Entanglement of Purification in Random Tensor Networks

The entanglement of purification $E_P(A\text{\hspace{0.17em}⁣}:B)$ is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: $E_P(A\text{\hspace{0.17em}⁣}:B)\ge \frac{1}{2}{S}_R^{(2)}(A\text{\hspace{0.17em}⁣}:B)$, where $S_R^{(n)}(A\text{\hspace{0.17em}⁣}:B)$ is the Renyi reflected entropy. Using this, we compute $E_P(A\text{\hspace{0.17em}⁣}:B)$ for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section $EW(A\text{\hspace{0.17em}⁣}:B)$, proving a previous conjecture motivated from AdS/CFT.