A New Theorem on the Nonclassicality of States

A new theorem on the non-classicality depth of states has been proved. We show that if $W_{\hat \rho} (\alpha_m,s_m)=0$ exist for some value of the ordering parameter $s$ at some phase-space point $\alpha_m$, and if $W_{\hat \rho} (\alpha,s_m)$ is an acceptable quasi-classical distribution, the non-classicality of $\hat \rho$ in parallel with Lee's non-classicality depth is then given by $\tau_m=(1 - s_m)/2$. In this way, a general examination of the effects of the single-photon-addition and -subtraction operations has been studied. The theorem, indeed, provides a theoretical background for generating quantum states of arbitrary non-classicality depth.