  Quantum coherence is an essential feature of quantum mechanics which is
responsible for the departure between classical and quantum world. The recently
established resource theory of quantum coherence studies possible quantum
technological applications of quantum coherence, and limitations which arise if
one is lacking the ability to establish superpositions. An important open
problem in this context is a simple characterization for incoherent operations,
constituted by all possible transformations allowed within the resource theory
of coherence. In this work, we contribute to such a characterization by proving
several upper bounds on the maximum number of incoherent Kraus operators in a
general incoherent operation. For a single qubit, we show that the number of
incoherent Kraus operators is not more than 5, and it remains an open question
if this number can be reduced to 4. The presented results are also relevant for
quantum thermodynamics, as we demonstrate by introducing the class of
Gibbs-preserving strictly incoherent operations, and solving the corresponding
mixed-state conversion problem for a single qubit.
