ISSN 0940-7677

ISBN 3-540-42082-7

The book gives a systematic, broad and concise account of the foundations of quantum theory from a probabilist's point of view, with particular emphasis on statistical aspects. It extends and develops ideas presented in author's monograph

„Probabilistic and Statistical Aspects of
Quantum Theory" , Nauka,
1980
(English translation - North Holland, 1982).

2d Russian edition, corrected and complemented. ICI, Moscow-Izhevsk,
2003. http://www.rcd.ru

Among the topics covered are: standard and generalized statistical
models
of quantum mechanics and hidden variables, quantum statistical decision
theory and state estimation, capacities of quantum communication
channels,
open system and continuous measurement dynamics, quantum processes in
the
Fock space and stochastic calculus. The text is concluded with a
comprehensive
bibliography.

The mathematical level is the Hilbert space operator theory
together with the basics of probability and statistics, while the
essence is diverse aspects of positivity and tensor products related to
the fundamental probabilistic aspects of quantum theory. Efforts are
made to make the text accessible
to a wide audience of readers -- from physicists to probabilists and
operator theorists and from research workers to graduate students.

The Russian translation is published in 2003 by "Regular and Chaotic
Dynamics" http://www.rcd.ru

Preface

During the last three decades the mathematical foundations of quantum

mechanics, related to the theory of quantum measurement, have undergone

profound changes. A broader comprehension of these changes is now
developing.

It is well understood that quantum mechanics is not a merely dynamical

theory; supplied with the statistical interpretation, it provides a new
kind

of probabilistic model, radically different from the classical one.

The statistical structure of quantum mechanics is thus a subject
deserving

special investigation, to a great extent different from the standard
contents

of textbook quantum mechanics.

The first systematic investigation of the probabilistic structure of
quantum

theory originated in the well-known monograph of J. von Neumann

``Mathematical Foundations of Quantum Mechanics'' (1932). A year later

another famous book appeared -- A. N. Kolmogorov's treatise on the

mathematical foundations of probability theory. The role and the impact
of

these books were significantly different. Kolmogorov completed the long

period of creation of a conceptual basis for probability theory,
providing it

with classical clarity, definiteness and transparency. The book of von

Neumann was written soon after the birth of quantum physics and was one
of

the first attempts to understand its mathematical structure in
connection

with its statistical interpretation. It raised a number of fundamental

issues, not all of which could be given a conceptually satisfactory
solution

at that time, and it served as a source of inspiration or a starting

point for subsequent investigations.

In the 1930s the interests of physicists shifted to quantum field theory

and high-energy physics, while the basics of quantum theory were left
in a

rather unexplored state. This was a natural process of extensive

development; intensive exploration required instruments which

had only been prepared at that time. The birth of quantum mechanics

stimulated the development of an adequate mathematical apparatus -- the

operator theory -- which acquired its modern shape by the 1960s.

By the same time, the emergence of applied quantum physics, such as
quantum

optics, quantum electronics, optical communications, as well as the
development

of high-precision experimental techniques, has put the issue of
consistent

quantitative quantum statistical theory in a more practical setting.
Such a

theory was created in the 1970s--80s as a far-reaching logical
extension of

the statistical interpretation, resting upon the mathematical
foundation of

modern functional analysis. Rephrasing the well-known definition of

probability theory (``Probability theory is a measure theory -- with a
soul'' (M. Kac)),

one may say that it is a theory of operators in a

Hilbert space given a soul by the statistical interpretation of

quantum mechanics. Its

mathematical essence is diverse aspects of positivity and tensor product

structures in operator algebras (having their roots, correspondingly,
in the

fundamental probabilistic properties of positivity and independence).

Key notions are, in particular, resolution
of identity (positive

operator valued measure) and completely
positive map, generalizing,

correspondingly, spectral measure and unitary evolution of the standard

quantum mechanics.

The subject of this book is a survey of basic principles and

results of this theory. In our presentation we adhere to a pragmatic

attitude, reducing to a minimum discussions of both axiomatic and

epistemological issues of quantum mechanics; instead we concentrate on

the correct formulation and solution of quite a few concrete problems
(briefly

reviewed in the Introduction), which appeared unsolvable or even
untreatable

in the standard framework. Chapters 3--5 can be considered as a

complement to and an extension of the author's monograph ``Probabilistic and

Statistical Aspects of Quantum Theory'',

in the direction of problems involving state changes and measurement

dynamics. However, unlike that book, they give a concise survey rather
than

a detailed presentation of the relevant topics: there are more
motivations

than proofs, which the interested reader can find in the references. A
full

exposition of the material considered would require much more space and
more

advanced mathematics.

We also would like to mention that a finite dimensional version of this

generalized statistical framework appears to be an adequate background
for

the recent investigations in quantum information and computing, but
these

important issues require separate treatment and were only partially
touched

upon here.