Reexamination Of Bloch-Messiah Reduction

The Bloch-Messiah (BM) reduction allows the decomposition of an arbitrarily complicated Gaussian unitary into a very simple scheme in which linear optical components are separated from nonlinear ones. It is based on the combined application of the eigendecomposition (EID) and singular value decomposition (SVD) of the two matrices associated to the Bogoliubov transformation of the given Gaussian unitary, with the constraint that the unitary matrices appearing in the SVDs must verify a stringent condition, called in the paper "rotation condition". Thus, for a correct evaluation of the BM reduction, one has to choose, among the possible SVD pairs, the pair that meets this condition. The paper develops an approach to this problem, which is based on the introduction of a balancing matrix in the SVDs to fulfill the rotation condition. The theory is illustrated with a few application examples.