Dependence of a quantum mechanical system on its own initial state and the initial state of the environment it interacts with

We present a unifying framework to the understanding of when and how quantum mechanical systems become independent of their initial conditions and adapt macroscopic properties (like temperature) of the environment.By viewing this problem from an quantum information theory perspective, we are able to simplify it in a very natural and easy way. We first show that for any interaction between the system and the environment, and almost all initial states of the system, the question of how long the system retains memory of its initial conditions 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 $t$. We discuss applications of our entropic criterion to thermalization times in systems with an effective light-cone and to quantum memories suffering depolarizing noise. We make a similar statement for almost all initial states of the environment, and finally provide a sufficient condition for which a system never thermalizes, but remains close to its initial state for all times.