Late-time ensembles of quantum states in quantum chaotic systems
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 although 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 at the level of finite statistical moments. Importantly, our results apply to initial states easy to prepare in ongoing experiments -- specifically, product states -- that lie in the middle of the spectrum of quantum chaotic systems. We show that these states typically exhibit not only 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 -- at the level of finite statistical moments that can tell that the 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 states have smaller variance of the symmetry operator than Haar-random states and evolve into non-universal 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. Our results reveal that an extremely high level of quantum state randomness can still be achieved even when dynamics is constrained by symmetries.