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

Abstract

We study the late-time properties of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories (RCFTs). The two non-orthogonal locally excited states used to construct the transition matrix are generated by acting different descendant operators on the vacuum. We prove that for the cases where two descendant operators are generated by a single Virasoro generator respectively acting on a primary operator, the late-time excess of pseudo-entropy and pseudo-Rényi entropy always coincides with the logarithmic of the quantum dimension of the corresponding primary operator. Furthermore, we consider two linear combination operators generated by the generic summation of Virasoro generators. We find their pseudo-Rényi entropy and pseudo-entropy may get additional contributions, as the mixing of holomorphic and anti-holomorphic parts of the correlation function enhances the entanglement. Finally, we assert the pseudo-Rényi entropy and pseudo-entropy are still the logarithmic quantum dimension of the primary operator when the correlation function of linear combination operators can be divided into the product of its holomorphic part and anti-holomorphic part. We offer some examples to illustrate the phenomenon.

Timelike entanglement entropy

“…Remarkably, in Euclidean time-dependent asymptotically AdS spaces, the pseudo entropy is related to the minimal surface area in the same formula as (1.3). See [18][19][20][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] for further related developments. Indeed, in both the timelike EE and the EE in dS/CFT, the reduced density matrices are not Hermitian and thus are not standard von Neumann entropies, but rather pseudo entropies instead.…”