Nonclassicality witnesses

Sufficient single-mode witnesses include a Fano factor $\mathrm{Var}n/⟨n⟩<1$, an optical quadrature variance below $1/4$ for $[X,P]=i/2$, or photon antibunching with $g^{(2)}(0)<1$. Classical mixtures obey $\mathrm{Var}n\ge ⟨n⟩$, and $g^{(2)}(0)=\mathbb{E}(|\alpha {|}^4)/\mathbb{E}(|\alpha {|}^2{)}^2\ge 1$ by nonnegative variance. The tests are sufficient rather than necessary and exclude the vacuum case when dividing by zero mean occupation.