Some Aspects of Entanglement Wedge Cross-Section

TL;DR

This work analyzes the holographic entanglement wedge cross section $E_W$ in Einstein gravity, exploring its finite-temperature behavior and its dependence on Lifshitz scaling and hyperscaling violation. By deriving analytic low- and high-temperature expansions for $E_W$ in relativistic and nonrelativistic backgrounds, the authors demonstrate that $E_W$ obeys an area law at finite temperature and exhibits temperature-driven disentanglement, with a distinctive phase structure in certain cases. They also uncover universal corner contributions to $E_W$ and extend the analysis to unions of kinks and creases, showing that $E_W$ tracks central-charge–dependent corner data and remains bounded by entanglement measures like $I/2$. Overall, the results reinforce $E_W$ as a meaningful probe of quantum correlations in mixed states, with clear dependence on dynamical exponent $z$, hyperscaling violation exponent $ heta$, and geometric singularities, and they point toward future work on quantum corrections and higher-curvature settings.

Abstract

We consider the minimal area of the entanglement wedge cross section (EWCS) in Einstein gravity. In the context of holography, it is proposed that this quantity is dual to different information measures, e.g., entanglement of purification, logarithmic negativity and reflected entropy. Motivated by these proposals, we examine in detail the low and high temperature corrections to this quantity and show that it obeys the area law even in the finite temperature. We also study EWCS in nonrelativistic field theories with nontrivial Lifshitz and hyperscaling violating exponents. The resultant EWCS is an increasing function of the dynamical exponent due to the enhancement of spatial correlations between subregions for larger values of $z$. We find that EWCS is monotonically decreasing as the hyperscaling violating exponent increases. We also obtain this quantity for an entangling region with singular boundary in a three dimensional field theory and find a universal contribution where the coefficient depends on the central charge. Finally, we verify that for higher dimensional singular regions the corresponding EWCS obeys the area law.

Figures (10)

• Figure 1: Schematic configurations for computing $S_A$ (left) and $S_{A\cup B}$ (right). Note that in the latter case we have two different extremal configurations denoted by $\Gamma_{A\cup B}^{\text{con.}}$ and $\Gamma_{A\cup B}^{\text{dis.}}$ corresponding to connected and disconnected RT surfaces respectively.
• Figure 2: Left: Schematic configuration for computing $E_W$ where the entanglement wedge (shaded region) is connected. In this case $E_W$ is proportional to the area of $\Sigma_{AB}^{\rm min}$. Right: For small entangling regions where the entanglement wedge is disconnected and $\Sigma_{AB}^{\rm min}$ becomes empty the corresponding $E_W$ vanishes.
• Figure 3: Schematic configuration for computing HEE (left) and HMI (right). In the latter case, we just demonstrate the connected configuration where the HMI is non-zero.
• Figure 4: Schematic configuration for computing $E_W(\rho_{AB})$.
• Figure 5: Left: parameter space for $d=1$ where $E_W$ is nonzero only in the red shaded region. Right: $E_W$ as a function of $hT$ for different values of $h/\ell$. In all these cases $E_W$ undergoes a discontinuous phase transition beyond which it is identically zero. Here we set $c=1$.