Stabilizer entropy in non-integrable quantum evolutions

TL;DR

This work investigates how entanglement, non‑stabilizerness via Stabilizer Rényi Entropy (SRE), and entanglement‑spectrum anti‑flatness evolve after global quenches in spin chains, comparing integrable (free‑fermion and Bethe‑Ansatz) to non‑integrable dynamics. Using three initial‑state ensembles (FR, FC, NFC) and Krylov time evolution, the authors show that free‑fermion dynamics exhibit a long‑time gap from Haar randomness in $S_{\alpha}$ and $\mathcal{M}_{\alpha}$, while non‑integrable dynamics saturate to Haar‑like values; NFC states drive universal Haar behavior irrespective of integrability. The study also highlights a strong link between anti‑flatness and non‑stabilizerness, demonstrating that anti‑flatness can jointly diagnose the entanglement and magic pathways to chaos, with BA integrable systems sometimes mimicking chaotic behavior. Overall, the results illuminate how initial resources of magic and entanglement shape scrambling, thermalization, and the simulability of quantum dynamics, and suggest avenues for probing SRE propagation and disordered systems in the quest for quantum chaos diagnostics.

Abstract

Entanglement and stabilizer entropy are both involved in the onset of complex behavior in quantum many-body systems. Their interplay is at the root of complexity of simulability, scrambling, thermalization and typicality. In this work, we study the dynamics of entanglement, stabilizer entropy, and the anti-flatness of the entanglement spectrum after a quantum quench in a spin chain. We find that free-fermion theories show a gap in the long-time behavior of these resources compared to their random matrix theory value while non-integrable models saturate it.

Figures (7)

• Figure 1: Schematic representation of the results for the Hamiltonian evolution of a global quantum quench protocol in the long-time limit. The initial states considered belong to three distinct ensembles: factorized random (FR), factorized Clifford (FC), and non-factorized Clifford (NFC) states. FR and FC ensembles display the same behavior and are grouped together by a common grey background color. A single colored rectangle (labeled "0" in the figure) indicates that the states possess zero amount of the particular resource. In contrast, five stacked rectangles denote the maximal value ("max") expected from a typical random pure state. Notations: FF – free fermions; BA – Bethe Ansatz.
• Figure 2: Illustration of states with flat and non-flat distribution of RDM spectrum.
• Figure 3: Panels (a-c): Half-chain von Neumann entanglement entropy Eq. \ref{['VNdef']} for short-time dynamics generated by TFIM-L Hamiltonian and different initial state ensembles (see Sect. \ref{['STS_InitialStates']}): Factorized Clifford (FC), Factorized Random (FR), and Non-Factorized Clifford (NFC). Parameters choices: $N = 16$, $M = 100$ realizations, and $h_{z} = 1.5$ and the time-step $\Delta t = 0.1$. The legends are shared between the panels (a-c). The shaded areas represent the standard deviation from the sample mean across $M$ realizations of the initial states. Panel (d): Average entanglement entropy for different subsystem partition sizes in the long-time limit where the initial states are sampled from the FC ensemble. Parameters choices: $N = 16$, $M = 50$ realizations, $\Delta t = 2$, $t^{\rm final} = 10^{4}$. Panel (e): Relative difference as defined by Eq. \ref{['relative_difference']} of the von Neumann entanglement entropy for parameter choices as in panel (d) but for increasing system sizes. Non-integrable dynamics in the long-time limit lead to Haar random states, while integrable do not. The legends are shared between panels (d-e).
• Figure 4: Panels (a-c): Stabilizer Rényi entropy $\mathcal{M}_{2}$ measuring the amount of non-stabilizerness (magic) for short-time $t \in [ 0, 10]$ dynamics generated by TFIM+L Hamiltonian and different initial state ensembles (see Sect. \ref{['STS_InitialStates']}): Factorized Clifford (FC), Factorized Random (FR), and Non-Factorized Clifford (NFC). Parameters choices: $N = 16$, $h_{z} = 1.5$, and the time-step $\Delta t = 0.1$ with $M = 100$ realizations. Entanglement entropy for the same parameters is given in Fig. \ref{['fig:short1']}. The shaded areas represent the standard deviation from the sample mean across $M$ realizations of the initial states.
• Figure 5: Ensemble averaged relative difference defined for different quantities in the case of the long-time quench dynamics generated by the TFIM+L Hamiltonian. Parameters choices: $M = 50$, $\Delta t = 2$, $t^{\rm final} = 10^{4}$ (long-time limit), and FR as the initial state ensemble.