Quantum Entanglement Growth Under Random Unitary Dynamics

TL;DR

This work investigates how entanglement entropy grows in many-body quantum systems subject to spatially local, unitary, random dynamics. By mapping entanglement growth to Kardar–Parisi–Zhang (KPZ) universality, the authors develop three complementary pictures—the stochastic surface growth, the minimal-cut directed polymer, and the hydrodynamics of operator spreading—to derive universal 1D scaling exponents and to propose deterministic pictures for higher dimensions. They validate KPZ behavior in 1D through solvable large-q models and extensive Clifford-circuit numerics, showing linear mean growth with t and fluctuations ∼ t^{1/3}, while the entanglement velocity v_E acts as a true growth speed distinct from operator-spreading velocity. In higher dimensions, a minimal membrane picture generalizes the coarse-grained entanglement dynamics, predicting universal fluctuations and a geometry-dependent growth that persists without noise. The findings offer a unifying framework for entanglement dynamics in generic noisy quantum quenches and suggest heuristic extrapolations to Hamiltonian dynamics without noise, with potential implications for thermalization and quantum information scrambling.

Abstract

Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the `entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar--Parisi--Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) a `minimal cut' picture, reminiscent of the Ryu--Takayanagi formula in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple `minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the `velocity' of entanglement growth in the 1D `entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.

Figures (26)

• Figure 1: The KPZ 'triumvirate' is made up of three very different problems in classical statistical mechanics which all map to the KPZ universality class. As we will discuss, each of them can be usefully related to entanglement in 1+1D.
• Figure 2: Spin chain with open boundary conditions. $S(x)$ denotes the entanglement entropy (von Neumann or Renyi depending on context) between the part of the chain to the left of bond $x$, indicated by the box, and the part to the right.
• Figure 3: Dynamical update in the solvable model: application of a random unitary $U$ to a randomly chosen pair of adjacent spins.
• Figure 4: Surface growth model for entanglement $S(x,t)$ across a cut at $x$, in the large $q$ limit. Applying a unitary to bond $x$ can increase the height of the surface locally (Eq. \ref{['large q dynamics']}), corresponding to dropping a 'block' of height $\Delta S = 1$ or $\Delta S = 2$.
• Figure 5: Entanglement growth in the large $q$ model: Effect of applying a random unitary to the central bond, for four choices of the initial local entropy configuration of three adjacent bonds.