Holographic Entanglement of Purification from Conformal Field Theories

TL;DR

This work examines the holographic entanglement of purification (HEoP) in AdS$_3$/CFT$_2$ through path-integral optimization, proposing that the HEoP equals the entanglement entropy $S_{A\tilde{A}}$ of a specially purified state obtained by a Weyl transformation that minimizes the path-integral complexity. By solving the Liouville equation and employing Weyl-rescaled geometries, the authors compute $S^{min}_{A\tilde{A}}$ for both single and double interval configurations and demonstrate exact or near-exact matches with the holographic entanglement wedge cross-section $E_W(\rho_{AB})$ in the regimes of validity. The results provide quantitative evidence for the conjectured equivalence $E_W=E_P$ in static AdS$_3$/CFT$_2$ and illuminate how path-integral complexity and purification underpin holographic measures of mixed-state correlations. The analyses highlight the limitations of Weyl-invariance-based methods at larger cross-ratios and point to future work on higher dimensions and a rigorous mapping between minimal purifications and the original EoP definition.

Abstract

We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, 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 the minimal path-integral complexity. We confirm this claim in several examples.

Figures (5)

• 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 $\Sigma$ (thick curve in the above picture) multiplied by ${1\over 4G_N}={c\over 6}$ gives the holographic EoP (\ref{['heop']}).
• 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 $S^{min}_{A\tilde{A}}$ (blue dashed) as a function of $x$. The first two get vanishing for $x>3-2\sqrt{2}$. Note that the plot of $S^{min}_{A\tilde{A}}$ is valid only when $x\ll 1$, where it coincides with HEoP.
• Figure 5: A sketch of regularized conformal transformations, when $AB$ are double intervals in the CFT vacuum. The grey colored region in the left is mapped to the same colored region in the right. The total plane with the two cuts on $A$ and $B$ eliminated, is mapped into two copies of the right picture, i.e. a cylinder.