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We investigate the action of local unitary operations on multimode (pure ormixed) Gaussian states and single out the minimal number of locally invariantparametres which completely characterise the covariance matrix of such states.For pure Gaussian states, central resources for continuous-variable quantuminformation, we investigate separately the parametre reduction due to theadditional constraint of global purity, and the one following by thelocal-unitary freedom. Counting arguments and insights from the phase-spaceSchmidt decomposition and in general from the framework of symplectic analysis,accompany our description of the standard form of pure n-mode Gaussian states.In particular we clarify why only in pure states with n<=3 modes all the directcorrelations between position and momentum operators can be set to zero bylocal unitary operations. For any n, the emerging minimal set of parametrescontains complete information about all forms of entanglement in thecorresponding states. An efficient state engineering scheme (able to encodedirect correlations between position and momentum operators as well) isproposed to produce entangled multimode Gaussian resources, its number ofoptical elements matching the minimal number of locally invariant degrees offreedom of general pure n-mode Gaussian states. We demonstrate that so-called"block-diagonal" Gaussian states, without direct correlations between positionand momentum, are systematically less entangled, on average, than arbitrarypure Gaussian states.

Standard forms and entanglement engineering of multimode Gaussian states under local operations