Sum rule for the pseudo-Rényi entropy

TL;DR

This work introduces an operator sum rule that relates the $n$-th power of the reduced transition matrix $\mathcal{T}_A^{\psi|\phi}$ to a linear combination of powers of reduced density matrices $\rho_A(k)$ for superposition states $|\xi(k)\rangle$, revealing a Fourier-analytic structure that connects off-diagonal and diagonal data. The authors provide a detailed proof, derive a replica-based expression in QFT, and demonstrate the rule in concrete models including two-qubit systems, perturbative states, locally excited 2D CFTs, and spin chains, with consistency checks against holographic expectations and quasiparticle pictures. They further show that pseudo-Rényi entropy is linked to the Rényi entropies of the superposition states through the same sum-rule framework and discuss broader implications for gravity duals and static-dynamic quantity relations. The results offer a unified perspective on transition matrices, pseudoentropy, and their holographic interpretations, while highlighting limitations and avenues for generalization, such as pseudoentropy sum rules and mixed-state extensions.

Abstract

By generalizing the density matrix to a transition matrix between two states, represented as $|φ\rangle$ and $|ψ\rangle$, 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 $|φ\rangle$ and $|ψ\rangle$. 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-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.

Figures (1)

• Figure 1: The real and imaginary parts of the function $\tilde{\mathcal{S}}(\theta)$ and pseudoentropy $S(\mathcal{T}_A^{\psi|\phi})$ as a function of $\theta$. In the region $\theta \ll 1$, $S(\mathcal{T}_A^{\psi|\phi}) \approx \tilde{\mathcal{S}}(\theta)$, which is consistent with the result that the pseudoentropy sum rule (\ref{['SMsumpseudo']}) is available in the leading order of the perturbation.