Dynamical purification and the emergence of quantum state designs from the projected ensemble

Abstract: Quantum thermalization in a many-body system is defined by the approach of local subsystems towards a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values. Recently, it was demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of a local projective measurement of the complementary subsystem. Specifically, this collection of pure quantum states -- called the projected ensemble -- can under certain conditions mimic the behavior of a maximally entropic, uniformly random ensemble, i.e., form a {\it quantum state-design}, representing a ``deeper'' form of quantum thermalization. In this work, 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 the the projected ensemble's approach towards the uniform distribution. We prove that absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state-designs for all moments $k$ at the same time, extending previous rigorous results [Ho and Choi, Phys. Rev. Lett. {\bf 128}, 060601 (2022)]. Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of timescales between the formation of a quantum state-design for moment $k=1$ (regular thermalization) and for high moments $k\gg 1$ (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.