Boson permutation and parity operators: Lie algebra and applications

We show that dichotomic permutation and parity operators for a two-dimensional boson system form an su(2)su(2) algebra with a unitary operator that relates, in quantum optics, to a balanced beamsplitter. The algebra greatly simplifies the input–output transformations of states through quantum nonlinear systems such as the Kerr interferometer or the kicked top.