An Introduction to AdS/CFT and the Holographic Entanglement Entropy

In this talk we begin by discussing our fundamental understanding of quantum states and systems and realize the unique peculiarities of the quantum world. We review the density matrix framework to quantitatively discuss quantum entanglement. We then introduce the gauge/gravity duality and lay down some concrete statements about the correspondence and establish a dictionary between the two sides of the duality. We will be mostly interested in the boundary to bulk approach. The last part of the review introduces the holographic entanglement entropy which mysteriously describes the emergent nature of spacetime in holographic theories. We illustrate the static and the time-dependent proposal along with some mathematical techniques that help us compute the entanglement entropy in relativistic quantum field theories.