Conditional quantum-state transformation at a beam splitter

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 in a closed form by 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 which can be recovered by simple specification of the non-unitary transformation operator. As an application, we consider the generation of Schrodinger-cat-like states. An extension to mixed states and imperfect detection is outlined.