Quantum dynamical entropy and non-Markovianity: A collisional model perspective

Giovanni Nichele, Fabio Benatti

Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.

Göran Lindblad

Arghya Maity, Ahana Ghoshal

The thermodynamic uncertainty relation (TUR) is a fundamental principle in nonequilibrium thermodynamics that relates entropy production to fluctuations in a system, establishing a trade-off between the precision of an observable and the thermodynamic cost. Investigating TUR violations challenges classical thermodynamic limits, offering the potential for improved precision-entropy trade-offs, which is crucial for enhancing performance and optimization in quantum technologies. In this work, we investigate the thermodynamic uncertainty relation within a quantum collisional model, which offers the advantage of discretizing interactions into successive collisions with auxiliaries, allowing for precise tracking of dynamics and the incorporation of memory effects and non-Markovian behavior. We consider three types of dynamics in the collisional model: one is Markovian evolution, achieved by taking the continuous-time limit and imposing the stability condition, while the other two are non-Markovian dynamics—one arising from increasing the collision time between the system and the auxiliaries, and the other from incorporating interactions between the auxiliaries. For the Markovian dynamics, we examine the classical and quantum TUR bounds in the nonequilibrium steady-state regime, and also the finite-time TUR bound. We identify two distinct regimes of classical TUR violation: in some cases, the maximum violation occurs in the steady state, while in others, it is necessarily transient—appearing only at early times and vanishing with further evolution. For the two non-Markovian approaches, we find that both the degree and type of non-Markovianity crucially affect TUR violations. The second approach shows more pronounced violations during transient times, while the first approach has much stronger violations in the steady-state regime for a certain parameter window.

Luigi Accardi, Masanori Ohya, Noboru Watanabe

Robert Alicki, M. Fannes

Zhong‐Xiao Man, Yun‐Jie Xia, Rosario Lo Franco

We study the validity of the Landauer principle in the non-Markovian regime by means of collisional models where the intracollisions inside the reservoir cause memory effects generating system-environment correlations. We adopt the system-environment correlations created during the dynamical process to assess the effect of non-Markovianity on the Landauer principle. Exploiting an exact equality for the entropy change of the system, we find the condition for the violation of the Landauer principle, which occurs when the established system-environment correlations become larger than a threshold. We then generalize the study to the nonequilibrium situation where the system is surrounded by many reservoirs at different temperatures. Our results, verified through collisional models with Heisenberg-type interactions, suggest that the complexity of the environment does not play a significant role in the qualitative mechanisms underlying the violation of the Landauer principle under non-Markovian processes.