  Methods to decompose nonlinear optical transformation vary from setting to
setting, leading to apparent differences in the treatments used to model photon
pair sources, compared to those used to model degenerate down-conversion
processes. The Bloch-Messiah reduction of Gaussian processes to single-mode
squeezers and passive (linear) unitaries appears juxtaposed against the
practicalities of the Schmidt-decomposition for photon pair sources into
two-mode squeezers and passive unitaries. Here, we present a general framework
which unifies these forms as well as elucidating more general structure in
multimode Gaussian transformations. The decomposition is achieved by
introducing additional constraints into the Bloch-Messiah reduction used to
diagonalise Gaussian processes, these constraints motivated by physical
constraints following from the inequivalence of different physical degrees of
freedom in a system, ie. the temporal-spectral degrees of freedom vs different
spatial modes in a transformation. The result is the emergence of the two-mode
squeezing picture from the reduction, as well as the potential to generalise
these constraints to accommodate spectral imperfections in a source generating
3-mode continuous variable GHZ-like states. Furthermore, we consider the
practical scenario in which a transformation aims to generate a multiphoton
entangled state, whereby spatial modes provide desirable degrees of freedom,
whilst undesired spectral mode structure contributes noise, and show that this
spectral impurity can be efficiently modeled by finding an optimal low
dimensional bases for its simulation.
