  In this brief note, we compare two frameworks for characterizing possible
operations in quantum thermodynamics. One framework considers Thermal
Operations---unitaries which conserve energy. The other framework considers all
maps which preserve the Gibbs state at a given temperature. Thermal Operations
preserve the Gibbs state; hence a natural question which arises is whether the
two frameworks are equivalent. Classically, this is true---Gibbs-Preserving
Maps are no more powerful than Thermal Operations. Here, we show that this no
longer holds in the quantum regime: a Gibbs-Preserving Map can generate
coherent superpositions of energy levels while Thermal Operations cannot. This
gap has an impact on clarifying a mathematical framework for quantum
thermodynamics.

Quantum Physics