Nonclassicality of Quantum States: A Hierarchy of Observable Conditions

A necessary and sufficient hierarchy of conditions is derived that is completely equivalent to the failure of the Glauber-Sudarshan $P$ function to be a probability density. The conditions are formulated in terms of experimentally accessible characteristic functions of quadratures.

Figure 1

First- and second-order nonclassicality conditions for the state (19). The dotted line shows ${\tilde{N}}_1\left(k\right)$, which never exceeds the classical limit of 1 (dashed line). The solid line displays ${\tilde{N}}_2\left(k\right)$, which exceeds the classical boundary of 1 over two semi-infinite intervals.

Figure 2

Observable conditions for first- and second-order nonclassicality for the state (19). The dotted line represents $N_1\left(k\right)$, which never exceeds the classical limit $G_{\mathrm{g}\mathrm{r}}\left(k\right)$ (dashed line). The solid line displays $N_2\left(k\right)$, which exceeds $G_{\mathrm{g}\mathrm{r}}\left(k\right)$ within two semi-infinite $k$ intervals.