  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 there 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 to determine all the covariance matrix elements of a
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 friendly
to be implemented in the most currently used continuous variable systems.
