  Using conditional measurement on a beam splitter, we study the transformation
of the quantum state of the signal mode within the concept of two-port
non-unitary transformation. Allowing for arbitrary quantum states of both the
input reference mode and the output reference mode on which the measurement is
performed, we show that the non-unitary transformation operator can be given as
an $s$-ordered operator product, where the value of $s$ is entirely determined
by the absolute value of the beam splitter reflectance (or transmittance). The
formalism generalizes previously obtained results that can be recovered by
simple specification of the non-unitary transformation operator. As an
application, we consider the generation of Schr\"odinger-cat-like states. An
extension to mixed states and imperfect detection is outlined.
