Gaussian states are the backbone of quantum information protocols with continuous-variable systems whose power relies fundamentally on the entanglement between the different modes. In the case of global pure states, knowledge of the reduced states in a given bipartition of a multipartite quantum system bears information on the entanglement in such bipartition. For Gaussian states, the reduced states are also Gaussian, so their determination requires essentially the experimental determination of their covariance matrix. Here we develop strategies to determine the covariance matrix of an arbitrary
n-mode bosonic Gaussian state through measurement of the total phase acquired when appropriate metaplectic evolutions, associated with quadratic Hamiltonians, are applied. Simply one-mode metaplectic evolutions, such rotations, squeezing, and shear transformations, in addition to a single two-mode rotation, allows us to determine all the covariance matrix elements of an
n-mode bosonic system. All the single-mode metaplectic evolutions are applied conditionally to a state in which an ancilla qubit is entangled with the
n-mode system. The ancillary system provides, after measurement, the value of the total phase of each evolution. The proposed method is experimentally suited to implement in the most currently used continuous-variable systems.
We consider two classes of non-Gaussian entangled states generated from the product of number states by the action of the beam splitter or the two-mode squeezer. It is shown that, for many of these states, the covariance matrix is compatible with the covariance matrix of separable Gaussian states and their separability cannot be verified by the measurements of the first and second moments of canonical variables. We identify a couple of continuous-variable entanglement criteria with higher-order moments to verify these non-Gaussian entanglements.