Abstract
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 and are spatially correlated over a distance . 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.
21 More- Received 1 September 2016
DOI:https://doi.org/10.1103/PhysRevX.7.031016
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum mechanics involves distinctive nonlocal correlations known as “entanglement,” which imply that the state of a system cannot be reduced to separate states of its constituent parts. Quantitative measures of entanglement have recently illuminated many areas of theoretical physics, ranging from the classification of phases of matter to the question of how our classical world emerges from the microscopic quantum-mechanical “soup.” But despite remarkable theoretical, computational, and experimental progress in studying entanglement, a theory explaining entanglement generation in generic many-body systems has been lacking. Here, we propose that general results for entanglement growth can be obtained by studying certain “minimal models” composed of quantum circuits made of random gates.
We present a detailed picture of entanglement growth in these minimal models, whose dynamics are random both temporally and spatially. We show that our findings can be mapped to paradigmatic problems in classical statistical mechanics, including the growth of a classical surface and an elastic energy minimization problem for a membrane. We then extract “universal” lessons that apply to generic many-body systems. These lessons include leading-order scaling forms for entanglement growth, information about the relation of the “speed” of thermalization to other characteristic speeds, and easily visualizable heuristics for entanglement growth.
We expect that our findings will pave the way for studies focused on entanglement in higher dimensions.