  In this thesis I present numerical studies of entanglement entropy and the
quantum null energy condition in strongly coupled far-from-equilibrium quantum
states using holography. I give a careful introduction into the numerical
methods used and provide the computer codes to compute entanglement entropy and
the quantum null energy condition. These methods are then applied to systems of
various degrees of complexity, including homogeneous and isotropic far-
from-equilibrium quantum quenches dual to Vaidya spacetimes, to homogeneous and
anisotropic finite temperature states dual to anisotropic black brane
geometries, and to inhomogeneous and anisotropic states of colliding walls of
energy dual to gravitational shock wave collisions in Anti-de Sitter space. For
all these scenarios I compute the fully non-linear dynamics of the dual
geometry, which requires to numerically solve five-dimensional Einstein
equations with negative cosmological constant and asymptotic Anti-de Sitter
boundary conditions. The numerical solutions for the geometries allow to
extract the time evolution of the holographic energy momentum tensor and
provides the background for computing two-point functions, entanglement entropy
and the quantum null energy condition. From the anisotropic system one learns
that the near-equilibrium dynamic of entanglement entropy has an accurate
description in terms of quasinormal modes. In the shock wave system I identify
characteristic features of entanglement entropy that allow to discriminate
between thick and narrow shocks. All my numerical studies confirm the quantum
null energy condition, also the shock wave system, which itself can violate the
classical null energy condition for sufficiently narrow shocks. My results also
show that the quantum null energy condition can saturate in the
far-from-equilibrium regime.
