The book "Probabilistic and Statistical Aspects of Quantum Theory" by Alexander Holevo, published by Scuola Normale Superiore Pisa, delves into the probabilistic and statistical foundations of quantum theory. The content is divided into several chapters, each focusing on different aspects of quantum mechanics and its statistical aspects. 1. **Statistical Models**: This chapter introduces the concepts of states and measurements, convex sets, and the definition of statistical models. It also discusses the classical statistical model and the statistical model of quantum mechanics, including the problem of hidden variables. 2. **Mathematics of Quantum Theory**: Here, the author covers operators in Hilbert space, quantum states and measurements, spectral representations of bounded and unbounded operators, and the realization of measurements. The chapter also explores uncertainty relations, trace-class operators, Hilbert-Schmidt operators, and the $\mathcal{L}^2$ spaces associated with quantum states. 3. **Symmetry Groups in Quantum Mechanics**: This chapter examines the statistical model and Galilean relativity, one-parameter shift groups, and uncertainty relations. It discusses the kinematics and dynamics of a quantum particle in one dimension, minimum-uncertainty states, and the completeness relation. The chapter also covers joint measurements of coordinate and velocity, the time observable, and the coherent-state representation. 4. **Covariant Measurements and Optimality**: This section focuses on parametric symmetry groups and covariant measurements, the structure of covariant measurements, and the covariant quantum estimation problem. It includes measurements of angular parameters, uncertainty relations for angular quantities, and the estimation of pure states. 5. **Gaussian States**: This chapter discusses quasiclassical states of the quantum oscillator, the CCR for many degrees of freedom, and the proof of the Stone-von Neumann uniqueness theorem. It also covers the characteristic function and moments of a state, the structure of general Gaussian states, and a characteristic property of Gaussian states. 6. **Unbiased Measurements**: This part addresses quantum communication channels, lower bounds for variance, and the estimation of force by measurements over a trial object. It includes bounds for the measurement covariance matrix based on symmetric and right logarithmic derivatives, and a general bound for the total mean-square deviation. The book also includes a supplement that discusses the statistical structure of quantum theory and the problem of hidden variables, providing a detailed exploration of these topics. The preface and forewords highlight the book's historical context, its contributions to the field, and its relevance to modern quantum mechanics and its applications in quantum information theory and quantum estimation theory.The book "Probabilistic and Statistical Aspects of Quantum Theory" by Alexander Holevo, published by Scuola Normale Superiore Pisa, delves into the probabilistic and statistical foundations of quantum theory. The content is divided into several chapters, each focusing on different aspects of quantum mechanics and its statistical aspects. 1. **Statistical Models**: This chapter introduces the concepts of states and measurements, convex sets, and the definition of statistical models. It also discusses the classical statistical model and the statistical model of quantum mechanics, including the problem of hidden variables. 2. **Mathematics of Quantum Theory**: Here, the author covers operators in Hilbert space, quantum states and measurements, spectral representations of bounded and unbounded operators, and the realization of measurements. The chapter also explores uncertainty relations, trace-class operators, Hilbert-Schmidt operators, and the $\mathcal{L}^2$ spaces associated with quantum states. 3. **Symmetry Groups in Quantum Mechanics**: This chapter examines the statistical model and Galilean relativity, one-parameter shift groups, and uncertainty relations. It discusses the kinematics and dynamics of a quantum particle in one dimension, minimum-uncertainty states, and the completeness relation. The chapter also covers joint measurements of coordinate and velocity, the time observable, and the coherent-state representation. 4. **Covariant Measurements and Optimality**: This section focuses on parametric symmetry groups and covariant measurements, the structure of covariant measurements, and the covariant quantum estimation problem. It includes measurements of angular parameters, uncertainty relations for angular quantities, and the estimation of pure states. 5. **Gaussian States**: This chapter discusses quasiclassical states of the quantum oscillator, the CCR for many degrees of freedom, and the proof of the Stone-von Neumann uniqueness theorem. It also covers the characteristic function and moments of a state, the structure of general Gaussian states, and a characteristic property of Gaussian states. 6. **Unbiased Measurements**: This part addresses quantum communication channels, lower bounds for variance, and the estimation of force by measurements over a trial object. It includes bounds for the measurement covariance matrix based on symmetric and right logarithmic derivatives, and a general bound for the total mean-square deviation. The book also includes a supplement that discusses the statistical structure of quantum theory and the problem of hidden variables, providing a detailed exploration of these topics. The preface and forewords highlight the book's historical context, its contributions to the field, and its relevance to modern quantum mechanics and its applications in quantum information theory and quantum estimation theory.