Probabilistic and Statistical Aspects of Quantum Theory

verfasst von: Alexander Holevo

Verlag: Edizioni della Normale

Buchreihe : CRM Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation relations and Gaussian states, uncertainty relations as well as new fundamental bounds concerning the accuracy of quantum measurements, are discussed in this book in an accessible yet rigorous way. Compared to the first edition, there is a new Supplement devoted to the hidden variable issue. Comments and the bibliography have also been extended and updated.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Statistical models

Abstract

Any theoretical model ultimately relies upon experience — the framework for a model is constituted by the array of experimental data relevant to the study of the object or phenomenon. Let us consider a very schematic and general description of an experimental situation and try to trace back the emergence of the principal components of a theoretical model.

Alexander Holevo

Chapter 2. Mathematics of quantum theory

Abstract

In the previous chapter the quantum statistical model was introduced in its simplest finite-dimensional form. However to describe many interesting and important properties of quantum-mechanical objects the infinitedimensional generalization of this model is needed, where matrices are replaced by operators in a Hilbert space.

Alexander Holevo

Chapter 3. Symmetry groups in quantum mechanics

Abstract

The spatially-temporal structure which we are going to introduce is a distinctive feature in the description of mechanical objects. In classical mechanics the motion of point masses is described in a frame of reference which consists of a spatial Cartesian coordinate system and a clock. The distinguished class is formed by the so called inertial frames which are characterized by the property that a free point mass performs uniform rectilinear motion with respect to such a frame. It follows that the coordinates (ξ, τ) and (ξ′, τ′) of a point mass in any two inertial frames of reference are related by the Galilei transformation

$$ \xi ' = R\xi + x + v\tau , \tau ' + \tau + t, $$

(3.1.1)

where R is an orthogonal matrix (rotation) which describes the orientation of the new spatial axes with respect to the old ones, x is the vector of the spatial shift of the origin, ν is the relative velocity, and τ is the time shift showing the difference between the readings of the two clocks.

Alexander Holevo

Chapter 4. Covariant measurements and optimality

Abstract

All symmetry groups considered in the previous chapter were parametric groups (Lie groups) of transformations. This means that a parametric set Θ = {θ}, i.e., a continuous manifold in a finite-dimensional space is given and elements of the group G = {g} act as continuous one-to-one mappings of the set Θ onto itself, g : θ → gθ. Moreover the group G is itself parametrized in such a way that the group product g₁g₂ is at least locally continuous in g₁, g₂.

Alexander Holevo

Chapter 5. Gaussian states

Abstract

Consider a quantum degree of freedom, e.g., an oscillator, described by the canonical observables q = Q, p = ℏP. The ground state |0)(0| is the minimum-uncertainty state in which Q and P have zero mean values. The state

$$ \left| {\left. {\overline P ,\overline Q } \right)\left( {\overline Q ,\overline P } \right.} \right| = W_{\overline Q ,\hbar \overline P } \left| {\left. 0 \right)\left( 0 \right.} \right|W_{\overline Q ,\hbar \overline P }^* $$

(5.1.1)

(where to simplify notations we put $ \left| {\left. {\overline P ,\overline Q } \right)} \right. \equiv \left| {\left. {\overline P ,\overline Q ;{\hbar \mathord{\left/ {\vphantom {\hbar {2\omega }}} \right. \kern-\nulldelimiterspace} {2\omega }}} \right)} \right. $ ω being the oscillator frequency) can be regarded as the result of an external influence onto the object in ground state, shifting the mean values of the canonical observables but leaving their uncertainties unchanged.

Alexander Holevo

Chapter 6. Unbiased measurements

Abstract

Consider the idealized scheme of information transmission presented at Figure 6.1. In the absence of signal the physical carrier of information C (say, electromagnetic field) is in a state S. Usually it is adopted that S is the equilibrium (Gibbs) state at the given temperature. Transmission of a signal is accomplished through an influence of the communication source L onto the system C, which forces a definite change in its state. If there are parameters of the source L which can be varied, then the resulting state S_θ will depend on the values of these parameters θ.

Alexander Holevo

Supplement

Statistical structure of quantum theory and hidden variables

Abstract

By the end of XVIII century scientists developed the picture of the material world as a huge mechanism, the evolution of which is subject to a rigid dynamical laws and in principle can be predicted with arbitrary detail and accuracy. This system of conceptions which acquired the name “determinism” was progressive for that time and was based on a triumphant success of the Newtonian mechanics which allowed to give a rational explanation to a number of earlier inexplicable physical facts.

Alexander Holevo

Backmatter

Titel: Probabilistic and Statistical Aspects of Quantum Theory
verfasst von: Alexander Holevo
Verlag: Edizioni della Normale
Electronic ISBN: 978-88-7642-378-9
Print ISBN: 978-88-7642-375-8
DOI: https://doi.org/10.1007/978-88-7642-378-9