On the Classical and Quantum Aspects of Memory Effects in Open Dynamical Systems

Open systems that are strongly coupled to their environments generally manifest memory effects in their irreversible evolution, such as revivals of distances or divergences, which can be interpreted as backflows of information from the environment. This interpretation is universal for both classical and quantum systems. For instance, we shall study how, by reducing a Markovian quantum evolution to a fixed commutative subalgebra, one generally obtains a non-Markovian classical dynamics experiencing backflow of information. The latter is assisted by coherences built up by the quantum evolution which effectively act as an environment for the classical subalgebra. We shall demonstrate that this effect can be driven by a dissipative dynamics, yet capable of building enough coherence with respect to a suitable basis. Conversely, there exist memory effects with no classical counterpart. This is the case of backflow of information superactivating in bipartite systems whose subsystems do not exhibit revivals when observed individually. Nonetheless, such phenomenon can be assisted by a classical memory, as occurs in a sufficiently correlated Markov spin chain acting as a collisional environment. In this framework, the physical origin of the superactivation effect is investigated through the study of system–environment correlations. Generally, though, it is quite a hard task to infer the mechanisms behind information flows from the reduced dynamics, since the latter only represents the one-time marginal of an underlying multi-time quantum stochastic process. A possible way out of this problem is proposed by investigating system–environment information flows in terms of the Alicki–Lindblad–Fannes dynamical entropy, which extends the classical entropy of Kolmogorov and Sinai to quantum dynamical systems. After introducing the appropriate symbolic construction in presence of an external environment, exact results are provided in the framework of collisional models. The interpretation of the open-system dynamical entropy is then discussed in the state purification scheme known as GNS construction, whereby further interesting connections emerge with the superactivation of memory effects.