Abstract
General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state ρ12 described by a 4 × 4 covariance matrix V, the Schur complement of a local covariance submatrix V1 of it can be interpreted as a new covariance matrix representing a Gaussian operator of party 1 conditioned to local parity measurements on party 2. The connection with a partial parity measurement over a bipartite quantum state and the determination of the reduced Wigner function is given and an operational process of parity measurement is developed. Generalization of this procedure to an n-partite Gaussian state is given, and it is demonstrated that the n − 1 system state conditioned to a partial parity projection is given by a covariance matrix such that its 2 × 2 block elements are Schur complements of special local matrices.
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