When does a quantum mechanical system depend on the initial conditions of the system or the environment

any interaction between the system and the environment, and almost all initial states of the system, the question of how long such memory lasts can be answered by studying the temporal evolution of just one special initial state. This special state thereby depends only on our knowledge of macroscopic parameters of the system. We provide a simple entropic inequality for this state that can be used to determine whether mosts states of the system have, or have not become independent of their initial conditions after time t. Analyzing the rate of entropy change over time for a particular kind of interaction then allows us to place rigorous bounds on such time scales. We make a similar statement for almost all initial states of the environment, and nally provide a sucient condition for which a system never thermalizes, but remains close to its initial state for all times.