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 $\tilde{A}$ adjoins $A$ and $\tilde{A}\tilde{B}$ 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 $|\psi\rangle_M$. We calculate the variance of entanglement entropy of $A\tilde{A}$ in the state $|\psi(\delta)\rangle:=e^{i\delta H_{\tilde{A}\tilde{B}}}|\psi\rangle_M$. We find a constraint on the state $|\psi\rangle_M$, $[K_{A\tilde{A},M},O_{\tilde{A}}]=0$, where $K_{A\tilde{A},M}$ is the modular Hamiltonian of $A\tilde{A}$ in the state $|\psi\rangle_M$, $O_{\tilde{A}}\in \mathscr{R}(\tilde{A})$ 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 $|\psi\rangle_M$, since the distance between these three states and $|\psi\rangle_M$ is very large.