Quantum Chernoff bound as a measure of nonclassicality for one-mode Gaussian states

We evaluate a Gaussian distance-type degree of nonclassicality for a single-mode Gaussian state of the quantum radiation field by use of the recently discovered quantum Chernoff bound. The general properties of the quantum Chernoff overlap and its relation to the Uhlmann fidelity are interestingly illustrated by our approach.

Figure 1

(Color online) Displaying the saddle-point evaluations. The function $Q_G(s,r^{\prime })$, Eq. (16), is plotted versus $r^{\prime }$ and $s$ for two nonclassical Gaussian states having the parameters (a) $r=2,\overline{n}=1$, for which we get the saddle-point values ${\tilde{Q}}_G=0.617$, ${\tilde{r}}^{\prime }=1.265$, $\tilde{s}=0.283$; (b) $r=1$, $\overline{n}=0.5$, with the saddle-point results ${\tilde{Q}}_G=0.893$, ${\tilde{r}}^{\prime }=0.653$, $\tilde{s}=0.387$.

Figure 2

(Color online) Chernoff degree of nonclassicality $D_0^{\left(C\right)}$ (full line), Eq. (11), Bures degree $D_0^{\left(B\right)}$ (dashed line), Eq. (18), versus the thermal mean occupancy $\overline{n}$ for nonclassical states with the squeeze factor $r=2$. We have also plotted the corresponding saddle-point values ${\tilde{r}}^{\prime }$ (double-dot dashed line) and $\tilde{s}$ (dot-dashed line).

Figure 3

(Color online) Displaying the inequalities (7) for the saddle-point value ${\tilde{Q}}_G$. Variation of the optimal Chernoff value ${\tilde{Q}}_G$ (full line), the optimal fidelity $\tilde{\mathcal{F}}$ (dotted line), and $\sqrt{\tilde{\mathcal{F}}}$ (dashed line) with the mixedness of the nonclassical Gaussian state ${\rho }_G$. The squeeze parameter is $r=2$.