Entanglement Wedges from Information Metric in Conformal Field Theories

TL;DR

A new method of deriving the geometry of entanglement wedges in holography directly from conformal field theories (CFTs) is presented and an information metric called the Bures metric of reduced density matrices for locally excited states is analyzed.

Abstract

We present a new method of deriving the geometry of entanglement wedges in holography directly from conformal field theories (CFTs). We analyze an information metric called the Bures metric of reduced density matrices for locally excited states. This measures distinguishability of states with different points excited. For a subsystem given by an interval, we precisely reproduce the expected entanglement wedge for two dimensional holographic CFTs from the Bures metric, which turns out to be proportional to the AdS metric on a time slice. On the other hand, for free scalar CFTs, we do not find any sharp structures like entanglement wedges. When a subsystem consists of disconnected two intervals we manage to reproduce the expected entanglement wedge from holographic CFTs with correct phase transitions, up to a very small error, from a quantity alternative to the Bures metric.

Figures (6)

• Figure 1: A sketch of entanglement wedge $M_A$ for an interval $A$ in AdS$_3/$CFT$_2$ and holographic computations of two point functions dual to geodesics. The blue (or green) geodesic does (or does not) intersect with $M_A$.
• Figure 2: A sketch of conformal transformation for the calculation of Tr$[\rho\rho']$. Green Points (or bule points) are outside (or inside) of the wedge (\ref{['outent']}).
• Figure 3: The profiles of $I(\rho,\rho')$ as a function of $x'$ (horizontal axis) and $\tau'$ (depth axis) for the choice $A=[0,2]$ (i.e.$L=2$). The left two ones are for a 2d holographic CFT while the right ones for a 2d free scalar CFT. In the upper two graphs we chose $h=1/2$ and $(x,\tau)=(1,0.1)$ and in the lower two, we chose $h=10$ and $(x,\tau)=(-1,0.1)$.
• Figure 4: A sketch of conformal transformation for Tr$[\rho_A\rho'_A]$ in the double interval case. We assumed the phase (i), where the entanglement wedge is connected, as depicted by the colored region. The lower picture describes the geometry after the transformation and is given by a torus by identifying the edges. Blue (or Green) points are outside (or inside) of $M_A$.
• Figure 5: The plots of the locations of the operator insertion on $\tilde{w}-$plane where the non-trivial Wick contraction is favored (blue colored regions in the left pictures). The upper pictures are for $\kappa=0.1$ where $M_A$ is connected, while the lower ones are for $\kappa=0.2$ where $M_A$ is disconnected. In the upper middle and right picture, blue curves are the borders between the non-trivial and trivial contraction, while orange curves describe the borders of the entanglement wedge. The same is true for the lower right picture.