By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function, i.e., tan α → tanh α, sin α → sinh α, we find the quantum mechanical fractional squeezing transformation (FrST) which satisfies additivity. By virtue of the integration technique within the ordered product of operators (IWOP) we derive the unitary operator responsible for the FrST, which is composite and is made of e iπa † a/2 and exp iα 2 a 2 + a †2 . The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches.

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A new complex function space whose basis is the single-variable Hermite polynomial H2jξ*+τξ2τ is constructed, which is related to both entangled state representation and spin coherent state in Schwinger bosonic realization. New binomial theorem involving two-variable Hermite polynomial is derived, which helps to constitute the new complex function space. We also present a new integration transformation of the basis H2jξ*+τξ2τ with its reciprocal transformation which is useful to deriving some operator identities.

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We discuss entanglement for a tripartite system by setting up a new state vector representation |p, χ 1 , χ 2 in three-mode Fock space. The Schmidt decomposition of |p, χ 1 , χ 2 is presented and its application in teleporting a bipartite entangled state or a two-mode squeezed state to a pair of receivers is analyzed.