Entanglement Entropy from a Holographic Viewpoint

Abstract

The entanglement entropy has been historically studied by many authors in order to obtain quantum mechanical interpretations of the gravitational entropy. The discovery of AdS/CFT correspondence leads to the idea of holographic entanglement entropy, which is a clear solution to this important problem in gravity. In this article, we would like to give a quick survey of recent progresses on the holographic entanglement entropy. We focus on its gravitational aspects, so that it is comprehensible to those who are familiar with general relativity and basics of quantum field theory.

Figures (6)

• Figure 1: Examples of choices of subsystem $A$.
• Figure 2: The calculation of holographic entanglement entropy.
• Figure 3: The holographic proof of strong subadditivity. In each of three figures, the vertical black line represents the boundary of the AdS, while the horizontal direction in the right is the $z$ direction in AdS. Though we are assuming the time slice of AdS$_3$ just for simplicity, this argument can be extended into higher dimension in a straightforward way. In the left picture, the red and blue curve represents the minimal surfaces $\gamma_{A\cup B}$ and $\gamma_{B\cup C}$. In the middle picture, we just recombine them into two surfaces (green and brown ones). The true minimal surfaces $\gamma_{A\cup B\cup C}$ and $\gamma_{B}$ are given by the right picture. Therefore the strong subadditivity is obvious.
• Figure 4: The left figure schematically describes the black hole creation and the extremal surface $\gamma_A$. The orange curve represents the time-evolving black hole. The right figures describe how the extremal surface $\gamma_A$ and $\gamma_B$ should be chosen. In the black hole creation spacetime, the right lower picture describes the correct choice i.e. $\gamma_A$ and $\gamma_B$ coincides. On the other hand, for eternal blackholes (i.e time-independent black holes), we need to distinguish $\gamma_A$ and $\gamma_B$ as in the right upper figure.
• Figure 5: The time evolution of the entanglement entropy in a two dimensional CFT.