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
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 coding
theorem, establishing the possibility of reliable data transmission and processing
at the rates not exceeding the definite value (capacity ) characterizing the
given information processing system (for definiteness one speaks of a
(communication channel). The issue of the information
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.