In Chapter 9, we presented the quantum rule of combination of subsystems through the tensor product. In this chapter, we will discuss a key elaboration of this rule that applies to composite systems with a specific symmetry, namely, invariance under exchange of identical particles.

In this chapter we develop a generalization of hidden Markov models valid for the evolution of a system in continuous time. That is, we describe how to model and analyze hidden Markov jump processes. In this context, we introduce the concept of uniformization to simulate continuous time trajectories and then use uniformization to develop a Monte Carlo strategy to sample trajectories from a posterior over trajectories. Having discussed how trajectories can be sampled from the posterior over all candidate trajectories, we then describe strategies for full posterior inference over trajectories and other model parameters. We end with strategies for trajectory marginalization and continuous time filtering.

Spin was introduced as part of the effort to understand the structure of atoms prior to the development of mature quantum theory. Variations of Bohr’s model described atoms in terms of three quantum numbers, roughly similar to $n,𝓁$, and $m$ that appear when solving the Schrödinger equation in central potentials. In this context, Pauli proposed that, in each atom, there exists at most one electron for each triplet of quantum numbers. This proposal is Pauli’s famous “exclusion principle,” which we will analyze in Chapter 15.

The central message of the introduction is one must understand science if one wants to do science well. This requires a holistic educational approach, one that not only teaches the whats and hows of science, but most critically, it's whys. Why is the sky blue? Why do normal cells turn into cancer cells? Why do we use the scientific method and from where did it come? Why would one want to be a scientist in the first place? Why is science done in the way it is, that is, what is the gestalt of science? The whats and whys of science are practical in nature. The whys, in contrast, encompass theoretical, philosophical, historical, and social underpinnings of science. The whys are particularly important now when the probity and veracity of science are being attacked, and people seek to replace actual facts with "alternative facts" (falsehoods) for political, religious, or economic purposes or out of plain ignorance.

This chapter introduces two commonly used methods of determining the local tensors of an MPS. The first is the variational optimization method, which determines an MPS by minimizing the energy expectation value. This method is equivalent to solving a generalized eigenequation around the extreme point of the ground-state energy. The second is an update method based on an imaginary time evolution, which cools down a quantum state from finite to zero temperature. We discuss three update approaches: update via canonicalization, full update, and simple update. For an MPS, the canonicalization approach is accurate and easy to implement. However, the full and simple update can be generalized to higher dimensions and applied to, for example, PEPS. The full update is a global minimization approach. It is accurate but has a higher computational cost than the simple update. The simple update is a local optimization approach based on an entanglement mean-field approximation and is easy to implement. Finally, we discuss the purification technique and apply it to evaluate the thermal density matrix or solve a quenched disorder problem in the framework of MPS.

In this chapter we extend our discussion of the previous chapter to model dynamical systems with continuous state-spaces. We present statistical formulations to model and analyze noisy trajectories that evolve in a continuous state space whose output is corrupted by noise. In particular, we place special emphasis on linear Gaussian state-space models and, within this context, present Kalman filtering theory. The theory presented herein lends itself to the exploration of tracking algorithms explored in the chapter and in an end-of-chapter project.

Often there are situations in which the solutions to the time-independent Schrödinger equation are known for a particular potential but not for a similar but different potential. Time-independent perturbation theory provides a means of finding approximate solutions using an expansion in the known eigenfunctions.

Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.

In this chapter we introduce and apply hidden Markov models to model and analyze dynamical data. Hidden Markov models are one of simplest of dynamical models valid for systems evolving in a discrete state-space at discrete time points. We first describe the evaluation of the likelihood relevant to hidden Markov models and introduce the concept of filtering. We then describe how to obtain maximum likelihood estimators using expectation maximization. We then broaden our discussion to the Bayesian paradigm and introduce the Bayesian hidden Markov model. In this context, we describe the forward filtering backward sampling algorithm and Monte Carlo methods for sampling from hidden Markov model posteriors. As hidden Markov models are flexible modeling tools, we present a number of variants including the sticky hidden Markov model, the factorial hidden Markov model, and the infinite hidden Markov model. Finally, we conclude with a case study in fluorescence spectroscopy where we show how the basic filtering theory presented earlier may be extended to evaluate the likelihood of a second-order hidden Markov model.

The history of the laser dates back to at least 1951 and an idea of Townes. He wanted to use ammonia molecules to amplify microwave radiation. Townes and two students completed a prototype device in late 1953 and gave it the name maser or microwave amplification by stimulated emission of radiation.

The first step towards quantum theory was a response to a problem that could not be addressed by the concepts and methods of classical physics: the radiation from black bodies.

The purpose of this chapter is to introduce the diagrammatic representation of tensors and tensor network states and discuss some basic properties or formulas of matrices or tensors used in various renormalization group methods. It includes, for example, the singular value decomposition and the polar decomposition of matrices, the higher-order singular value decomposition of tensors, and the automatic differentiation and its backpropagation scheme. Several Trotter-Suzuki decomposition formulas of non-commutate operators are also introduced.