A. S. Holevo. An Introduction to Quantum Information
Theory (In Russian).

Moscow Center of
Continuous Mathematical Education (MCCME), Moscow 2002.

Some
50 years ago two revolutionary discoveries were made,

which
to a great extent determined the image of the contemporary
world.

These were -- invention of transistor,

opening way
to miniaturization of digital

computers and radical reduction of
material and

power expenses in information processing systems, and
-- creation of

the mathematical foundations of information theory,
giving the principles of

rational and reliable design of such
systems and arrays of data.

At present we are witnessing
emergence of

theoretical and experimental foundations of the
*quantum information science*. Principles of

quantum
cryptography were already practically implemented
and

successfully demonstrated.

Under discussion is the idea of
quantum computer, promising

overwhelming perspectives (a
practical

realization of this project, however, would require a
technological

revolution comparable with the discovery of
transistor.)

Independently of how soon such an ambitious project
could be realized, the

quantum information theory represents a new
exciting field

of key importance to a number of fundamental
problems,

which up to recent time were out of the scope of
scientific

research. It also stimulates the development of
experimental techniques

and high precision technologies for
manipulation of microsystems, potentially

important for new
efficient applications.

A central result of the classical
information theory is *codingtheorem*, establishing
the possibility of reliable data transmission and
processing

at the rates not exceeding the definite value (

given information processing system (for definiteness one speaks of a

(

capacity of quantum communication channels arose soon after publication of the

pioneering Shannon's paper and goes back

to even earlier classical papers of Gabor and Brillouin, asking for

fundamental physical limits on the rate and quality of information

transmission. This work laid a physical foundation and raised the question of

consistent quantum information treatment of the problem. Important steps in

this direction were made in the seventies when quantum statistical decision

(detection and estimation) theory was created, making a quantum

probabilistic frame for this circle of problems. At that time the quantum

entropy bound and strict superadditivity of classical information in quantum

communication channels were established.

A substantial progress has been achieved during the past years, when a number

of quantum coding theorems was discovered, proving the achievability

of the entropy bound. Moreover, it was realized that quantum channel is

characterized by the whole spectrum of capacities depending on the nature

of the information resources and the specific protocols used for the

transmission. To a great extent this progress was stimulated by an interplay

between the quantum communication theory and quantum information ideas

related to recent development in quantum computing. This new age

of quantum information science

is characterized by emphasis to the new possibilities (rather than restrictions)

opened by the quantum nature of the information processing agent.

On the other hand, the

question of information capacity is important for the theory of quantum

computer, particularly in connection with quantum error-correcting codes,

communication and algorithmic complexities and a number of other important issues.

This book is intended to give a systematic, self-contained and rigorous

treatment of the aspects of quantum information theory related to the notion of quantum

communication channels and their capacities. It is preceeded with

a detailed introduction to the statistical structure of quantum theory

providing a general basis for quantum probability, statistics and information,

therefore a reader need not have a profound preliminary background in quantum

mechanics. Knowledge of basic mathematical courses will suffice. The book was

not aimed to be all-embracing: e. g., we do not touch quantum cryptography and

entanglement quantification, which now undergo fast development,

and our consideration of quantum computing is quite fragmentary. An

interested reader can find discussion of these problems in other sources

given in the references; on the other hand, a reader will find some topics in our

lectures, such as e.g. entanglement-assisted classical communication,

not covered by other books. Further, this book is addressed to a more

mathematically oriented reader and are more concise.

The proofs of several simple auxiliary results

in the lectures are left as a useful exercise; on the other hand, we emphasize

a number of important open problems still awaiting for their solution.