Sum rule for the pseudo-R\'enyi entropy

TL;DR

This paper introduces an operator sum rule for pseudo-Renyi entropy based on transition matrices, linking off-diagonal operator elements to superposition states, with proofs in finite and quantum field systems, and explores implications in gravity duals.

Contribution

It establishes a novel operator sum rule for pseudo-Renyi entropy involving transition matrices and superposition states, expanding understanding of pseudoentropy in quantum systems.

Findings

Proved the operator sum rule in finite-dimensional and quantum field systems.

Linked off-diagonal operator elements to superposition state expectations.

Highlighted implications for transition matrices, pseudoentropy, and gravity duals.

Abstract

By generalizing the density matrix to a transition matrix between two states, represented as $|\phi\rangle$ and $|\psi\rangle$, 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 $|\phi\rangle$ and $|\psi\rangle$. 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-R\'enyi entropy and the R\'enyi 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.