  We propose and investigate bounds on quantum process fidelity of quantum
filters, i.e. probabilistic quantum operations represented by a single Kraus
operator K. These bounds generalize the Hofmann bounds on quantum process
fidelity of unitary operations [H.F. Hofmann, Phys. Rev. Lett. 94, 160504
(2005)], and are based on probing the quantum filter by pure states forming two
mutually unbiased bases. Determination of these bounds therefore requires much
less measurements than full quantum process tomography. We find that it is
particularly suitable to construct one of the probe basis from the right
eigenstates of K, because in this case the bounds are tight in the sense that
if the actual filter coincides with the ideal one then both the lower and upper
bounds are equal to one. We theoretically investigate application of these
bounds to a two-qubit optical quantum filter formed by interference of two
photons on a partially polarizing beam splitter. For experimentally convenient
choice of factorized input states and measurements we study the tightness of
the bounds. We show that more stringent bounds can be obtained by more
sophisticated processing of the data using convex optimization and we compare
our methods for different choice of the input probe states.
