Entanglement of purification and projection operator in conformal field theories

We investigate entanglement of purification (EoP) in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for rho(AB), where rho(AB) is reduced density matrix for subregion AB of a global state rho. The set can be approximated by acting all the unitary observables, located in the complement of subregion AB, on the global state rho, as long as the global state rho is cyclic for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state correspondence, we give an explanation of holographic EoP formula. We also explore the projection operator with the conformal basis in conformal field theory. In some limit we may produce the holographic EoP results by using the projection operator. Finally, we discuss the similarity and difference between the projection operator and unitary operations for calculating EoP. (C) 2019 The Author(s). Published by Elsevier B.V.