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Quantum Entanglement Growth Under Random Unitary Dynamics
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 and
are spatially correlated over a distance . 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
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