Estimating the reflected entropy from random matrices

The reflected entropy S_R(ρ_AB) of a density matrix ρ_AB is a useful tool to estimate the bounds for the entropy of purification. Because of this, it further determines the size of a tensor network we need to represent the purification of the state ρ_AB. When the dimension of the Hilbert space is large, calculating the entropy of purification of ρ_AB becomes impractical. So we approximate the entropy of purification with the average reflected entropy<S_R(ρ_AB)>, where ρ is a random density matrix. But random matrices obey the volume law rather than the area law, we cannot directly apply this approximation to a physics model. To overcome this problem, we introduce the double restriction Tr_A(ρ) = ρ_B and Tr_B(ρ) = ρ_A to the random density matrix ρ, and calculate<S_R(ρ_AB)>for a given mutual information. We then compare our results with the actual reflected entropies on small random systems and spin chain models, e.g., the 1-dimensional Ising model and XXZ model.