„Probabilistic and Statistical Aspects of Quantum Theory"

2^nd English edition - Edizioni della Normale, Pisa, 2011 (1^st English edition, - North Holland, 1982).

Foreword to the second Russian edition

When this book was first published in 1980 (the English edition
appeared in 1982 in North Holland), the author addressed it to a
broad audience of readers, both mathematicians and physicists having
intention to make them acquainted with the new prospects and
possibilities which emerge from the interaction of ideas of the
mathematical statistics and the quantum theory. During the past
period this approach became even more demanded. On one side, its
advantages in the questions of foundations of quantum theory related
to quantum measurements became more apparent and widely
acknowledged. On the other hand, one should stress that these
theoretical findings were not an end in itself: in modern
high-precision physical experiments researchers become able to
operate with single ions, atoms and photons which leads to
potentially important applications, such as quantum communications,
computation and cryptography. Of great importance is the question of
extraction of maximal possible information from the state of a
given quantum system. For example, in currently discussed proposals
for quantum computing information is written in the states of
elementary quantum memory cells -- qubits, and then is read off by
means of quantum measurement. From the statistical point of view,
measurement gives an estimate for the quantum state -- as a whole,
or for some of its parameters. In this way a new interest emerges to
quantum estimation theory, the fundamentals of which are presented
in this book.

One of essential consequences of penetration of the ideas of
mathematical statistics into the theory of quantum measurement is
the wide use of the mathematical notion of (non-orthogonal)
resolution of the identity in the system Hilbert space
(in the Western literature -- POVM, positive operator-valued measures),
describing the statistics of decision procedures. During the time
passed resolutions of the identity became a standard tool both in
mathematical and in physical literature on quantum measurements.
All this, in my opinion, justifies publication of the second Russian
edition of the book, moreover as the first one became a rare book.

The present edition includes Supplement discussing in some detail
the problem of hidden variables in quantum mechanics which continues
to provoke a keen interest. Moreover, the edition is complemented
with Comments reflecting new results and achievements.

Academician K. A. Valiev