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Entanglement is usually quantified by von Neumann entropy, but its propertiesare much more complex than what can be expressed with a single number. We showthat the three distinct dynamical phases known as thermalization, Andersonlocalization, and many-body localization are marked by different patterns ofthe spectrum of the reduced density matrix for a state evolved after a quantumquench. While the entanglement spectrum displays Poisson statistics for thecase of Anderson localization, it displays universal Wigner-Dyson statisticsfor both the cases of many-body localization and thermalization, albeit theuniversal distribution is asymptotically reached within very different timescales in these two cases. We further show that the complexity of entanglement,revealed by the possibility of disentangling the state through aMetropolis-like algorithm, is signaled by whether the entanglement spectrumlevel spacing is Poisson or Wigner-Dyson distributed.

Entanglement Complexity in Quantum Many-Body Dynamics, Thermalization and Localization