  We introduce a complete description of a multi-mode bosonic quantum state in
the coherent-state basis (which in this work is denoted as "$K$" function ),
which---up to a phase---is the square root of the well-known Husimi "$Q$"
representation. We express the $K$ function of any $N$-mode Gaussian state as a
function of its covariance matrix and displacement vector, and also that of a
general continuous-variable cluster state in terms of the modal squeezing and
graph topology of the cluster. This formalism lets us characterize the non
Gaussian state left over when one measures a subset of modes of a Gaussian
state using photon number resolving detection, the fidelity of the obtained
non-Gaussian state with any target state, and the associated heralding
probability, all analytically. We show that this probability can be expressed
as a Hafnian, re-interpreting the output state of a circuit claimed to
demonstrate quantum supremacy termed Gaussian boson sampling. As an
example-application of our formalism, we propose a method to prepare a two-mode
coherent-cat-basis Bell state with fidelity close to unity and success
probability that is fundamentally higher than that of a well-known scheme that
splits an approximate single-mode cat state---obtained by photon number
subtraction on a squeezed vacuum mode---on a balanced beam splitter. This
formalism could enable exploration of efficient generation of cat-basis
entangled states, which are known to be useful for quantum error correction
against photon loss.
