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arXiv:2504.11139 · Physical Review Research

Nonstabilizerness in open XXZ spin chains: Universal scaling and dynamics

Doru Sticlet, Balázs Dóra, Dominik Szombathy, Gergely Zaránd, Cătălin Paşcu Moca

Published Sep 26, 2025• 11 citations
arXivHTMLPDF

Abstract

Magic, or nonstabilizerness, is a crucial quantum resource, yet its dynamics in open quantum systems remain largely unexplored. We investigate magic in the open XXZ spin chain under either boundary gain and loss or bulk dephasing using the stabilizer Rényi entropy M 2 . To enable scalable simulations of large systems, we develop a unique, highly efficient algorithm for computing M 2 within the matrix product state formalism while maintaining constant bond dimension—an advancement over existing methods. For boundary driving, we uncover universal scaling laws, M 2 ( t ) ∼ t 1 / z , linked to the dynamical exponent z for several distinct universality classes. We also disentangle classical and quantum contributions to magic by introducing a mean-field approximation for magic, thus emphasizing the prominent role of quantum-critical fluctuations in nonstabilizerness. For bulk dephasing, dissipation can transiently enhance magic before suppressing it and drive it to a nontrivial steady-state value. These findings position magic as a powerful diagnostic tool for probing universality and dynamics in open quantum systems.

DOI
10.1103/96bk-xf8p

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Magic, or nonstabilizerness, is a crucial quantum resource, yet its dynamics in open quantum systems remain largely unexplored. We investigate magic in the open XXZ spin chain under either boundary gain and loss or bulk dephasing using the stabilizer Rényi entropy M 2 . To enable scalable simulations of large systems, we develop a unique, highly efficient algorithm for computing M 2 within the matrix product state formalism while maintaining constant bond dimension—an advancement over existing methods. For boundary driving, we uncover universal scaling laws, M 2 ( t ) ∼ t 1 / z , linked to the dynamical exponent z for several distinct universality classes. We also disentangle classical and quantum contributions to magic by introducing a mean-field approximation for magic, thus emphasizing the prominent role of quantum-critical fluctuations in nonstabilizerness. For bulk dephasing, dissipation can transiently enhance magic before suppressing it and drive it to a nontrivial steady-state value. These findings position magic as a powerful diagnostic tool for probing universality and dynamics in open quantum systems.
Nonstabilizerness, also known as "magic," stands as a crucial resource for achieving a potential advantage in quantum computing. Its connection to many-body physical phenomena is poorly understood at present, mostly due to a lack of practical methods to compute it at large scales. We present a novel approach for the evaluation of nonstabilizerness within the framework of matrix product states (MPSs), based on expressing the MPS directly in the Pauli basis. Our framework provides a powerful tool for efficiently calculating various measures of nonstabilizerness, including stabilizer Rényi entropies, stabilizer nullity, and Bell magic, and enables the learning of the stabilizer group of an MPS. We showcase the efficacy and versatility of our method in the ground states of Ising and XXZ spin chains, as well as in circuits dynamics that has recently been realized in Rydberg atom arrays, where we provide concrete benchmarks for future experiments on logical qubits up to twice the sizes already realized.
Stabilizer Rényi entropy has emerged as a probe of nonstabilizerness in quantum many-body systems, but its scaling structure beyond critical points remains poorly understood compared with entanglement entropy. Recent field-theory approaches indicate that stabilizer entropy contains universal critical data and boundary-sensitive terms, raising the question of how these structures extend into massive and crossover regimes. We address this problem for a broad class of finite-range spin chains at Rényi index one-half. We derive exact finite-size formulas for both full periodic chains and finite intervals of the infinite chain, making the universal crossover from critical to noncritical behavior analytically accessible. In periodic geometry, the entropy obeys a volume law away from criticality and exhibits a universal finite-size crossover controlled by the competition between system size and correlation length. We also show that the large-scale SRE density develops a cusp across the field-tuned critical line, while the XX endpoint is governed by a distinct scaling regime associated with the saturation point. In the subsystem geometry, the interval entropy separates bulk critical behavior from boundary contributions generated by the way the finite region cuts the infinite chain. The crossover from critical to massive behavior is then encoded in boundary constants and universal functions controlled by the correlation length. Through exact stabilizer-entropy correspondences, the scaling theory extends to internal XY reductions, Finite-range spin chains, and Cluster--Ising representatives. Our results provide an exact lattice benchmark for the emerging QFT description of stabilizer entropy beyond isolated conformal points.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.
We investigate nonstabilizerness, also known as ``magic,'' to understand criticality and exceptional points in non-Hermitian quantum many-body systems. Our focus is on parity-time (PT\mathcal{PT}PT)-symmetric spin chains, specifically the non-Hermitian transverse-field Ising and XX models. We calculate stabilizer Rényi entropies in their ground states using non-Hermitian matrix product state methods. Our findings show that magic exhibits unique and model-specific signs of phase transitions. In the Ising chain, it peaks near the regular Hermitian-like critical line but disappears across exceptional points. In contrast, in the XX chain, it reaches its maximum at the exceptional line where PT\mathcal{PT}PT symmetry is broken. Finite-size scaling reveals that these effects become more pronounced with larger systems, highlighting nonstabilizerness as a sensitive marker for both quantum criticality and non-Hermitian spectral degeneracies. We also investigate magic in momentum space for the XX model analytically and find that it reaches a minimum around exceptional points. Our results indicate that magic takes extremal values at the exceptional points and serves as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter.
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters K=1/3K=1/3K=1/3 and K=3K=3K=3. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.
While nonstabilizerness (''magic'') is a key resource for universal quantum computation, its behavior in many-body quantum systems, especially near criticality, remains poorly understood. We develop a spectral transfer-matrix framework for the stabilizer Rényi entropy (SRE) in infinite matrix product states, showing that its spectrum contains universal subleading information. In particular, we identify an SRE correlation length -- distinct from the standard correlation length -- which diverges at continuous phase transitions and governs the spatial response of the SRE to local perturbations. We derive exact SRE expressions for the bond dimension χ=2χ=2χ=2 MPS ''skeleton'' of the cluster-Ising model, and we numerically probe its universal scaling along the Z2\mathbb{Z}_2Z2​ critical lines in the phase diagram. These results demonstrate that nonstabilizerness captures signatures of criticality and local perturbations, providing a new lens on the interplay between computational resources and emergent phenomena in quantum many-body systems.