Nonstabilizerness in open XXZ spin chains: Universal scaling and dynamics

TL;DR

This work investigates the dynamics of nonstabilizerness (magic) in open quantum systems, focusing on the open XXZ spin chain with boundary driving and bulk dephasing, quantified by the stabilizer Rényi entropy $M_2$. It introduces a scalable algorithm that computes $M_2$ in the matrix product state formalism while keeping the bond dimension fixed, enabling large-system simulations. The authors uncover universal dynamical scaling $M_2(t) \sim t^{1/z}$ across transport regimes with dynamical exponent $z$ (ballistic $z=1$, KPZ $z=3/2$ at $\Delta=1$, diffusive $z=2$) and show a mean-field decomposition to separate classical from quantum contributions. Under bulk dephasing, magic can transiently increase before decaying, with two timescales $\tau_d \sim 1/\gamma_z$ and $\tau_c \sim 1/(J|\Delta-1|)$; in certain sectors magic reaches a nonzero steady state, while in zero-magnetization sectors it exhibits power-law decay, demonstrating the intricate role of symmetry and dissipation in open quantum dynamics.

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 novel, highly efficient algorithm for computing $M_2$ within the matrix product states formalism while maintaining constant bond dimension--an advancement over existing methods. For boundary driving, we uncover universal scaling laws, $M_2(t) \sim t^{1/z}$, 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.

Figures (4)

• Figure 1: (a) Illustration of the MPS structure for the vectorized density matrix, with explicit site labels shown. (b) Depiction of the Pauli vector construction and its corresponding MPS structure. Red dots represent the Pauli tensors. (c) After contracting the physical indices, the Pauli vector is expressed as an MPS with the same bond dimension $\chi$ as the original vectorized density matrix.
• Figure 2: (a,b,c) Time evolution of magic for different anisotropy parameters $\Delta$ and for various system sizes. The insets in each panel present the light-cone formation in magnetization for the respective $\Delta$ as a function of position $x=2j/L-1$, with $j$ numbering the sites. The dashed lines in the insets mark the boundaries of the emerging light cones, defined as the points where the magnetization reaches $1/4$ of its maximum value. The initial density matrix corresponds to the infinite-temperature state, $|\rho_0\rangle\!\rangle = |\mathbb{1}_\infty\rangle\!\rangle$. For $\Delta=0$, the magic and magnetization light cone exhibit a linear growth, characteristic for the ballistic transport regime. At $\Delta=1$, the growth follows a $t^{2/3}$ scaling, indicative of the KPZ universality class, while for $\Delta=2$, it becomes diffusive with a dynamical exponent $z=2$. The legend and color bar are shared for all panels.
• Figure 3: (a) Sketch of the boundary driven XXZ spin chain in the infinite temperature limit. (b) The steady-state values of the total magic, $M_2(t_\infty)$, and the mean-field magic, $M_2^{\rm{MF}}(t_\infty)$, as functions of $\Delta$. The hashed region represents the contribution to magic arising from quantum correlations. (c) Steady-state magnetization profile $S_z(t_\infty)$ for three different values of $\Delta$. In both panels, the system size is set to $L=32$.
• Figure 4: (a) Sketch of the bulk dephasing modeled with sites coupled to an external reservoir via the jump operator $S^z$. (b, c) Time evolution under dephasing affecting all qubits at the isotropic point $\Delta=1$. (b) Starting from an initial density matrix with zero magic, $\rho(0) = (|+\rangle\langle +|)^{\otimes L}$, the magic initially increases before undergoing an exponential decay to zero at late times. The inset displays the magic density in the NESS state $M_2(t_\infty, \theta)$ as a function of the initial polarization angle $\theta$. (c) When the initial density matrix is $\rho(0) = (|T\rangle\langle T|)^{\otimes L}$, the system possesses a high degree of magic. Due to conservation of the total $S_z$ spin component there is a nonzero magic in the NESS $M_2(t_\infty)=\log_2(6/5)$, subtracted in (c) from $M_2(t)$. The dashed line denotes the analytical result from Eq. \ref{['eq:M2_1q']}. (d, e) Power-law decay of magic $M_2$ for various initial states within the $\langle S_z \rangle = 0$ sector: (d) a product state $\rho(0) = (|+\rangle\langle +|)^{\otimes L}$ and (e) a Néel state, both evaluated for $\gamma_z = 1$ and $L = 96$. At the isotropic point $\Delta = 1$, the data in (d) recovers the exponential decay already presented in panel (b).