  Quantum coherence is a fundamental property that can emerge within any
quantum system. Incoherent operations, defined in terms of the Kraus
decomposition, take an important role in state transformation. The maximum
number of incoherent Kraus operators has been presented in [A. Streltsov, S.
Rana, P. Boes, J. Eisert, Phys. Rev. Lett. 119. 140402 (2017)]. In this work,
we show that the number of incoherent Kraus operators for a single qubit can be
reduced from 5 to 4 by constructing a proper unitary matrix. For qutrit systems
we further obtain 32 incoherent Kraus operators, while the upper bound in the
research of Sterltsov gives 39 Kraus operators. Besides, we reduce the number
of strictly incoherent Kraus operators from more than 15 to 13. And we consider
the state transformation problem for these two types of operations in single
qutrit systems.
