Non-classical Effects of the Photon-Added Hadamard Transformed Vacuum State

Theoretical analysis are given for the non-classical effects of the photon-added Hadamard transformed vacuum state (PAHTVS) generated by repeatedly acting photon addition operation on a vacuum state which passing through a Hadamard gate. It is shown that the normalization constant of the PAHTVS is a Legendre polynomial. Furthermore, we study the analytical expressions of several quasi-probability distributions for the PAHTVS. We discuss the negative values of the Wigner function for the PAHTVS, which implies the non-classical properties of the PAHTVS.