Of Darwin, Markov, and the complexity of quantum dynamics: Quantum Darwinism

State of the art and motivations In quantum mechanics, the Schrödinger equation regulates the evolution of the state vector of a given system. As we all know, this gives rise to a unitary evolution. However, an often unspoken assumption behind the validity of this approach to the dynamics of a quantum system is its emphclosed nature: there is no other system interacting with the one we are interested in. However, this is often too strong an assumption to be realistic. Even the best possible quantum experiment cannot guarantee the absolute isolation of the system to study from the surrounding world. Therefore, it is normally the case that open-system dynamics should be considered, where the effects of the interactions with the outside environment are taken in consideration, in a way or another. A full characterisation of open-system dynamics is a very challenging problem is modern quantum mechanics, in particular in those cases where the environment that interacts with the quantum system of our interest is, per se, a dynamical object that is able to undergo its own evolution and, possibly, retain memory of the effects that its coupling to our quantum system has. This case is often referred to as non-Markovian open system dynamics and is, at the sam