Phase-Space Inequalities Beyond Negativities

We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.

Figure 1

The Wigner and $Q$ function of a squeezed state are shown (top) together with the left-hand side of Eq. (6) (bottom). Negativities in the latter certify the nonclassicality of the state.

Figure 2

The phase-space relation in Eq. (6) (solid line) and the Wigner function (dashed line) of a lossy photon state are plotted at the origin of phase space ($\alpha =0$) as functions of the loss parameter $q$. Negative values indicate nonclassicality.

Figure 3

Nonclassicality certification of the lossy SPATS in terms of losses ($1-\epsilon$) and thermal photon number ($\overline{n}$). Nonclassicality is detected via negativities of the Mandel $Q_{\mathrm{M}}$ parameter for $\overline{n}<0.707$ (dotted line) and via negative Wigner function for $\epsilon >0.5$ (dashed line). Above the solid curve, inequality Eq. (6) is violated.

Figure 4

Schematics of the correlation measurement technique.

Figure 5

Left-hand side of inequality (5) ($s=-3$ and $k=1/2$) for a cat state with coherent amplitude $\omega =0.7$ sampled from the correlation measurement depicted in Fig. 4.