Holographic Entanglement of Purification from Conformal Field Theories

We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of the entanglement wedge cross section. We argue that, in ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has minimal path-integral complexity. We confirm this claim in several examples.

Figure 1

Conformal transformation between a complex plane with one slit and an upper half plane.

Figure 2

Calculation of Holographic EOP. The length of $\mathrm{\Sigma }$ (thick curve in the above picture) multiplied by $(1/4G_N)=(c/6)$ gives the holographic EOP, Eq. (3).

Figure 3

Conformal map from complex plane with two cuts to a cylinder and its path-integral optimization.

Figure 4

Plots of HEOP (red), $I(A:B)/2$ (green), and $S_{A\tilde{A}}^{\min }$ (blue dashed) as a function of $x$. The first two vanish for $x>3-2\sqrt{2}$. Note that the plot of $S_{A\tilde{A}}^{\min }$ is valid only when $x\ll 1$, where it coincides with HEOP.