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Probabilistic and Statistical Aspects of Quantum Theory (Statistics and Probability): A.S. Holevo

The mathematical language of modern quantum mechanics is operator theory. Operators play there a role similar to functions in classical mechanics, probability theory and statistics. However, while the use of functions in classical theories is founded on premises which seem intuitively quite clear, in quantum theory the situation with operators is different.

Historically the 'matrix mechanics' of Heisenberg and the 'wave mechanics' of SchrSdinger which gave rise to the contemporary form of quantum theory, originated from ingenious attempts to fit mathematical objects able to reflect some unusual (from the macroscopic point of view) features of microparticle behaviour in particular, a peculiar combination of continuous and discrete properties. The 'probabilistic interpretation' developed later by Born and others elucidated the meaning of operator formalism by postulating rules connecting mathematical objects with observable quantities. However a good deal of arbitrariness remained in these postulates and the most convincing argument for quantum-theoretical explanations was still the 'striking' coincidence of theoretical predictions with experimental data. This state of affairs gave rise to numerous attempts, on one hand, to find classical alternatives to quantum theory which would give an equally satisfactory description of the experimental data, and on the other hand, to find out physical and philosophical arguments for justifying the inevitability of the new mechanics.

Notwithstanding the impressive philosophical achievements in this field there was and still is a need for the structural investigation of quantum theory from a more mathematical point of view aimed at elucidating the connections between the entities of the physical world and the elements of operator formalism. The present book is essentially in this line of research opened by the classical von Neumann's treatise "Mathematical Foundations of Quantum Mechanics". However it differs from most subsequent investigations by the strong emphasis on the statistical rather than 'logical' essence of quantum theory; it gives an account of recent progress in the statistical theory of quantum measurement, stimulated by the new applications of quantum mechanics, particularly in quantum optics.

The first three chapters give an introduction to the foundations of quantum mechanics, addressed to the reader interested in the structure of quantum theory and its relations with classical probability. In spite of the mathematical character of the presentation it is not 'axiomatic'. Its purpose is to display the origin of the basic elements. of operator formalism resting, as far as possible, upon the classical probabilistic concepts.

The present revision is not an end in itself---it emerged from the solution of concrete problems concerning the quantum limitations to measurement accuracy, arising in applications. So far there has been no general approach to such kind of problems. The methods of mathematical statistics adapted for classical measurements required radical quantum modification. The last chapters of the book are devoted to the recently developed quantum estimation theory, which is an analog of the corresponding branch of mathematical statistics.

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