\[fig\_plot01\](a)(i), \[fig\_plot01\](a)(iii), \[fig\_plot01\](b)(i), and \[fig\_plot01\](b)(iii), we see that $\chi_q$ is greater than $\chi_s$, and $A_q$ is greater than $A_s$, showing that quantum illumination with joint measurement does indeed have an advantage over single-mode illumination. In the communication context, Alice can communicate with Bob with a higher bit rate if Bob uses a probe entangled with an idler instead of a coherent state probe.

From Fig. \[fig\_plot01\], we see that the performance of a coherent state probe is approximately equal to performance of an EPR probe when a local Gaussian measurement is performed on the mode B. However, $A_s$ is slightly higher than $A_c$ (and $\chi_s$ slightly higher than $\chi_c$) because $A_s$ is a concave function of energy (see Appendix \[appendix2\]). By considering the ratio of $A_s$ and $A_c$, we find that their relative difference approaches zero in both the limits $\epsilon\to0$ and $\bar n\to0$. This indicates that there is no advantage to using an EPR state for illumination, over a coherent state probe, if a Gaussian measurement is first made on mode B of the EPR state. A local Gaussian measurement on mode B of an EPR state will cause mode A to collapse to a single-mode Gaussian state. Hence, 