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Quantum thermalization in a many-body system is defined by the approach oflocal subsystems towards a universal form, describable as an ensemble ofquantum states wherein observables acquire thermal expectation values.Recently, it was demonstrated that the distribution of these quantum states canalso exhibit universal statistics, upon associating each state with the outcomeof a local projective measurement of the complementary subsystem. Specifically,this collection of pure quantum states -- called the projected ensemble -- canunder certain conditions mimic the behavior of a maximally entropic, uniformlyrandom ensemble, i.e., form a {\it quantum state-design}, representing a``deeper'' form of quantum thermalization. In this work, we investigate thedynamical process underlying this novel emergent universality. Leveraging aspace-time duality mapping for one-dimensional quantum circuits, we argue thatthe physics of dynamical purification, which arises in the context of monitoredquantum systems, constrains the the projected ensemble's approach towards theuniform distribution. We prove that absence of dynamical purification in thespace-time dual dynamics (a condition realized in dual-unitary quantum circuitswith appropriate initial states and final measurement bases) generically yieldsexact state-designs for all moments $k$ at the same time, extending previousrigorous results [Ho and Choi, Phys. Rev. Lett. {\bf 128}, 060601 (2022)].Conversely, we show that, departing from these conditions, dynamicalpurification can lead to a separation of timescales between the formation of aquantum state-design for moment $k=1$ (regular thermalization) and for highmoments $k\gg 1$ (deep thermalization). Our results suggest that the projectedensemble can probe nuanced features of quantum dynamics inaccessible to regularthermalization, such as quantum information scrambling.

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