A. S. Holevo "Statistical Structure of Quantum Theory," Lect. Notes Phys. m67, Springer, Berlin 2001.
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


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.