Towards fully quantum second laws of thermodynamics: limitations on the evolution of quantum coherences

Piotr Ćwikliński, Michał Studziński, Michał Horodecki, Jonathan Oppenheim

Published 2014-05-20, updated 2015-01-30Version 2

The second law of thermodynamics places a limitation on what states a system can evolve into. For closed systems, it says that a state can be transformed into another state, only if the course grained entropy increases. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Here, the free energy is written in terms of the fine-grained entropy. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture more severe constraints exist. The results are obtained in the paradigm of Thermal Operations, and we introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to Thermal Operations.