Pseudoentropy is a complex-valued generalization of entanglement entropy defined from a reduced transition matrix. We study the pseudoentropy 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 pseudoentropy is governed by the correlation between the physical Hamiltonian $H$ and the modular Hamiltonian $K_A=-log{\rho }_A$ of the initial reduced state, $S_A(t,0)=S_A\left(0\right)-it\langle K_A(H-\langle H\rangle )\rangle +O\left(t^2\right)$. 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 pseudoentropy 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 pseudoentropy, derive an all-order expression in a Schmidt-diagonal model, recover thermal pseudoentropy as a special case, illustrate the covariance or 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 $\mathcal{PT}$-symmetric toy-model illustration of the non-Hermitian extension.

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• Entanglement in field theory
• PT-symmetric quantum mechanics
• Quantum information theory