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## Details

We study the entanglement dynamics of quantum many-body systems at long times.

For upper bounds, we prove the following: (I) For any geometrically local Hamiltonian on a lattice, starting from a random product state the entanglement entropy almost never approaches the Page curve. (II) In a spin-glass model with random all-to-all interactions, starting from any product state the average entanglement entropy does not approach the Page curve. We also extend these results to any unitary evolution with charge conservation and to the Sachdev-Ye-Kitaev model.

For lower bounds, we say that a Hamiltonian is an ``extensive entropy generator'' if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. These results imply ``unbounded growth of entanglement’’ in many-body localized systems.

References: arXiv:2102.07584 & 2104.02053