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    Probabilistic and Statistical Aspects of Quantum Theory

    ISBN-10: 8876423753
    ISBN-13: 9788876423758
    Edition: 2011
    Description: 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  More...
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    Copyright Year: 2011
    Publisher: Scuola Normale Superiore
    Publication Date: 3/24/2011
    Binding: Paperback
    Pages: 334
    Size: 6.25" wide x 9.50" long x 1.00" tall
    Weight: 1.430
    Language: English

    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.

    Foreword to the second English edition
    Preface to the second Russian edition
    Statistical models
    States and measurements
    Some facts about convex sets
    Definition of a statistical model
    The classical statistical model
    Reduction of statistical model. Classical model with a restricted class of measurements
    The statistical model of quantum mechanics
    On the problem of "hidden variables"
    Mathematics of quantum theory
    Operators in a Hilbert space
    Quantum states and measurements
    Spectral representation of bounded operators
    Spectral representation of unbounded operators
    On realization of measurement
    Uncertainty relations and compatibility
    Trace-class operators and Hilbert-Schmidt operators
    L2 spaces associated with a quantum state
    Uncertainty relations for measurements with finite second moments
    Matrix representation of square-summable operators. The commutation operator of a state
    Symmetry groups in quantum mechanics
    Statistical model and Galilean relativity
    One-parameter shift groups and uncertainty relations
    Kinematics of a quantum particle in one dimension
    Uniqueness theorem. The Schrodinger and the momentum representations
    Minimum-uncertainty states. The completeness relation
    Joint measurements of coordinate and velocity
    Dynamics of a quantum particle in one dimension
    Time observable. The "time-energy" uncertainty relation
    Quantum oscillator and phase measurement
    The coherent-state representation
    Representations of the rotation group and angular momenta
    Measuring the angle of rotation
    Covariant measurements and optimality
    Parametric symmetry groups and covariant measurements
    Structure of covariant measurements
    The covariant quantum estimation problem
    Measurements of angular parameters
    Uncertainty relations for angular quantities
    Covariant measurements of angular parameter in the case of arbitrary representation of the group T
    Covariant measurements of a shift parameter
    The case of irreducible representation
    Estimation of pure state
    Measuring parameters of orientation
    Gaussian states
    Quasiclassical states of the quantum oscillator
    The CCR for many degrees of freedom
    Proof of the Stone-von Neumann uniqueness theorem. The Weyl transform
    Characteristic function and moments of state
    Structure of general Gaussian states
    A characteristic property of Gaussian states
    Unbiased measurements
    Quantum communication channel
    A lower bound for the variance in one-dimensional case
    The case of shift parameter
    Estimation of force by measurements over a trial object
    A bound for the measurement covariance matrix based on symmetric logarithmic derivatives
    A bound based on right logarithmic derivatives
    A general bound for the total mean-square deviation
    Linear measurements
    Mean-value estimation for Gaussian states
    Supplement Statistical structure of quantum theory and hidden variables
    The structure of statistical theories
    Axiomatic approaches in quantum mechanics
    The classical picture of statistical experiment
    The main features of statistical description
    Statistical model of quantum mechanics
    Compatibility and complementarity
    Classical and nonclassical models
    The problem of hidden variables
    "No-go" proofs and the minimal statistical requirements on hidden variables
    A hidden variable model for a single quantum system
    Hidden variables and evolutions of quantum system
    Composite systems, EPR paradox and the Bell inequality

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