As we have remarked earlier, it is rather remarkable that an intrinsically quantum concept such as entanglement has a very simple geometric dual. Part of the reason of course is that for planar field theories with ceff ≫ 1, one essentially attains a classical limit. Nevertheless, it is intriguing that there is a close connection between geometric concepts in the bulk and quantum features of the boundary theory. One therefore naturally wonders whether this fact can be leveraged to learn how the holographic map between quantum field theories and gravitational dynamics actually works.

The holographic RT and HRT prescriptions allow us to explore general properties of entanglement entropy in a class of QFTs. We will first examine the consistency of holographic entanglement entropy with expectations that follow from the basic definition as detailed in Sect. 2.4 We will also see that there are certain features that are peculiar to holographic systems, in part owing to the fact that we are working in the large ceff limit. We reiterate that the holographic entanglement entropy prescriptions are geared to capturing the leading semiclassical part of entanglement in terms of geometric data. Subleading corrections require ascertaining the bulk entanglement, as discussed in the previous section. All in all, this leads to some unexpected features, which at first sight seem unconventional, but are easily understood once one fully appreciates the implications of the limit ceff ≫ 1 being effectively a semiclassical regime of the QFT.

A simple but quite non-trivial class of non-equilibrium processes of a quantum many-body system is quantum quenches. We start with a ground state | φ0⟩ of a certain Hamiltonian H0. At time t = 0, we suddenly change the Hamiltonian from H0 to a new one H. The original state | φ0⟩ no longer stays at the ground state and starts to experience the time evolution for t > 0. Such a process is called a quantum quench. In particular, when the Hamiltonian changes homogeneously over the whole space, it is called a global quench [131, 132, 133], while if the change is localized in a certain small region, it is called a local quench [134].

In an attempt to find a quasilocal measure of quantum entanglement, we introduce the concept of entanglement density in relativistic quantum theories. This density is defined in terms of infinitesimal variations of the region whose entanglement we monitor and in certain cases can be mapped to the variations of the generating points of the associated domain of dependence. We argue that strong subadditivity constrains the entanglement density to be positive semidefinite. Examining this density in the holographic context, we map its positivity to a statement of integrated null energy condition in the gravity dual. We further speculate that this may be mapped to a statement analogous to the second law of black hole thermodynamics for the extremal surface.