The notion of modeling complex physical systems using randomness goes back to the beginnings of statistical mechanics. Since then, randomized physical systems have been a subject of study from many-body physics to quantum information. In this thesis, we will examine random quantum systems in different contexts.
To start, we discuss the emergence of equilibrium statistical mechanics in isolated quantum systems, more specifically, the phenomena of equilibration and thermalization. How these two very common and intuitive phenomena arise from unitary evolution is a riddle that, despite intense research, is not as of yet fully solved. We will argue that while it is hard to derive this phenomenology from the microscopic description of a specific physical system, such phenomena are \emph{typical}, in the sense that they occur in the vast majority of physical systems, and under some assumptions their absence is sensitive to small, hard to detect changes in the underlying Hamiltonian.
We will then move on to systems which seem to avoid thermalization by retaining memory of their initial conditions due to a phenomenon known as many-body localization. Most systems known to exhibit such a behavior are so called disordered systems, described by an ensemble of Hamiltonians in which certain couplings are drawn randomly. We will discuss the notion of local integrals of motion in many-body localized systems and describe a method to measure their properties by observing the spread of entanglement in the system. Following this, we will more closely examine the role of randomness and disorder in the emergence of many-body localization by examining the local integral of motions of disorder-free systems which appear to resist thermalization.
Finally, we will discuss how randomizing a quantum system can allow one to extract information about its state. More specifically, we describe an algorithm which interpolates between two known extreme schemes, one allowing to capture only local properties, and the other only globally, and show that this intermediate version is able to perform efficiently in both regimes.
