By generalizing the density matrix to a transition matrix between two states, represented as $|\phi\rangle$ and $|\psi\rangle$, one can define the pseudo entropy analogous to the entanglement entropy. In this paper, we derive an operator sum rule involving the reduced transition matrix and density matrix of the superposition states of $|\phi\rangle$ and $|\psi\rangle$. As a special application, we demonstrate that the pseudo R\'enyi entropy is related to the R\'enyi entropy of the superposition states. We provide a proof of the operator sum rule and verify its validity in both finite-dimensional systems and quantum field theory. Our results have potential applications in physics, particularly involving off-diagonal matrix elements of local and non-local observables.
Comment: 4+8 pages