Looking at Shadows of Entanglement Wedges

TL;DR

The paper develops a purely CFT-based method to reconstruct entanglement wedges by introducing CFT wedges defined through distinguishability measures of reduced density matrices. It shows that the Bures/fidelity-based CFT wedges reproduce the holographic entanglement wedges across single and double interval setups, AdS/BCFT, higher dimensions, and time dependence, while some Renyi-like measures (I(ρ,ρ')) can yield small deviations. The work demonstrates that sharp wedges arise in holographic CFTs due to large-N factorization, whereas non-holographic theories (e.g., c=1 free scalar) lack sharp wedges. It also discusses the roles of low-energy versus high-energy content, compares multiple distance measures, and suggests HKLL-type operators could further refine the CFT probing of bulk geometry, creating a strong link between boundary data and bulk entanglement structure.

Abstract

We present a new method of deriving shapes of entanglement wedges directly from CFT calculations. We point out that a reduced density matrix in holographic CFTs possesses a sharp wedge structure such that inside the wedge we can distinguish two local excitations, while outside we cannot. We can determine this wedge, which we call a CFT wedge, by computing a distinguishability measure. We find that CFT wedges defined by the fidelity or Bures distance as a distinguishability measure, coincide perfectly with shadows of entanglement wedges in AdS/CFT. We confirm this agreement between CFT wedges and entanglement wedges for two dimensional holographic CFTs where the subsystem is chosen to be an interval or double intervals, as well as higher dimensional CFTs with a round ball subsystem. On the other hand if we consider a free scalar CFT, we find that there are no sharp CFT wedges. This shows that sharp entanglement wedges emerge only for holographic CFTs owing to the large N factorization. We also generalize our analysis to a time-dependent example and to a holographic boundary conformal field theory (AdS/BCFT). Finally we study other distinguishability measures to define CFT wedges. We observe that some of measures lead to CFT wedges which slightly deviate from the entanglement wedges in AdS/CFT and we give a heuristic explanation for this. This paper is an extended version of our earlier letter arXiv:1908.09939 and includes various new observations and examples.

Figures (24)

• Figure 1: We sketched an entanglement wedge $M_A$ for an interval $A$ in AdS$_3/$CFT$_2$. We also show holographic computations of two point functions dual to geodesics. The blue (or green) geodesic does (or does not) intersect with $M_A$ at $P$.
• Figure 2: We sketched the conformal mapping for the calculation of Tr$[\rho\rho']$. Green Points (or bule points) describe the local excitations in the CFT which are dual to bulk local excitations outside (or inside) of the CFT wedge.
• Figure 3: The value of $I(\rho,\rho')$ as a function of $\text{Re}[w]$ (horizontal axis) and $\text{Im}[w]$ (depth axis) when $w'$ is fixed inside the CFT wedge. In particular, we chose $h_\alpha=1/2$ and $w'=1+0.1i$ and $A=[0,2]$ (i.e.$L=2$). The left and right graph describe the result for the holographic CFT and the $c=1$ free scalar CFT, respectively.
• Figure 4: The value of $I(\rho,\rho')$ as a function of $\text{Re}[w]$ (horizontal axis) and $\text{Im}[w]$ (depth axis) when $w'$ is fixed outside the CFT wedge. In particular, we chose $h_\alpha=10$ and $A=[0,2]$ (i.e.$L=2$). The upper two graphs are for $w'=-1+0.1i$ and the lower ones are for $w'=1+2i$, both of which are outside of the wedge. The left and right graphs describe the result for the holographic CFT and the $c=1$ free scalar CFT, respectively. We find that the wedge structure is sharp only in the holographic CFT. For free scalar CFT, we can detect an excitation even outside of the wedge.
• Figure 5: The complex plane which describes the path-integral which calculates the trace $A_{n,m}=\hbox{Tr}[(\rho^m\rho'\rho^m)^n].$ i.e. (\ref{['amn']}), where we performed the conformal transformation (\ref{['confglk']}). Here we choose $m=1$ and $n=3$ for convenience.