Bosonic linear unitary Bogoliubov transformation reduction theorem

We show that bosonic linear unitary Bogoliubov transformations may be reduced to a standard form. This leads to a better understanding of the action of general quadratic Hamiltonians. We show that this reduction theorem has a simple physical interpretation in quantum optics. Using this reduction we derive a no-go theorem for superpositions of macroscopically distinct states from single-photon detection. Finally, we construct a few minimal quantum optical circuits