ABSTRACT
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. The testing and exploitation of quantum non-locality (stronger than classical correlations that imply the presence of quantum entanglement) with Gaussian states does instead require a departure from Gaussian measurements. Certain non-Gaussian measurements are customarily implemented with current technology, and it is hence still relevant to study in detail the creation of such a resource for Gaussian states. This chapter introduces Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. The treatment highlights the privileged role played by quadratic canonical i.e'., symplectic' transformations in this context, and hinges on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. It also discusses another equivalent parametrisation of a generic Gaussian state, deriving a more direct rendition of the spectrum of a Gaussian state in terms of observable parameters.
