Late-time ensembles of quantum states in quantum chaotic systems

Initially unentangled states undergoing quantum chaotic dynamics are expected to evolve into featureless states at late times. While this expectation holds true on an averaged, coarse-grained level, it is unclear if this expectation applies to higher statistical moments, as symmetries present in physical systems constrain the exploration of Hilbert space. Here, we study the universal structure of late-time ensembles obtained from unitary dynamics in quantum chaotic systems with symmetries, such as charge or energy conservation. We find that while quantum states do not ergodically explore the entire Hilbert space at late times, the late-time ensemble typically becomes indistinguishable from Haar random states in the thermodynamic limit and at the level of finite statistical moments. Importantly, our results apply to initial states that are easy to prepare in ongoing experiments---specifically, product states---that lie in the middle of the spectrum of quantum chaotic systems. We show that typically these states not only exhibit the same late-time ensemble average as Haar random states, but also the same state-to-state fluctuations and higher statistical moments. In other words, there is no measurement---whether local or nonlocal---that one can perform at the level of finite statistical moments to tell that states are not exploring the entire Hilbert space. Interestingly, within the class of low-entanglement initial states, we also find atypical initial conditions in the middle of the spectrum of Hamiltonians known to be ``maximally chaotic''. Such atypical initial states have smaller variance of the symmetry operator relative to that of Haar random states, and evolve into nonuniversal ensembles that can be distinguished from the Haar ensemble by simple measurements or subsystem properties. In the limiting case of initial states with negligible variance of the symmetry operator (e.g., states with fixed particle number or energy eigenstates), the late-time ensemble has universal behavior captured by constrained random state ensembles. On the one hand, our results reveal that an extremely high level of quantum state randomness can typically be achieved even when dynamics are constrained by symmetries; on the other hand, our results show that one can still achieve strongly nonergodic dynamics at ``infinite temperature'' within quantum chaotic regimes.