Universal Spreading of Nonstabilizerness and Quantum Transport

TL;DR

This work demonstrates that in $ ext{U}(1)$-symmetric many-body dynamics, the growth of quantum resources—coherence (PE) and nonstabilizerness (SRE)—exhibits universal power-law behavior governed by the underlying transport regime. Using the XXZ spin chain, the authors connect transport exponents (ballistic, diffusive, KPZ-like superdiffusive) to resource growth with $\mathcal{S}_k, \mathcal{M}_k \propto t^{1/z}$, and they establish analytical bounds linking PE and SRE under symmetry constraints. Numerically, they develop collision MPS and Pauli-MPS methods to efficiently compute these quantities in large systems, corroborating the universal scaling across clean and disordered dynamics. The results highlight a deep, operational link between hydrodynamic transport and quantum complexity, suggesting that quantum resources can serve as robust diagnostics beyond entanglement for symmetry-constrained many-body systems.

Abstract

We investigate how transport properties of $U(1)$-conserving dynamics impact the growth of quantum resources characterizing the complexity of many-body states. We quantify wave-function delocalization using participation entropy (PE), a measure rooted in the coherence theory of pure states, and assess nonstabilizerness through stabilizer Rényi entropy (SRE). Focusing on the XXZ spin chain initialized in domain-wall state, we demonstrate universal power-law growth of both PE and SRE, with scaling exponents explicitly reflecting the underlying transport regimes, ballistic, diffusive, or KPZ-type superdiffusive. Our results establish a solid connection between quantum resources and transport, providing insights into the dynamics of complexity within symmetry-constrained quantum systems.

Figures (5)

• Figure 1: Sketch of the investigated setup: An $N$-qubit system is prepared in a product state $\ket{\psi_0}$ in a domain-wall configuration. Coherent evolution under a $U(1)$-symmetric Hamiltonian $H$ leads to the melting of the domain wall and the restoration of a uniform magnetization profile, represented by $\braket{Z_j} = \braket{\psi_t|Z_j|\psi_t}$, where $Z_j$ is proportional to the $z$-component of the spin-1/2 operator. The initial state has vanishing participation entropy, $\mathcal{S}_k(\ket{\psi_0}) = 0$, and stabilizer Rényi entropy, $\mathcal{M}_k(\ket{\psi_0}) = 0$. Time evolution under $H$ increases these resources as $\mathcal{S}_k, \mathcal{M}_k \propto t^{1/z}$, where $z$ is the dynamical exponent characterizing transport in the system: $z = 2$ for diffusion, $z = 3/2$ for superdiffusion, and $z = 1$ for ballistic transport.
• Figure 2: Ballistic and superdiffusive regime: Time evolution of (a) PE and (b) SRE for $J_z=0.25$ and different system sizes $L$. Both PE and SRE exhibit a linear increase in time, consistent with ballistic transport dynamics characterized by a dynamical exponent $z=1$. In (c) and (d), we show PE and SRE for $J_z=1$ and various system sizes $L$. Both measures exhibit power-law growth consistent with superdiffusive spin transport characterized by a KPZ scaling exponent $z \sim 3/2$. The insets in (b,d) shows finite-size scaling analysis of the extracted dynamical exponent $z$, confirming ballistic and robust KPZ universality, while in (a,c), we present the dynamics of the entanglement entropy.
• Figure 3: Diffusive regime in disordered XXZ model ($h=0.2$): Time evolution of (a) PE and (b) SRE for different system sizes $L$. Both complexity measures exhibit clear diffusive scaling, growing as $t^{1/2}$. This diffusive behavior emerges due to the introduction of weak disorder, which breaks integrability and enforces a diffusive transport regime.
• Figure 4: Sketch of Replica TN for PE and SRE: (a) Definitions of tensors used to construct the MPO to calculate the PE. (b) Definitions of tensors used for the construction of Pauli-MPS. In particular, we show how to construct the tensors of the Pauli-MPS and MPO $B$ and $B^{\alpha \alpha^{\prime}}$. (c) The PE and SRE are represented as contractions of a two-dimensional tensor network.
• Figure 5: Convergence of collision MPS for $N=80$: (a)Participation entropy $\mathcal{S}_2$ as a function of the bond dimension cutoff $\chi_C$ for different time $t=1$ and $t=10$ in the balistic regime ($J_z=0.25$), (b) $\mathcal{S}_2$ in the superdiffusive regime ($J_z=1.0$) indicating that convergence is preserved across dynamical evolution. The saturation of $\mathcal{S}_2$ with increasing $\chi_C$ confirms that the iterative compression strategy in the collision MPS method reliably captures the participation entropy even in large-scale, time-evolved states.