Quantum entanglement growth under random unitary dynamics

Adam Nahum, Jonathan Ruhman, Sagar Vijay, Jeongwan Haah

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium 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 ðtimeÞ1=3 and are spatially correlated over a distance ∝ ðtimeÞ2=3. We derive KPZ universal behavior 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 coarse grained 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.

Original languageEnglish
Article number031016
JournalPhysical Review X
Issue number3
StatePublished - 24 Jul 2017
Externally publishedYes

Bibliographical note

Funding Information:
We thank M. Kardar for useful discussions. A. N. and J. R. acknowledge the support of fellowships from the Gordon and Betty Moore Foundation under the EPiQS initiative (Grant No. GBMF4303). A. N. also acknowledges support from EPSRC Grant No. EP/N028678/1. S. V. is supported partly by the KITP Graduate Fellows Program and the U.S. DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-SC0010526. J. H. is supported by the Pappalardo Fellowship in Physics at MIT.


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