Timelike Entanglement Signatures of Ergodicity and Spectral Chaos

TL;DR

This paper introduces timelike entanglement measures derived from the spacetime density kernel in the Rosenzweig-Porter model, which effectively diagnose eigenvector ergodicity and spectral chaos through various entropy and norm diagnostics.

Contribution

It develops new timelike entanglement diagnostics based on the spacetime density kernel that distinguish ergodic, fractal, and localized phases in spectral models.

Findings

Imagitivity and Frobenius norm grow rapidly in ergodic regime

Second Tsallis entropy shows a dip-ramp-plateau pattern in chaotic regime

Kernel negativity decreases across ergodic-fractal-localized crossover

Abstract

We investigate timelike entanglement measures derived from the spacetime density kernel in the Rosenzweig-Porter model and show that they sharply diagnose both eigenvector ergodicity and spectral chaos. For several Hilbert-space bipartitions, we compute the second Tsallis entropy, the entanglement imagitivity that quantifies non-Hermiticity, and Schatten-norm diagnostics of the kernel. The imagitivity and Frobenius norm exhibit rapid growth and high late-time plateaus in the ergodic regime, are suppressed in the localized regime, and show intermediate behavior in the fractal phase. The real part of the second Tsallis entropy displays a spectral form factor-like dip-ramp-plateau throughout the chaotic window and a suppressed ramp in the localized regime. We further introduce a kernel negativity, defined as the negative spectral weight of the Hermitian part of the kernel. This negativity equals the trace-norm distance to the set of positive semidefinite operators and the maximal witnessable negative quasiprobability, and its time-averaged value decreases across the ergodic-fractal-localized crossover in close correspondence with the fractal dimension.