  In this paper, we establish a sum rule that connects the pseudoentropy and
entanglement entropy of a superposition state. Through analytical continuation
of the superposition parameter, we demonstrate that the transition matrix and
density matrix of the superposition state can be treated in a unified manner.
Within this framework, we naturally derive sum rules for the (reduced)
transition matrix, pseudo R\'enyi entropy, and pseudoentropy. Furthermore, we
demonstrate the close relationship between the sum rule for pseudoentropy and
the singularity structure of the entropy function for the superposition state
after analytical continuation. We also explore potential applications of the
sum rule, including its relevance to understanding the gravity dual of
non-Hermitian transition matrices and establishing upper bounds for the
absolute value of pseudoentropy.
