„**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