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Both conservation laws and practical restrictions impose symmetry constraintson the dynamics of open quantum systems. In the case of time-translationsymmetry, which arises naturally in many physically relevant scenarios, thequantum coherence between energy eigenstates becomes a valuable resource forquantum information processing. In this work we identify the minimum amount ofdecoherence compatible with this symmetry for a given population dynamics. Thisyields a generalisation to higher-dimensional systems of the relation $T_2 \leq2 T_1$ for qubit decoherence and relaxation times. It also enables us towitness and assess the role of non-Markovianity as a resource for coherencepreservation and transfer. Moreover, we discuss the relationship betweenergodicity and the ability of Markovian dynamics to indefinitely sustain asuperposition of different energy states. Finally, we establish a formalconnection between the resource-theoretic and the master equation approaches tothermodynamics, with the former being a non-Markovian generalisation of thelatter. Our work thus brings the abstract study of quantum coherence as aresource towards the realm of actual physical applications.

Markovian evolution of quantum coherence under symmetric dynamics