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We introduce a complete description of a multi-mode bosonic quantum state inthe 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 afunction of its covariance matrix and displacement vector, and also that of ageneral continuous-variable cluster state in terms of the modal squeezing andgraph topology of the cluster. This formalism lets us characterize the nonGaussian state left over when one measures a subset of modes of a Gaussianstate using photon number resolving detection, the fidelity of the obtainednon-Gaussian state with any target state, and the associated heraldingprobability, all analytically. We show that this probability can be expressedas a Hafnian, re-interpreting the output state of a circuit claimed todemonstrate quantum supremacy termed Gaussian boson sampling. As anexample-application of our formalism, we propose a method to prepare a two-modecoherent-cat-basis Bell state with fidelity close to unity and successprobability that is fundamentally higher than that of a well-known scheme thatsplits an approximate single-mode cat state---obtained by photon numbersubtraction on a squeezed vacuum mode---on a balanced beam splitter. Thisformalism could enable exploration of efficient generation of cat-basisentangled states, which are known to be useful for quantum error correctionagainst photon loss.

Efficient representation of Gaussian states for multi-mode non-Gaussian quantum state engineering via subtraction of arbitrary number of photons