  We investigate the single qubit transformations under several typical
coherence-free operations, such as, incoherent operation (IO), strictly
incoherent operation (SIO), physically incoherent operation (PIO), and
coherence preserving operation (CPO). Quantitative connection has been built
between IO and SIO in single qubit systems. Moreover, these coherence-free
operations have a clear hierarchical relationship in single qubit systems: CPO
$\subset$ PIO $\subset$ SIO=IO. A new and explicit proof for the necessary and
sufficient condition of single qubit transformation via IO or SIO has been
provided, which indicates that SIO with only two Kraus operators are enough to
realize this transformation. The transformation regions of single qubits via
CPO and PIO are also given. Our method provides a geometric illustration to
analyze single qubit coherence transformations by introducing the Bloch sphere
depiction of the transformation regions, and tells us how to construct the
corresponding coherence-free operations.
