Dynamical purifcation and deep thermalization

Quantum thermalization in a many-body system is defined by the approach of local expectation values towards universal, equlibrium values. Recently, it was demonstrated that universal statistics can emerge not just in expectation values, but also in distributions of states on a subsystem obtained by projectively measuring the complementary subsystem. Specifically, this ensemble of states, known as the projected ensemble, can under certain conditions mimic the behavior of a uniformly random ensemble, a phenomenon dubbed deep thermalization. We investigate the dynamical process underlying this novel emergent universality. Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains deep thermalization. In particular we show that dynamical purification can lead to a separation of timescales between the equilibration of different moments of the projected ensemble: moment k=1 corresponding to regular thermalization, and high moments k»1 corresponding to deep thermalization. Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling.