Entanglement Wedges from the Information Metric in Conformal Field Theories

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 the 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 anti–de Sitter 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 two disconnected intervals, we manage to reproduce the expected entanglement wedge from holographic CFTs with the correct phase transitions, up to a very small error, from a quantity alternative to the Bures metric.

Figure 1

A sketch of entanglement wedge $M_A$ (red colored region) for an interval $A$ in ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$ and holographic computations of two-point functions dual to geodesics.

Figure 2

A sketch of conformal transformation for the calculation of $\mathrm{Tr}[\rho {\rho }^{\prime }]$.

Figure 3

The profiles of $I(\rho ,{\rho }^{\prime })$ as a function of $x^{\prime }$ (horizontal axis) and ${\tau }^{\prime }$ (depth axis) for the choice $A=[0,2]$ (i.e., $L=2$). The two left 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 $\mathrm{Tr}[{\rho }_A{\rho }_A^{\prime }]$ 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 the $\tilde{w}$ plane where the nontrivial 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 pictures, blue curves are the borders between the nontrivial and trivial contraction, while orange curves describe the borders of the entanglement wedge. The same is true for the lower right picture.