  Gaussian quantum states of bosonic systems are an important class of states.
In particular, they play a key role in quantum optics as all processes
generated by Hamiltonians up to second order in the field operators (i.e.
linear optics and quadrature squeezing) preserve Gaussianity. A powerful
approach to calculations and analysis of Gaussian states is using phase-space
variables and symplectic transformations. The purpose of this note is to serve
as a concise reference for performing phase-space calculations on Gaussian
states. In particular, we list symplectic transformations for commonly used
optical operations (displacements, beam splitters, squeezing), and formulae for
tracing out modes, treating homodyne measurements, and computing fidelities.
