Theorems on Entanglement Typicality in Non-equilibrium Dynamics

To investigate the typicality of entanglement in dynamical non-equilibrium systems, we consider a situation where a system in a pure state starts to share entanglement with its environment system due to the interaction between them. Assuming the initial state is randomly chosen from an ensemble of pure states, a criterion for the typicality of the Renyi entropies is presented. In addition, it is proven that the second Renyi entropy has a typical behavior in two cases. The first one is an energy dissipation process in a multiple-qubit system which is initially in a random pure state in an energy shell. Since the typical behavior is qualitatively the same as the prediction of the Page curve conjecture, it gives an example where the Page curve conjecture works in a dynamical process. In the second case, the typicality is proven for any dynamics described by a multiple-product of a single-qudit channel when the system is initially in a pure state randomly chosen from the whole Hilbert space. This result shows that entanglement typicality is not a specific feature of energy dissipating processes.