Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.

The analyses we have discussed in previous chapters include the use of base functions, such as sinusoidal functions with specified frequencies, i.e. harmonic analysis; sinusoidal base functions with a frequency range from 0 to the Nyquist frequency with an interval inversely proportional to the total length of time of the data, i.e. Fourier analysis; and wavelet base functions for wavelet analysis. These base functions, however, are chosen regardless of the nature of the variability of the data themselves. In this chapter, we will discuss a different method, in which the base functions are determined empirically, that is dependent on the nature of the data. In other words, this method will find the base functions from the data and these base functions describe the nature of the data. The method is applicable to many types of data, especially to time series data at multiple locations, e.g. a sequence of weather maps or satellite images. There are several variants of the method, but here we will only provide an introduction for the basics.

Many scientific questions lead to hypotheses about random vectors. For instance, the question of whether global warming has occurred over a geographic region is a question about whether temperature has changed at each spatial location within the region. One approach to addressing such a question is to apply a univariate test to each location separately and then use the results collectively to make a decision. This approach is called multiple testing or multiple comparisons and is common in genomics for analyzing gene expressions. The disadvantage of this approach is that it does not fully account for correlation between variables. Multivariate techniques provide a framework for hypothesis testing that takes into account correlations between variables. Although multivariate tests are more comprehensive, they require estimating more parameters and therefore have low power when the number of variables is large. Multivariate statistical analysis draws heavily on linear algebra and includes a generalization of the normal distribution, called the multivariate normal distribution, whose population parameters are the mean vector and the covariance matrix.

This chapter covers the analysis of static systems under probabilistic input uncertainty. The first part of the chapter is devoted to analyzing linear and nonlinear static systems when the first and second moments of the input vector are known, and it provides techniques for characterizing the first and second moments of the state vector. For the linear case, the techniques provide the exact moment characterization, whereas for the nonlinear case, the characterization, which is based on a linearization of the system model, is approximate. The second part of the chapter provides techniques for the analysis of both linear and nonlinear static systems when the pdf of the input vector is known. The techniques included provide exact characterizations of the state pdf for both linear and nonlinear systems. In both cases, the inversion of the input-to-state mapping is required, which in the linear case involves the computation of the inverse of a matrix; however, for the nonlinear, it involves obtaining an analytical expression for the input-to-state mapping. The chapter concludes by utilizing the techniques developed to study the power flow problem under active power injection uncertainty.