Interplay of entanglement structures and stabilizer entropy in spin models

TL;DR

This work investigates the quantum complexity of spin-1/2 models by jointly analyzing entanglement structure and nonstabilizerness (magic) through entanglement spectrum antiflatness, capacity, and stabilizer entropy measures. The authors deploy multiple magic monotones, including Stabilizer Rényi Entropy (SRE), Robustness of Magic (RoM), and Mana, alongside entanglement diagnostics in a suite of spin models (XXZ, XY, XY with Dzyaloshinskii–Moriya interaction, Cluster Ising, and Cluster XY) using tensor-network techniques. They demonstrate that entanglement spectral properties and magic measures consistently identify quantum phases and phase transitions, with distinctive finite-size scaling behaviors and, in the XY family, a separability circle that clarifies the relation between classical-like ground states and quantum complexity. The findings deepen our understanding of quantum complexity in many-body systems and point to robust diagnostics for quantum phases that could inform future experimental explorations in trapped ions, Rydberg arrays, and superconducting qubits.

Abstract

Understanding the interplay between nonstabilizerness and entanglement is crucial for uncovering the fundamental origins of quantum complexity. Recent studies have proposed entanglement spectral quantities, such as antiflatness of the entanglement spectrum and entanglement capacity, as effective complexity measures, establishing direct connections to stabilizer Rényi entropies. In this work, we systematically investigate quantum complexity across a diverse range of spin models, analyzing how entanglement structure and nonstabilizerness serve as distinctive signatures of quantum phases. By studying entanglement spectra and stabilizer entropy measures, we demonstrate that these quantities consistently differentiate between distinct phases of matter. Specifically, we provide a detailed analysis of spin chains including the XXZ model, the transverse-field XY model, its extension with Dzyaloshinskii-Moriya interactions, as well as the Cluster Ising and Cluster XY models. Our findings reveal that entanglement spectral properties and magic-based measures serve as intertwined, robust indicators of quantum phase transitions, highlighting their significance in characterizing quantum complexity in many-body systems.

Figures (13)

• Figure 1: Rényi entropies, entanglement capacity and log-ratio of moments of the reduced state: We present the variation of the entanglement capacity, Rényi entropies, and the log-ratio ($\log \Lambda$) of the moments of the single-qubit reduced state derived from the two-qubit state in Eq. \ref{['eq:max_entangled_state']}, as a function of $u$. Notably, the behavior of $C_E$de2019aspectsNandy2021 and $\log \Lambda$ differs significantly from that of the Rényi entropies. While the Rényi entropies attain their maximum value at $u = 0.5$ (corresponding to the maximally entangled state), where both the capacity and $\log \Lambda$ vanish, the latter two exhibit peaks at intermediate values of $u$, corresponding to partially entangled states.
• Figure 2: Antiflatness, capacity of entanglement and stabilizer Rényi entropy: (\ref{['fig:magic_theta']}) The stabilizer Rényi entropy (SRE) as a function of the parameter $\theta$ for the initial state defined in Eq. \ref{['eq:magic_initial_state']}. The behaviour of SRE is periodic in $\theta$ and it reaches a maximum for $\theta=\pi/4$. Moreover, we plot the average over different realizations and time of $\Lambda$ and the average of the $C_E$ as a function of parameter $\theta$. (\ref{['fig:magic_theta2']}) We plot the average over the trajectories and time of the maximum value of $\log\langle\Lambda\rangle$ and of $C_E$ as a function of parameter $\theta$.
• Figure 3: SRE and antiflatness in the XXZ model: (\ref{['fig:M2_XXZ']}) SRE vs the interaction $\Delta$ for $h_z=0.5$. (\ref{['fig:Lamb_XXZ']}-\ref{['fig:CE_XXZ']}) The antiflatness $\log (\Lambda)$ and $C_E$ as a function of the interaction $\Delta$ for $h_z=0.5$.
• Figure 4: SRE on the XY model's separability circle: The 2-SRE ($M_2$) of the state $\ket{\phi_i^{XY}}$ (Eq. \ref{['eq:XY_CGS_single_qubit_2_SRE']}) with respect to $\gamma \in [0,1]$, $\theta(\gamma) = \arccos{\sqrt{(1-\gamma)/(1+\gamma)}}$ and $h = \sqrt{1 - \gamma^2}$.
• Figure 5: SRE in the XY model: (\ref{['fig:M2_XY']}) SRE vs the magnetic field $h_z$ for $\gamma=0.7$. (\ref{['fig:M2_XY_separable']}) SRE vs the magnetic field $h_z$ along the separable line $\gamma=\sqrt{1-h^2_z}$.