Sum rule for the pseudo-Rényi entropy

By generalizing the density matrix to a transition matrix between two states, represented as $\mathrm{\mid }\phi ⟩$ and $\mathrm{\mid }\psi ⟩$, one can define the pseudoentropy analogous to the entanglement entropy. In this paper, we establish an operator sum rule that pertains to the reduced transition matrix and reduced density matrices corresponding to the superposition states of $\mathrm{\mid }\phi ⟩$ and $\mathrm{\mid }\psi ⟩$. It is demonstrated that the off-diagonal elements of operators can be correlated with the expectation value in the superposition state. Furthermore, we illustrate the connection between the pseudo-Rényi entropy and the Rényi entropy of the superposition states. We provide proof of the operator sum rule and verify its validity in both finite-dimensional systems and quantum field theory. We additionally demonstrate the significance of these sum rules in gaining insights into the physical implications of transition matrices, pseudoentropy, and their gravity dual.