Adaption of fractional squeezing transformation to optical Wigner transformation

We show that our recently introduced fractional squeezing transformation performed on a function can naturally lead to a rotation with an angle of the associated Wigner distribution function, i.e., there exists algorithmic isomorphism between symplectic transformation of the Wigner distribution function and the fractional squeezing transformation. This adaption for both one-mode and two-mode cases are demonstrated in our work, while the complex fractional squeezing transformation based on the entangled state representation is introduced for fitting the discussion of the entangled one.