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 $K_A=-\logρ_A$ 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 $\mathcal{PT}$-symmetric toy-model illustration of the non-Hermitian extension.

• The paper introduces real-time pseudo entropy as a complex extension of entanglement entropy to capture time-oriented quantum responses.
• It establishes that the short-time expansion of pseudo entropy is governed by the covariance between the subsystem modular Hamiltonian and the system Hamiltonian.
• Numerical and analytical models, including two-qubit systems and Ising chain quenches, validate modular correlations as indicators of irreversibility and critical behavior.

Real-Time Pseudo Entropy and Modular-Hamiltonian Correlations: Technical Summary

Background and Motivation

Entanglement entropy is central in quantifying non-classical correlations in quantum subsystems, but its unitary evolution prohibits a microscopic arrow of time. "Pseudo entropy" generalizes entanglement entropy to complex values, using reduced transition matrices between two possibly nonorthogonal states. This framework enables amplitude-level diagnostics of time-oriented responses in quantum systems, probing emergent irreversibility prior to any coarse-graining or probabilistic measurement. The work under discussion systematically develops the real-time pseudo entropy—constructed from the transition between a pure state and its time-evolved image—and establishes its leading order response as directly governed by modular-Hamiltonian correlations.

Definition and Central Results

The real-time pseudo entropy for a subsystem $A$ is constructed from the reduced transition matrix between the initial state $|\Psi\rangle$ and its time-evolved form $|\Psi(t)\rangle = e^{-iHt}|\Psi\rangle$: $$\tau_A(t,0) = \operatorname{Tr}_{\bar A} \frac{|\Psi(t)\rangle\langle\Psi|}{\langle\Psi|\Psi(t)\rangle},\qquad S_A(t,0) = -\operatorname{Tr}_A\left[ \tau_A(t,0) \log \tau_A(t,0) \right].$$ Unlike ordinary entanglement entropy, $S_A(t,0)$ is generically complex due to the non-Hermitian nature of the transition matrix.

The core result provides a universal short-time expansion: $$S_A(t,0) = S_A(0) - it\, \langle K_A(H-\langle H\rangle)\rangle + \mathcal{O}(t^2),$$ with $K_A = -\log\rho_A$, the modular Hamiltonian of the initial reduced density matrix $$\rho_A = \operatorname{Tr}_{\bar A} |\Psi\rangle\langle\Psi|$$.

For Hermitian $H$ and $K_A$, this expansion decomposes: $|\Psi\rangle$0 Thus, the imaginary part of the pseudo entropy encodes the symmetrized covariance between the subsystem modular Hamiltonian and the system Hamiltonian, manifesting time orientation even at infinitesimal time—distinct from branch-related artifacts. The real part captures the lack of commutation between $|\Psi\rangle$1 and $|\Psi\rangle$2.

All-Order Solution: Schmidt-Diagonal Model and Thermal Limit

For states and Hamiltonians diagonal in the same Schmidt basis, the pseudo entropy admits an exact solution for all times, expressing it in terms of a complex-tilted distribution. The explicit formula: $|\Psi\rangle$3 where $|\Psi\rangle$4 and averages are computed with respect to the complex tilted Boltzmann weights.

Expanding at small $|\Psi\rangle$5 yields the short-time behavior: $|\Psi\rangle$6 recovering for thermal states ($|\Psi\rangle$7) that

$|\Psi\rangle$8

This links the imaginary pseudo entropy rate directly to the energy variance in thermal ensembles.

Two-Qubit Example and Numerical Verification

In a minimal noncommuting example with two qubits,

$|\Psi\rangle$9

the explicit calculation demonstrates the above covariance-commutator decomposition. The results show, for real initial states, vanishing real response and pure imaginary slope—confirming the modular covariance prediction.

Figure 1: Real and imaginary parts of the real-time pseudo entropy in the two-qubit model; the imaginary part is linear with a slope set by the modular covariance.

Many-Body Dynamics: Ising Chain Quenches

The analysis generalizes to non-integrable many-body systems by numerically studying quantum quenches in finite transverse-field Ising chains. After a field quench, the initial slope of the imaginary component of the pseudo entropy matches the modular covariance, while the real part remains quadratic due to time-reversal symmetry.

Figure 2: Real-time pseudo entropy after a transverse-field Ising-chain quench; the imaginary part demonstrates a linear dependence driven by modular covariance.

Additionally, by examining small parameter quenches, the paper introduces a modular susceptibility—the modular covariance between the subsystem modular Hamiltonian and the Hamiltonian's driving operator—which exhibits finite-size scaling indicative of sensitivity to quantum criticality.

Figure 3: Finite-size modular susceptibility $|\Psi(t)\rangle = e^{-iHt}|\Psi\rangle$0 in the open transverse-field Ising chain; peak enhancement near the critical point signals critical modular response.

Non-Hermitian Extension and PT-Symmetry

The framework is generalized to non-Hermitian quantum mechanics using biorthogonal transition matrices. The modular-covariance formula persists, with expectation values replaced by biorthogonal analogues. The analytic structure across $|\Psi(t)\rangle = e^{-iHt}|\Psi\rangle$1-breaking transitions is examined, showing a shift from hyperbolic to trigonometric and branch-cut dominated pseudo entropy behavior. This connects modular-oriented response to phenomena associated with exceptional points and non-Hermitian topology.

Implications and Future Perspectives

Theoretical Implications: The imaginary component of real-time pseudo entropy defines a subsystem-oriented, time-asymmetric response sensitive to quantum correlations between system evolution and information structure. This bridges amplitude-level quantum dynamics to emergent irreversibility, prefiguring the appearance of entropy production once a probabilistic or measurement-based framework is introduced. The approach brings structural insight to irreversibility in closed quantum systems and is directly linked to modular Hamiltonians central to quantum information and holography.

Numerical Results: Strong agreement is found between analytic modular covariance predictions and exact calculations in both finite-size two-qubit and many-body settings, establishing the reliability of the response formulas.

Potential Applications: Modular susceptibilities may serve as operational diagnostics for criticality, complementing fidelity susceptibilities and quantum geometric tensors. In non-Hermitian systems, biorthogonal modular responses provide a framework for systematic studies of entropy and irreversibility near exceptional points.

Speculation and Outlook: Prospective research directions include a rigorous mapping from complex, amplitude-level pseudo entropy to classical entropy production in open or measured systems, possible universal scaling of modular susceptibilities in larger systems, and holographic derivations of modular-oriented responses as complex extremal surface areas. These results have direct bearing on the intersection of quantum thermodynamics, non-equilibrium field theory, and AdS/CFT.

Conclusion

This work establishes that the leading short-time, imaginary part of real-time pseudo entropy in closed quantum systems is universally governed by the symmetrized covariance between the modular Hamiltonian of a subsystem and the system Hamiltonian. This result clarifies the microscopic origin of oriented, time-asymmetric responses in quantum dynamics and provides a modular-based analytic tool for exploring irreversibility, critical phenomena, and non-Hermitian physics in a unified language.