Landauer’s Principle in Repeated Interaction Systems

Abstract

We study Landauer's Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system S in contact with a structured environment E made of a chain of independent quantum probes; S interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer's lower bound, which relates the energy variation of the environment E to a decrease of entropy of the system S during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment E displaying small variations of order T −1 between the successive probes encountered by S, after n ≃ T interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of S in this regime, in order to tackle the adiabatic limit of Landauer's bound. We find that saturation of Landauer's bound is related to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer's bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.

Landauer’s Principle for Trajectories of Repeated Interaction Systems

“…We will often omit the parentheses and composition symbols. For more details about the dynamics of RIS processes in various regimes, see [BJM14, HJPR17]. We now turn to energetic and entropic considerations on RIS, at the root of Landauer's principle.…”

“…In [HJPR17], the present authors analyzed the Landauer principle and its saturation in the framework of an adiabatic limit of RIS that we briefly recall here. We introduce the adiabatic parameter T ∈ N and consider a repeated interaction process with T probes, such that the parameters governing the kth probe and its interaction with S, namely (h E k , β k , v k ), are chosen by sampling sufficiently smooth functions as described by the following assumption.…”

“…In [HJPR17], the present authors used a suitable adiabatic theorem to characterize the large T behaviour of the total entropy production term (7), which monitors the saturation of the Landauer bound in the adiabatic limit (note that the terms in the sum (7) are T -dependent through ADRIS). Briefly, under suitable assumptions, convergence of σ tot T is characterized by the fact that the term X(s) defined in (18) below vanishes identically.…”

“…The behaviour of this averaged quantity was investigated in [HJPR17]. For a RIS satisfying the assumptions ADRIS and Prim, the condition lim sup…”

“…, T . This is the setup that was studied in [HJPR17]; there we showed that the total entropy production lim T →∞ T k=1 σ k was finite only under the condition X(s) = 0 for all s ∈ [0, 1], where X(s) is a quantity depending on the probe parameters at time s ∈ [0, 1] which we discuss below, see (18). The proof of this result relied mostly on a new discrete, non-unitary adiabatic theorem that allowed us to control a product of T slowly varying completely positive, trace-preserving (CPTP) maps that represent the reduced dynamics acting on S.…”