Reflected Entropy and Entanglement Wedge Cross Section with the First Order Correction

We study the holographic duality between the reflected entropy and the entanglement wedge cross section with the first order correction. In the field theory side, we consider the reflected entropy for $\rho_{AB}^m$, where $\rho_{AB}$ is the reduced density matrix for two intervals in the ground state. The reflected entropy in the 2d holographic conformal field theories is computed perturbatively up to the first order in $m-1$ by using the semiclassical conformal block. In the gravity side, we compute the entanglement wedge cross section in the backreacted geometry by cosmic branes with tension $T_m$ which are anchored at the AdS boundary. Comparing both results we find a perfect agreement, showing the duality works with the first order correction in $m-1$.