Real-time pseudo entropy and modular-Hamiltonian correlations
Abstract
Pseudo entropy is a complex-valued generalization of entanglement entropy defined from a reduced transition matrix. We study the pseudo entropy associated with a real-time transition matrix between an initial pure state and its unitary time evolution. For a subsystem A, we show that the short-time behavior of real-time pseudo entropy is governed by the correlation between the physical Hamiltonian H and the modular Hamiltonian KA=−logρA of the initial reduced state, $ S_A(t,0)=S_A(0)-it \langle K_A(H-\langle H\rangle)\rangle + \mathcal{O}(t^2)$. For Hermitian dynamics, the initial imaginary response is controlled by the symmetrized covariance of H and KA with an overall minus sign, while the initial real response is governed by their commutator. Thus the imaginary part of real-time pseudo entropy is not merely a branch artifact: it is a time-oriented modular response generated by the correlation between microscopic time evolution and subsystem coarse graining. We clarify the relation of this result to the known first law of pseudo entropy, derive an all-order expression in a Schmidt-diagonal model, recover thermal pseudo entropy as a special case, illustrate the covariance/commutator decomposition in a two-qubit model, and confirm the covariance response in transverse-field Ising-chain quenches, including a finite-size study of a modular susceptibility near the Ising critical region. We discuss how this amplitude-level oriented response can be related to ordinary entropy production, and also give a concrete 𝒫𝒯-symmetric toy-model illustration of the non-Hermitian extension.
Versions
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BibTeX
@misc{misumi2026realtime,
title = {{Real-time pseudo entropy and modular-Hamiltonian correlations}},
author = {Tatsuhiro Misumi},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.14208},
eprint = {2606.14208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14208},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Real-time pseudo entropy and modular-Hamiltonian correlations
AU - Tatsuhiro Misumi
PY - 2026
DA - 2026/06/22
PB - arXiv
DO - 10.48550/arxiv.2606.14208
UR - https://arxiv.org/abs/2606.14208
AB - Pseudo entropy is a complex-valued generalization of entanglement entropy defined from a reduced transition matrix. We study the pseudo entropy associated with a real-time transition matrix between an initial pure state and its unitary time evolution. For a subsystem A, we show that the short-time behavior of real-time pseudo entropy is governed by the correlation between the physical Hamiltonian H and the modular Hamiltonian K_A=−logρ_A of the initial reduced state, $ S_A(t,0)=S_A(0)-it \langle K_A(H-\langle H\rangle)\rangle + \mathcal{O}(t^2)$. For Hermitian dynamics, the initial imaginary response is controlled by the symmetrized covariance of H and K_A with an overall minus sign, while the initial real response is governed by their commutator. Thus the imaginary part of real-time pseudo entropy is not merely a branch artifact: it is a time-oriented modular response generated by the correlation between microscopic time evolution and subsystem coarse graining. We clarify the relation of this result to the known first law of pseudo entropy, derive an all-order expression in a Schmidt-diagonal model, recover thermal pseudo entropy as a special case, illustrate the covariance/commutator decomposition in a two-qubit model, and confirm the covariance response in transverse-field Ising-chain quenches, including a finite-size study of a modular susceptibility near the Ising critical region. We discuss how this amplitude-level oriented response can be related to ordinary entropy production, and also give a concrete 𝒫𝒯-symmetric toy-model illustration of the non-Hermitian extension.
ER -