Superpositions of Quantum Gaussian Processes

We generalise the Gaussian formalism of Continuous Variable (CV) systems to describe their interactions with qubits/qudits that result in quantum superpositions of Gaussian processes. To this end, we derive a new set of equations in closed form, which allows us to treat hybrid systems' unitary and open dynamics exactly (without truncation), as well as measurements (ideal and noisy). The $N$-qubits $n$-modes entangled states arising during such processes are named Gaussian-Branched Cat States (GCSs). They are fully characterised by their superposed phase-space quantities: sets of generalised complex first moments and covariance matrices, along with the qubit reduced density matrix (QRDM). We showcase our general formalism with two paradigmatic examples: i) measurement-based entanglement of two qubits via a squeezed, leaking, and measured resonator; ii) the generation of the Wigner negativity of a levitated nanoparticle undergoing Stern-Gerlach interferometry in a diffusive environment.