Pseudo-entropy for descendant operators in two-dimensional conformal field theories

Abstract: We study the late-time behaviors of pseudo-(R\'enyi) entropy of locally excited states in rational conformal field theories (RCFTs). To construct the transition matrix, we utilize two non-orthogonal locally excited states that are created by the application of different descendant operators to the vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudo-entropy and pseudo-R\'enyi entropy corresponds to the logarithmic of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(R\'enyi) entropy compared to that for (R\'enyi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(R\'enyi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(R\'enyi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.