Non-Markovianity vs athermality: perturbation-enhanced information backflow

Non-Markovianity and athermality are useful resources in quantum technologies, and it is therefore important to understand the relations between the two, for general quantum dynamics. We propose three measures of non-Markovianity, first within the ambit of thermal operations, and then beyond it, that result from unavoidable perturbations in system's Hamiltonian and that leads to violations of conservation of total energy characterizing any thermal operation. The proposed measures are based respectively on system-environment entanglement, total correlation in the system-environment partition, and on a concept of distance defined on the sets of usual and approximate thermal operations. We investigate the response of non-Markovianity to the athermality-inducing perturbations, using all the three measures. For the entanglement and distance-based measures, we derive upper bounds on the response by a quantity that depends on the perturbative Hamiltonian. For the total correlation-based measure, we are able to compute the exact response. We present examples of qubit-qubit and qubit-qutrit systems for which perturbation leads to enhancement of non-Markovianity, as quantified by the entanglement and total correlation-based measures.