**Multi-partite squeezed states and SU(1,1) symmetry**

One goal of quantum information science is quantum information processing using complex quantum optical networks comprising passive and active linear optical elements, such as beam splitters and squeezers. Such networks can be described mathematically as Sp(2n, R) transformations on n modes, which correspond to mappings that preserve Gaussian states.
Recently, tripartite squeezed states have been produced experimentally and are quite useful for quantum information tasks such as quantum state sharing and quantum teleportation. Theoretically, such states have been characterized based on the type of input states, but we have developed a simple and elegant mathematical framework, which is three-boson realization of SU(1, 1), and characterized all squeezed states of this type as SU(1, 1) coherent states. Inspired by the elegance of this theory, we have generalized it to multiboson realization of SU(1, 1) that characterizes any multi-port linear optical system constructed from a two-mode squeezer and several passive optical elements, or by concatenating such multi-port systems to each other. Thus, this theory gives us new insight into the properties of a large class of multipartite squeezed states generated in any complex optical network with concatenated sections each with one two-mode squeezer.