  We present strategies how to reconstruct (estimate) properties of a quantum
channel described by the map E based on incomplete measurements. In a
particular case of a qubit channel a complete reconstruction of the map E can
be performed via complete tomography of four output states E[rho_j ] that
originate from a set of four linearly independent test states j (j = 1, 2, 3,
4) at the input of the channel. We study the situation when less than four
linearly independent states are transmitted via the channel and measured at the
output. We present strategies how to reconstruct the channel when just one, two
or three states are transmitted via the channel. In particular, we show that if
just one state is transmitted via the channel then the best reconstruction can
be achieved when this state is a total mixture described by the density
operator rho = I/2. To improve the reconstruction procedure one has to send via
the channel more states. The best strategy is to complement the total mixture
with pure states that are mutually orthogonal in the sense of the Bloch-sphere
representation. We show that unitary transformations (channels) can be uniquely
reconstructed (determined) based on the information of how three properly
chosen input states are transformed under the action of the channel.
