the next sections.

Accessible information and Holevo information calculations {#sec_results_info}
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The accessible information and Holevo information quantities $A_q$, $\chi_q$, $A_c$, $\chi_c$, $A_s$ and $\chi_s$ were calculated numerically for typical settings of quantum illumination. Due to finite computational resources, the states must be approximated to a Hilbert space with finite dimensions. Under this restriction, the highest noise mean photon number that does not result in significant error is $\bar n_{\rm env}=4$. Plots are shown in Fig. \[fig\_plot01\] of the information quantities for noise mean photon number $4$ and probe mean photon number $\bar n=(0.01,0.5)$. We will now review the information quantities for each case listed in Sec. \[sec\_three\_cases\].

*Case 1*. The Holevo information $\chi_q$ and Fuchs upper and lower bounds for the accessible information $A_q$ for quantum illumination with joint measurement are shown in Fig. \[fig\_plot01\]. The difference between the upper and lower bounds of $A_q$ is, at most, 0.7%, implying that the true accessible information is close to the Fuchs bounds. As evident in the plot, there is a substantial difference between the $\chi_q$ and $A_q$.\
*Case 2*. $\chi_c$ and $A_c$: In the previous section, we assume that a heterodyne measurement is the optimal local Gaussian measurement to make 