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

We consider a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this paper is denoted as the "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, reinterpreting 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.