Unitarity breaking in self-averaging spectral form factors

TL;DR

This paper investigates how filtering and averaging techniques in spectral form factors lead to unitarity breaking, revealing open quantum dynamics and conditions under which the SFF becomes self-averaging at long times.

Contribution

It demonstrates that filtering and averaging of the spectral form factor induce unitarity breaking, connecting these processes to open quantum system dynamics and providing a unified framework for understanding their effects.

Findings

Averaging over Hamiltonian ensembles and filters involve mixed-unitary quantum channels.

Frequency filters relate to a master equation generalizing energy dephasing.

Frequency and energy filters cause the SFF to become self-averaging at long times.

Abstract

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, 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.