measurements. It was found that the optimal point occurs when the measurement is a heterodyne measurement. The two discord values $\delta^{G (0)}({\rm A}|{\rm B})$ and $\bar \delta^{\rm G}({\rm A}|{\rm B})$ are then substituted into Eq.  to obtain the consumed Gaussian discord.

Due to the optimality of the Gaussian discord of state $\rho^{(0)}$, and the fact that Gaussian discord is an upper bound for the discord for state $\bar\rho$, the consumed Gaussian discord is a lower bound of the consumed discord, i.e., $\delta^{\rm G}_{\rm con}({\rm A}|{\rm B})\le\delta_{\rm con}({\rm A}|{\rm B})$. A plot of the $\delta^{\rm G}_{\rm con}({\rm A}|{\rm B})$ compared to the information differences is shown in Fig. \[fig\_plot02\].

As discussed in Sec. \[sec\_results\_analytic\], since a heterodyne measurement on mode B optimizes $\delta^{(0)}({\rm A}|{\rm B})$, and numerical results show that this is the case for $\bar \delta({\rm A}|{\rm B})$ and $\chi_c$, from Theorem \[theorem1\], $\delta^{(0)}({\rm A}|{\rm B})=\chi_q-\chi_c$. Numerical calculation of $\delta^{(0)}({\rm A}|{\rm B})$ and $\chi_q-\chi_c$ agree within the precision of the calculation, further verifying the theorem.

From Fig. \[fig\_plot02\], we see that the difference in Holevo information between quantum illumination ($\chi_q-\chi_c$) and single-mode illumination ($\chi_q-\chi_s$) is $1.3\%$ for $\bar n=0.5$ and $0.005 \%$ for $\bar n=0.01$ when $\epsilon=0.3$. The percentage difference 