Eigenstates from time-averaged evolution: a sinc filter, phase projection, and a squared-Hamiltonian comparison

Quantum many-body states with sufficiently low energy variance can serve as approximations to thermal states, and they may be prepared by energy filtering simple pure states. In this work, we examine how narrow the filter width must be to guarantee thermal behavior. To this end, we analyze the problem in the Floquet regime, where filtered states are found to be equivalent to time averages. This equivalence allows us to reproduce the distinct Rényi- $α$ entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.Quantum many-body states with sufficiently low energy variance can serve as approximations to thermal states, and they may be prepared by energy filtering simple pure states. In this work, we examine how narrow the filter width must be to guarantee thermal behavior. To this end, we analyze the problem in the Floquet regime, where filtered states are found to be equivalent to time averages. This equivalence allows us to reproduce the distinct Rényi-$α$ entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.

quant-pharXiv:2607.06847Jul 9, 20260 citations

Projection of a quantum state on eigenstates of average Hamiltonian

Jae‐Seung Lee, Anatoly K. Khitrin

arXiv:0801.4098Journal of Magnetic ResonanceMar 6, 20095 citations

The Eigenvalue Equation and the Evolution Operator

Michele Cini

UNITEXT for physicsJan 1, 20181 citation

Unitarity breaking in self-averaging spectral form factors

Apollonas S. Matsoukas-Roubeas, Mathieu Beau, Léa Ferreira dos Santos +1 more

The complex Fourier transform of the two-point correlator of the energy spectrum of a quantum system is known as the spectral form factor (SFF). It constitutes an essential diagnostic tool for phases of matter and quantum chaos. In black hole physics, it describes the survival probability (fidelity) of a thermofield double state under unitary time evolution. However, detailed properties of the SFF of isolated quantum systems with generic spectra are smeared out by large temporal fluctuations, whose minimization requires disorder or time averages. This requirement holds for any system size, that is, the SFF is non-self-averaging. Exploiting the fidelity-based interpretation of this quantity, we prove that using filters and disorder and time averages of the SFF involve unitarity breaking, i.e., open quantum dynamics described by a quantum channel that suppresses quantum noise. Specifically, averaging over Hamiltonian ensembles, time averaging, and frequency filters can be described by the class of mixed-unitary quantum channels in which information loss can be recovered. Frequency filters are associated with a time-continuous master equation generalizing energy dephasing. We also discuss the use of eigenvalue filters. They are linked to non-Hermitian Hamiltonian evolution without quantum jumps, whose long-time behavior is described by a Hamiltonian deformation. We show that frequency and energy filters make the SFF self-averaging at long times.The complex Fourier transform of the two-point correlator of the energy spectrum of a quantum system is known as the spectral form factor (SFF). It constitutes an essential diagnostic tool for phases of matter and quantum chaos. In black hole physics, it describes the survival probability (fidelity) of a thermofield double state under unitary time evolution. However, detailed properties of the SFF of isolated quantum systems with generic spectra are smeared out by large temporal fluctuations, whose minimization requires disorder or time averages. This requirement holds for any system size, that is, the SFF is non-self-averaging. Exploiting the fidelity-based interpretation of this quantity, we prove that using filters and disorder and time averages of the SFF involve unitarity breaking, i.e., open quantum dynamics described by a quantum channel that suppresses quantum noise. Specifically, averaging over Hamiltonian ensembles, time averaging, and frequency filters can be described by the class of mixed-unitary quantum channels in which information loss can be recovered. Frequency filters are associated with a time-continuous master equation generalizing energy dephasing. We also discuss the use of eigenvalue filters. They are linked to non-Hermitian Hamiltonian evolution without quantum jumps, whose long-time behavior is described by a Hamiltonian deformation. We show that frequency and energy filters make the SFF self-averaging at long times.

arXiv:2307.04791Physical Review ADec 1, 202322 citations

The Eigenvalue Problem in Phase-Space

Leon W. Cohen

Springer eBooksNov 30, 20120 citations

Energy Translation and Proper-Time Eigenstates

Zhiyong Wang, Cai-Dong Xiong, Ailin Zhang

arXiv:quant-ph/0602100Foundations of Physics LettersDec 1, 20063 citations

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entropy scalings as reported in [Morettini et al., Physical Review Letters 133, 240401 (2024)]. Crucially, we show that under the Floquet eigenstate thermalization hypothesis, the trace distance between Floquet-filtered states and thermal states is bounded by the square root of filter width. We further demonstrate that these results extend naturally to the conventional Hamiltonian setting by mapping Hamiltonian-filtered states to their Floquet counterparts.