Entanglement of purification and disentanglement in CFTs

We study the entanglement of purification (EoP) of subsystem A and B in conformal field theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem (A) over tilde adjoins A and (A) over tilde(B) over tilde is the complement of AB. We estimate the amount of the disentanglement by using the holographic EoP conjecture as well as the inequality of Von Neumann entropy. Denote the state that produces the EoP by vertical bar psi >(M). We calculate the variance of entanglement entropy of A (A) over tilde in the state vertical bar psi(d)delta > := e(i delta H)(A) over tilde(B) over tilde vertical bar psi >(M). We find a constraint on the state vertical bar psi >(M), [K-A (A) over tilde ,K-M,O-(A) over tilde]=0, where KA (A) over tilde ,M is the modular Hamiltonian of A (A) over tilde in the state vertical bar psi >(M), O-(A) over tilde is an element of R((A) over tilde) is an arbitrary operator. We also study three different states that can be seen as disentangled states. Two of them can produce the holographic EoP result in some limit. But we show that none of they could be a candidate of the state vertical bar psi >(M), since the distance between these three states and vertical bar psi >(M) is very large.