branched to generate a whole "zoo" of the information characteristics of the channel depending on the type of transmitted information (quantum or classical), as well as from the additional resources used in transmission. Let us bring briefly on the four main inhabitants of this zoo. The channel is defined by a positive display T, converting the state at the input to the output state. This mapping is a compressed statistical description of the results of the system interaction at the entrance with its environment (noise). The property of positiveness guarantees from the appearance of negative probabilities, and the adverb "quite" means that positivity should be performed not only for the channel T, but also for its extensions of the TT type ", where T" - any other channel that in particular allows us to consider multiple use Channel. The most important characteristic of the quantum channel is its classical bandwidth C (T), i.e. Limit maximum speed of unmistakable classical messaging when using optimal encoding / decoding of long messages. From the above-mentioned entropy inequality implies that the number of classical information transmitted cannot be greater than Log D, where D is the dimension of the space of the quantum media information. Thus, the fact that any Hilbert space contains infinitely many different vectors of states, it does not help to transfer an unlimited amount of information: the more states are used to transmit, they are closer to each other and, therefore, indistinguishable.

However, as the American scientists, Charles Bennett and Peter Shore, the classical bandwidth of the channel T can be increased by using additional coupling between the input and channel output. At the same time, the adhesion itself does not allow to transmit information, because This would mean instantaneous transfer to the final distance. Capture plays the role of a "catalyst" that detects the hidden information resources of the quantum system. If the T is a channel without noise, then the gain in bandwidth provided by super-delicate coding, dvorat. The stronger the channel is different from the ideal, the benefits are more, and for the channels with a very big noise can be great. The classic bandwidth using the CEA (T) clutch state is the largest. At the European Congress of Mathematics in Amsterdam, the quantum theory of information is highlighted in a special direction

When transferring classical information on the quantum channel, the message is written in the quantum state. However, the whole full of information content cannot be reduced to the classical message and deserves a special term - quantum information, because Quantum state contains information about the statistics of all sorts, including mutually exclusive (additional) system measurements. The number of quantum information is measured by the value of the entropy of the state. The fundamental difference between quantum information from the classical lies in the impossibility of copying. Simple reasoning, based on the linearity of the equations of quantum evolution, shows that there is no "quantum xerox", i.e. physical device that allows you to copy an arbitrary quantum state. However, the theory predicts the possibility of a non-trivial way to transmit quantum information, in which the state carrier is not physically transmitted, and only some classical information is sent (the so-called telepatch quantum state). The necessary additional resource becomes the clutch between the input and the output of the communication channel. To reduce the transmission of an arbitrary quantum state only to the transfer of classical information without using an additional quantum resource is not possible: Since the classical information is copied, it would mean the possibility of copying and quantum information.

Quantum bandwidth Q (T) is the limit maximum number of quantum information, which can be an arbitrarily transmitted to the T. T. There is a deep analogy between the quantum channel and the channel with a cutter, and the environment of the system under consideration is played in the quantum case. The value of Q (T) is closely related to the cryptographic characteristics of the channel, such as bandwidth for the secret transmission of classical information CP (T) and the rate of distribution of a random key. It is the smallest of bandwidth, because Presents the highest requirements for the channel.

Calculation or valuation of quantities Q (T), CP (T), C (T), CEA (T) is an important and difficult mathematical task. At one time, the appearance of quantum mechanics had a powerful mutually engraving effect on a number of mathematics regions: primarily on the theory of operators, operator algebras, group representations.

The process continues now, and in it, the achievements of the quantum theory of information play an increasing role. Thus, the study of coupling stimulated progress in understanding the geometry of tensor products, and the channels and coding theorems were closely related to the structures of positiveness in operator spaces and algebras. The new impetus received a noncommutative analysis; Even in such a seemingly well-studied area, like the theory of matrices, new bright results and new difficult and interesting problems appeared. At the European Congress of Mathematics 2008 in Amsterdam, the quantum theory of information was highlighted in a special direction, which is devoted to a number of invited reports.

"Meso": on the border "Micro" and "Macro"

The progress of microelectronics and nanotechnology is approaching the frontier, followed by ignoring the quantum nature of the information carriers will not be possible. Elements of modern computing equipment are only two to three orders of magnitude excellent characteristic atomic dimensions. The Honorary Chairman of the Board of Directors and the founder of Intel Corporation Gordon Moore believes that only 10-15 years will leave this difference. Then the will will have to look for new solutions, and the fundamental results of the quantum information theory can play a decisive role.

A quantum computer is a hypothetical computing device using specific quantum effects and therefore much superior in its capabilities of any classical computing machine. Its memory (quantum register) should consist of a plurality of elementary cells - cubes, which are in the detached state, and the operations suggest controlled quantum mechanical interaction between them. Data in the process of computing is quantum information, which, at the end of the process, is converted into the classical way to measure the final state of the quantum register. Winning in quantum algorithms is achieved due to the fact that when using one quantum operation big number The coefficients of the superposition of quantum states, which in virtual form contain classical information, is converted simultaneously (quantum parallelism).

The quantum computer is on the verge between the micro and macromir, which causes the difficulties of its incarnation. The main technical obstacle to the implementation of a quantum computer is decorativeization - the decomposition of quantum superpositions, due to ultra-sensitivity of microsystems to the external influences of the macromir. If the decorative speed does not exceed a certain threshold, the use of quantum codes that correct errors, theoretically allows you to make quantum computing noise-resistant. However, while the size of the quantum register should be increased by order. Intensive searches for solving these problems are underway: theoretical methods of optimizing the architecture of the quantum computer are developed; the schemes of adiabatic calculations, quantum cellular automata, calculations based on measurements are proposed; The idea of \u200b\u200ba topological quantum computer is discussed, physically resistant to errors. Experimentally examined models of qubits based on the principles of nuclear magnetic resonance, quantum optics and electrodynamics, semiconductor quantum dots, ion traps, superconducting meso structures, etc.

Quantum informatics has become a new interdisciplinary scientific direction at the junction of physics, computer science and mathematics, which raises new important issues and gives the key to understanding some fundamental patterns of nature, until recently remaining explorers. Its theoretical developments stimulate both new achievements in the field of mathematics and the development of experimental physics, significantly expanding the possibilities of manipulating the states of microsystems and is potentially important for the emergence of new effective technologies.

ADDITIONAL LITERATURE

1. Bor N. Nuclear Physics and human knowledge, M.: Il, 1961.
2. Valiev K.A., Kokin A.A. Quantum computers: hopes and reality (2nd ed.). M.: Iki, 2004.
3. Valiev K.A. Research in the field of quantum technologies in computer science and metrology // Herald of the Russian Academy of Sciences, 2003. T. 73. N 5. P. 400-405.
4. Kadomtsev B.B. Dynamics and information. M.: UFN, 1999.
5. Nielsen MA, Chang I. Quantum calculations and quantum information (per. From English). M.: Mir, 2006.
6. Kholevo A.S. Introduction to quantum information theory. M.: MCNMO, 2002.
7. Kholevo A.S. Probabilistic and statistical aspects of quantum theory (2nd ed.). M.: Iki, 2003.

Alexander Semenovich Kholevo - Professor, Doctor of Physical and Mathematical Sciences, works in the Mathematical Institute. V.A. Steklov wounds.

The area of \u200b\u200bscientific interests is a quantum information theory, quantum calculations; noncommutative theory of probabilities, quantum random processes, dynamic (Markov) semigroups; Statistical structure of quantum theory, quantum measurements.

October 31, 2015 at 22:52

"Quanta" here and now (part 3)

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In previous articles, I briefly spoke about the prerequisites in the development and, which led to the emergence of quantum information and quantum calculations as such. Today I wanted to consider in a similar way another direction that made a significant contribution: theory of information.

Information theory.

In the 40s Simultaneously with the development of informatics, cardinal changes took place in understanding the concept of communication. In 1948. Claude Shannon published several outstanding work, which laid the foundations modern theory Information and communications. Most likely the most an important stepShannon made in that he introduced the mathematical definition of the concept of information. Here, try to think based on the simplest, solar considerations, over the next question: how would you come to the mathematical definition of the concept of "source of information?" In the world, several decisions of this issue appeared at that time, however, Shannon's answer was the most fruitful improved improvement. Its use led to a number of certain serious results, and the creation of a theory that adequately displays many real communication problems.
Shannon was interested in two key questions that are directly related to the exchange of information on the communication channel. Firstly, what resources are required to transmit information on the channel? Secondly, can the information transmits in such a way as to be protected from noise in the communication channel?And he answered both of these questions, proving two fundamental theorems. First - coding theorem for channel without noise - Determining the number of physical resources required to store the output data source of information. Second - channel coding theorem with noise - Shows the amount of information that can be securely transferred through the channel in the presence of noise. Shannon showed to achieve reliable transmission in the presence of noise. It is possible to use codes that correct errors. The Channel Theorem for the noise channel sets the upper limit of the information protection providing similar codes. Unfortunately, the theorem does not give an explicit type of codes to help achieve this limit in practice. However, there is a challenging theory that allows you to develop a good code that corrects errors. Such codes are widely used, for example, in computer modems and satellite communications systems.

Quantum information theory.

Quantum information theory developed approximately similarly. In 1995. Ben Schumacher Proved an analogue of the Channon theorem on coding in the absence of noise, by defining a quantum bit (qubit) as a real physical resource. But it is worth noting that there is still no analogue of the Channon theorem about coding for the channel with noise in relation to quantum informatics. Despite this, the theory of fixing quantum errors was developed, allowing quantum computers to effectively calculate in the presence of noise, as well as securely transmit information.
Classic error correction ideas were very important and useful in the development and understanding of codes that allow you to correct quantum. In 1996, working independently Robert Kalderbank with Peter Shor and Andrew Styn. Opened important class Quantum codes called now CSS codes according to the first letters of their surnames. Later, these codes were attributed to the category of sylpectic, or stabilizing, codes. Discovery data was based on a large extent on the ideas of the classical linear coding theory, which significantly contributed to the rapid understanding of the codes of correction of quantum errors and their further use in the field of quantum computing and quantum information.
This theory was developed to protect quantum states from noise, but what about the transfer of classical information on the quantum channel? Is it effective at all, and if so, how much? And here it was already waiting for several surprises. In 1992. Charles Bennet and Steve Wisner They explained to the world how to transfer two classic bits of information by transferring only one qubit. This was called superproof coding.
Even more questions and, accordingly, more interest are the results in the field of distributed quantum calculations. Imagine that you have two computers connected to the network on which some task is solved. How many gears on the network will need to solve it? The answer to this question is not so important, another thing is important. Not so long ago, it was shown that for a similar quantum system, an exponentially fewer time may be required to solve the problem than for classic network computers. This is definitely a very significant result, but there is one minus - unfortunately, these tasks do not pose a special interest in real conditions.

Network quantum information theory.

The classical information theory begins with studying the properties of a single communication channel, while in practice we often deal with a network of multiple channels, and not with one. Properties are just such networks and studies the network theory of information that developed into extensive and complex science.
Network quantum theory of information, on the contrary, is largely born. We are extremely little knowing only about the possibilities of transmission in quantum networks, not to mention everything else. In recent years, a large number of results and developments have been obtained, even some quantum networks are created, but there are no single network theory for quantum channels. And then everything again rests against the conflicting intuition properties, illustrating the strange nature of quantum information.

Conclusion.

In this way, you can sum up the following: everything is not at all smoothly in the available quantum theory of information and a lot more about what to do. All the most important issue remains the proof of the theorem of a similar chennon theorem on coding for a channel with noise. In addition, it is necessary to search for practically important tasks for which distributed quantum calculations have a significant advantage over distributed classic. Well, and as I said, it is necessary to create a single network quantum information theory, since we still hope to create any less global quantum network. All this is the most important areas of research in this area.

Quantum information theory, section of mathematics, in which the general patterns of transmission, storage and transformation of information in systems subject to the laws of quantum mechanics are studied. Quantum information theory uses mathematical models to study the potential capabilities of such systems, and also develops the principles of their rational and noise-resistant construction. Quantum information theory leads to a new understanding of the fundamental patterns of quantum theory, its foundations and reality relationships, and also stimulates the development of experimental physics.

The quantum theory of information was formed as an independent discipline in the 1990s, but its origin refers to the 1950s and is associated with the emergence of the foundations of the classical theory of information and noise-resistant communication in the works of V. A. Kotelnikov and K. Shannon. At the initial stage (1950-80s), the main issue of quantum theory of information was to find out the fundamental restrictions on the possibility of transferring and processing information caused by the quantum-mechanical nature of its carrier. The development of information technologies in the direction of microminiature, the use of achievements of quantum optics and quantum electronics, supramolecular chemistry, exploring the cybernetic properties of molecular compounds, lead to the conclusion that in the foreseeable future, these restrictions will become the main obstacle to further development existing technologies and information processing principles. On the other hand, the appearance of a quantum computer, quantum cryptography and new communication protocols in the 1980s -90s, and new communication protocols allows not only about the restrictions, but also about the new features concluded in the use of specific quantum resources, the so-called quantum parallelism, coupling ( confidence) quantum states and additionalness between measurement and indignation.

In a quantum information theory, the information carrier is the state of the Chanteer system H, which is an information resource, since it has statistical uncertainty. The mathematical description of the pure state is the design operator (projector) p ψ on the vector ψ from the Hilbert Space Space System N. The mixed states are also considered, representing a statistical ensemble of pure states of Pψ I with probabilities P i. This state is described by the density operator ρ \u003d σ i p i ρ ψ, which is characterized by the following properties: ρ is a positive operator; ρ has a single trace. Thus, the eigenvalues \u200b\u200bλ j of the density operator form the probability distribution. Entropy of this distribution

the entropy von Neymann, like the entropy of Shannon of the classic source of messages, is a measure of uncertainty, that is, the content of the state described by the operator p.

When transmitting a classic (non-quantum) message via a quantum communication channel, it is written in a quantum state by setting the values \u200b\u200bof the parameters of the device forming the state. However, the whole full of information content of the quantum state cannot be reduced to the classical message, and therefore for the information contained in the quantum state, the special term "quantum information" is used. This is due to the fact that it contains the statistics of all sorts, including mutually exclusive (so-called additional), measurements over the system. The most striking difference between quantum information from the classical is the impossibility of copying, the linearity of the quantum evolution equations leads to the impossibility of a "quantum xerox", that is, a physical device that allows you to copy arbitrary quantum information.

Similarly, the number of classical information can be measured by the minimum number of binary characters (bits) required for encoding (compression) of the message, the number of quantum information can be defined as the minimum number of elementary quantum systems with two levels (Q-bits, qubits) required For storing or transferring this ensemble of quantum states with optimal coding. For asymptotically unmistakable coding of a quantum message of the length n, in which the states of the Pψ I appear with probabilities P i, the required number of Q-bits asymptotically (at n → ∞) is equal to NN (ρ). This means that the dimension of the quantum system in which the optimal compression of quantum information is carried out in a state ρ is asymptotically equal to 2 NH (ρ), which gives an informational interpretation of entropy von Neuman.

The size of the quantum state adhesion is based on unusual (for classical systems) properties of compound quantum systems, which are described by tensor (and not Cartesome, as in classical mechanics) by the work of the subsystems. The space of the composite system AB, along with the vectors of the form ψ A ⊗ψ in, contains all sorts of their linear combinations σ j ψ j a ⊗ψ j b. The cunt of the composite system defined by vectors is called non-depleted, and not reduced to those with adhesive. The clutchness is a purely quantum property, partly the kindred classical correlation, but it is not reduced to it (they talk about Einstein's correlations - Podolsky - Rosen). It is the presence of adhesive states that contradicts a hypothesis about the possibility of a classical statistical description of quantum systems that satisfy the so-called physical requirement of locality. The quantitative theory of adhesion is a kind of combinatorial geometry of tensor products of Hilbert spaces.

A dual way in composite quantum systems exist clutch and non-depleted observed (measurements). If the quantum systems A and B are in a non-depleted state, then the maximum Shannon amounts of information ι α, ι in, ι AV on states A, and the composite AB system satisfy in the general case the ratio of I AV\u003e IA + I B. This non-classical phenomenon The strict superadditimity of information plays an important role in the theory of bandwidth of the quantum communication channel.

The concept of communication channel and its bandwidth, which gives the limit speed of unmistakable transmission, plays a central role in the theory information. The mathematical approach gives these concepts universal significance: for example, the memory of a computer (classical or quantum) can be considered as a channel from the past to the future, then the bandwidth gives a quantitative expression for the limiting capacity capacity when error correction. The importance of consideration by quantum communication channels is due to the fact that every physical channel, ultimately, is a quantum and such an approach allows the fundamental quantum-mechanical patterns. It is essential that in the quantum case, the concept of bandwidth branches, generating a whole range of channel information characteristics, depending on the type of information transmitted (quantum or classical), as well as additional resources used in transmission.

In the quantum information theory, the quantum communication channel is set by the mapping F, which translates the state at the input state at the output state, ρ → f [ρ], which gives a compressed statistical description of the results of the system interaction at the entrance with its environment (noise). The classic bandwidth C (F) is defined as the maximum transfer rate of classical messages through the channel F ⊗ N with n blocks with asymptotically (with N → ∞) disappearing error and equal maximum quantity Shannon Information, which can be obtained by using arbitrary coding of classical messages to the state at the input and quantum measurements - decoding at the channel output. For the value of C (F), an explicit expression was obtained through the entropic characteristics of the channel, which constitutes the content of the coding theorem of the coding - Schumacher - Westmorend.

The classical bandwidth of the channel F can be increased by using the coupling between the input and channel output, while only the clutch does not allow to transmit information, the clutch plays the role of a "catalyst", which detects the hidden information resources of the quantum system. If f is the perfect channel, i.e. the channel without noise, then the gain in bandwidth delivered by the so-called superphoto encoding, dvorat. The more channel differs from the ideal, the benefits more and asymptotically (for channels with very large noise) may be great. The corresponding maximum transmission rate with EA (F) is called the classical bandwidth using a linked state; For it, there is also an obvious formula, obtained by American scientists. Bennett, P. Shor, J. Smolin and A. Tapulov.

The conversion of the quantum state ρ → f [ρ] can be considered as the transfer of quantum information. The theory predicts the possibility of a non-trivial transmission method, in which the states are not physically shipped, but only some classical information is transmitted (the so-called quantum teleportation). At the same time, the necessary additional resource is again the clutch between the input and output of the communication channel. To reduce the transmission of an arbitrary quantum state only to the transfer of classical information, without using an additional quantum resource, it is impossible: since the classical information is copied, it would mean the possibility of copying and quantum information.

In connection with the development of quantum codes that correct errors, the question arose about asymptotically (at n → ∞) error-free transmission of quantum information channel F ⊗ N. In this case, the quantum bandwidth Q (f) is defined as the maximum speed of the quantum information. The study of quantum bandwidth is based on analogies between the quantum channel and the classic channel with the interception, and in the quantum case, the role of the information interceptor plays the environment of the system under consideration. It turned out that the value of Q (f) is associated with the cryptographic characteristics of the channel, such as the bandwidth for the secret transmission of classical information with P (F) and the rate of distribution of a random key. The bandwidths of the channel F are associated with the relations Q (φ) ≤ s p (f) ≤ С (φ) ≤ s (f).

The large section of the quantum information theory is associated with the research of systems with continuous variables based on the principles of quantum optics. For them, a number of results relating to bandwidth, state curbral and other information characteristics were obtained. Many experiments on quantum processing of information, including super-proper coding and teleportation of photon states, as well as quantum cryptographic protocols, are implemented precisely in such systems. See also quantum connection.

Lit.: Bennett S. N., SHOR R. W. Quantum Information Theory // Transactions On Information Theory. 1998. Vol. 44. No. 6; K. A. Valiev, Kokin A. A. Quantum computers: hopes and reality. M., 2001; Quantum information physics / edited by D. Boumerster, etc. M., 2002; Kholevo A. S. Introduction to quantum theory of information. M., 2002; He is Probabilistic and statistical aspects of quantum theory. 2nd ed. M., 2003; Nielsen M. A., Chuang I. Quantum calculations and quantum information. M., 2006; Hayashi M. Quantum Information: An Introduction. IN.; N. Y., 2006.