Theorems on Entanglement Typicality in Non-equilibrium Dynamics

The notion of typicality in statistical mechanics is essential to characterize a macroscopic system. An overwhelming majority of the pure state looks almost identical if we neglect macroscopic non-local correlations, suggesting that thermal equilibrium is the collection of the typical properties. Quantum entanglement, which characterizes a non-local correlation, also has a typical behavior in equilibrium systems. However, it remains elusive whether there is a typical behavior of entanglement in dynamical non-equilibrium systems. To investigate the typicality, 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 criteria for the typicality of the R\'enyi entropies is presented. In addition, it is analytically proven that the second R\'enyi 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 the first proof of the Page curve conjecture 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.