Timelike Entanglement Signatures of Ergodicity and Spectral Chaos

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.

Figures (3)

• Figure 1: (a) Frobenius norm $\|T_{AB}(t)\|_2^2$ vs $t$: faster growth and higher plateau at small $\gamma$, suppressed at large $\gamma$, with the fractal regime interpolating. (b) $\mathrm{Re}\,\mathrm{Tr}[T_{AB}(t)^2]$ vs $t$: SFF-like dip--ramp--plateau in the ergodic regime; ramp weakens in the fractal regime and is absent when localized. (c) $2$-imagitivity $\mathcal{I}_2(t)$ vs $t$: non-Hermiticity grows and saturates most strongly for small $\gamma$ and is reduced for large $\gamma$. (d) Kernel negativity $\mathcal{N}_H(t)$ vs $t$: negative spectral weight develops at late times; its plateau decreases with increasing $\gamma$. (e) $\overline{F_2(t,\beta=0)}$ vs $t$: $\overline{F_2(t,0)}$ behaves as the normalised spectral form factor (SFF) showing dip--ramp--plateau in the ergodic regime; ramp weakens in the fractal regime and vanishes when localized. (f) $D_2$ (star) and rescaled time-averaged $\overline{\mathcal{N}_H(T_{AB})}$ (diagonal-cross) and $\overline{\|T_{AB}\|_2^2}$ (triple-cross) vs $\gamma$ (dashed lines at $\gamma=1,2$): $\overline{\mathcal{N}_H(T_{AB})}$ and $\overline{\|T_{AB}\|_2^2}$ decrease in tandem with the loss of eigenvector ergodicity---tracking the same ergodic--fractal--localized crossover signaled by $D_2$.
• Figure 2: Top row: second timelike Tsallis entropy, $\mathrm{Re}\,\mathrm{Tr}[T_{AB}(t)^2]$. Bottom row: averaged two-point correlator $\overline{F}_2(t,\beta)$ versus $t$. Each curve corresponds to a RP parameter $\gamma$ (step $0.1$), with the same rainbow color scheme as Fig. \ref{['fig:1']} (warm: small $\gamma$, cool: large $\gamma$); the time axis is logarithmic. Left column ($0.1\le \gamma\le 1$, ergodic): clear dip--ramp--plateau in both quantities. Middle ($1\le \gamma\le 2$, fractal): ramp weakens and becomes $\gamma$ dependent, with a delayed approach to the plateau. Right ($2\le \gamma\le 3$, localized): ramp is strongly suppressed, and both curves approach the plateau directly. Overall, $\mathrm{Re}\,\mathrm{Tr}[T_{AB}(t)^2]$ displays comparable dynamics to SFF by reproducing the dip--ramp--plateau across the RP phase diagram. The colour scheme used here is the same as in Fig. \ref{['fig:1']}.
• Figure 3: Top row (a--c): Frobenius norm $\|T_{AB}(t)\|_2^2$ for three bipartitions of a six-qubit RP system. In all cases, $\|T_{AB}(t)\|_2^2$ grows more rapidly and saturates to a higher plateau for small $\gamma$ (ergodic), while both the growth and plateau are suppressed at large $\gamma$ (localized), with the fractal regime interpolating between these limits. Bottom row (d--f): kernel negativity $\mathcal{N}_H(t)$ for the same bipartitions. For the disjoint partitions (d), negativity is generated dynamically and saturates to a $\gamma$-dependent value that decreases with increasing $\gamma$; the separated disjoint partition (e) exhibits the no growth over the time window. For the overlapping partition (f), $\mathcal{N}_H(t)$ is nonzero already at $t=0$ and relaxes to a late-time value that again decreases with increasing $\gamma$. Overall, the phase-sensitive trends of both $\|T_{AB}(t)\|_2^2$ and $\mathcal{N}_H(t)$ are robust, while their dynamical behavior and absolute magnitude depend on whether the subsystems overlap. The colour scheme used here is the same as in Fig. \ref{['fig:1']}.