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arxiv: 2606.14208 · v2 · pith:LU6BUGN7new · submitted 2026-06-12 · ✦ hep-th · cond-mat.mes-hall· quant-ph

Real-time pseudo entropy and modular-Hamiltonian correlations

Tatsuhiro Misumi This is my paper

Pith reviewed 2026-06-27 05:10 UTC · model grok-4.3

classification ✦ hep-th cond-mat.mes-hallquant-ph
keywords pseudo entropymodular Hamiltonianreal-time evolutionentanglement entropyshort-time expansioncovariancequantum quenchesIsing model
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The pith

Real-time pseudo entropy's short-time change is set by the correlation of physical and modular Hamiltonians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies real-time pseudo entropy, a complex generalization of entanglement entropy constructed from the transition matrix between an initial pure state and its unitary time evolution. It shows that for short times the pseudo entropy of any subsystem A expands as S_A(0) minus i t times the expectation value of K_A times (H minus its average), plus higher orders, where K_A is the modular Hamiltonian of the initial reduced state. For Hermitian dynamics this makes the imaginary part arise from the symmetrized covariance of H and K_A while the real part arises from their commutator. A sympathetic reader would care because the result turns the imaginary component into a concrete signature of how full-system time evolution couples to subsystem coarse-graining rather than an arbitrary feature of the branch cut.

Core claim

The central claim is that the real-time pseudo entropy S_A(t,0) for subsystem A expands at short times as S_A(0) minus i t times the expectation value of K_A times (H minus its average), plus O(t squared), where K_A equals minus log of the initial reduced density matrix. For Hermitian evolution the imaginary response is controlled by the symmetrized covariance of H and K_A while the real response is governed by their commutator. The relation follows from the definition of the transition matrix, recovers the thermal case as a special instance, admits an all-order form in a Schmidt-diagonal model, and is illustrated by explicit two-qubit calculations and transverse-field Ising quenches includi

What carries the argument

The expectation value of the product of the modular Hamiltonian K_A equals minus log rho_A and the deviation of the physical Hamiltonian H, which fixes the linear term in the short-time expansion of the real-time pseudo entropy.

If this is right

  • The imaginary part of real-time pseudo entropy encodes a physical time-oriented modular response rather than a branch artifact.
  • For Hermitian dynamics the initial imaginary response equals minus the symmetrized covariance of H and K_A.
  • The initial real response is governed by the commutator of H and K_A.
  • The result recovers thermal pseudo entropy as a special case and supplies an all-order expression in Schmidt-diagonal models.
  • The covariance response is confirmed numerically in transverse-field Ising-chain quenches including finite-size modular susceptibility near the critical point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Short-time pseudo entropy dynamics could serve as an experimental probe for the structure of modular Hamiltonians in interacting quantum systems.
  • The covariance-commutator decomposition might connect to existing measures of quantum information scrambling or chaos in time-dependent settings.
  • Relating the amplitude-level response to ordinary entropy production could open a quantitative bridge between microscopic transition matrices and thermodynamic irreversibility.

Load-bearing premise

The real-time transition matrix is formed between an initial pure state and its unitary time-evolved state.

What would settle it

A direct short-time expansion of the pseudo entropy computed in the two-qubit model or the transverse-field Ising chain that fails to reproduce the predicted covariance term with the modular Hamiltonian would disprove the governing relation.

Figures

Figures reproduced from arXiv: 2606.14208 by Tatsuhiro Misumi.

Figure 1
Figure 1. Figure 1: FIG. 1. Real and imaginary parts of the real-time pseudo [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size modular susceptibility [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original 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 $K_A=-\log\rho_A$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript studies real-time pseudo entropy defined from the reduced transition matrix between an initial pure state and its unitary time evolution. The central claim is the short-time expansion S_A(t,0)=S_A(0)−it⟨K_A(H−⟨H⟩)⟩+O(t²) for a subsystem A, with the linear coefficient given by the indicated correlation between the physical Hamiltonian H and the modular Hamiltonian K_A=−logρ_A. The work decomposes the response into symmetrized covariance (imaginary part) and commutator (real part) for Hermitian dynamics, recovers the thermal case as a special limit, supplies an all-order Schmidt-diagonal expression, illustrates the decomposition in a two-qubit model, and confirms the covariance coefficient via transverse-field Ising-chain quenches together with a finite-size modular-susceptibility study near criticality. A PT-symmetric non-Hermitian toy model is also discussed.

Significance. If the derivation holds, the result supplies a direct, first-principles link between real-time unitary evolution and the modular structure of a subsystem at the level of pseudo entropy. The explicit covariance/commutator decomposition, the demonstration that the imaginary part is a physical modular response rather than a branch artifact, the recovery of the thermal limit, and the numerical checks in solvable models are concrete strengths. The all-order Schmidt-diagonal formula and the finite-size critical-region study further strengthen the contribution to the literature on dynamical entanglement measures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of its contributions, and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct from definitions

full rationale

The short-time expansion S_A(t,0)=S_A(0)-it ⟨K_A(H−⟨H⟩)⟩ + O(t²) is obtained by direct first-order Taylor expansion of the pseudo-entropy functional applied to the normalized real-time transition matrix T_A(t) = Tr_B |ψ⟩⟨ψ(t)| / ⟨ψ|ψ(t)⟩. The covariance term is the exact linear coefficient arising from the definition of K_A = −log ρ_A and the global-phase normalization; no parameter is fitted and then relabeled as a prediction. The relation to the first law of pseudo entropy is presented as a clarification of a known result rather than a load-bearing self-citation chain. Explicit all-order Schmidt-diagonal expressions, two-qubit checks, and Ising-chain numerics are independent verifications, not reductions to the input. The manuscript is therefore self-contained with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of pseudo entropy via the reduced transition matrix and on unitary quantum evolution; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Quantum mechanics with unitary time evolution generated by a Hamiltonian H
    Invoked to define the real-time transition matrix between initial state and evolved state.
  • domain assumption Pseudo entropy is defined from the reduced transition matrix between two states
    This is the starting definition for the real-time pseudo entropy studied in the paper.

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