\\
J({\rm A}|{\rm B})&=S({\rm A})-\min_{\{\Pi_b\}}\sum p_b S({\rm A}|b)\end{aligned}$$ where $\Pi_b$ is the POVM element corresponding to the outcome $b$, $p_b$ is the probability of that outcome, and $S({\rm A}|b)$ is the entropy of the state conditioned on the outcome $b$. The discord is then $$\begin{aligned}
\label{eq_discord}
\delta({\rm A}|{\rm B}) &= I({\rm A}:{\rm B})-J({\rm A}|{\rm B}) \nonumber \\
&= S({\rm B}) - S({\rm AB}) + \min_{\{\Pi_b\}}\sum p_b S({\rm A}|b) \ ,\end{aligned}$$ where the minimization is done over all possible POVMs on mode B. In the special case that the domain of this minimization is restricted to Gaussian measurements, the discord is known as the Gaussian discord [@PhysRevLett.105.020503; @PhysRevLett.105.030501]. It was recently shown that for a large class of Gaussian states, Gaussian quantum discord is equal to quantum discord [@PhysRevLett.113.140405]. Henceforth, we denote the Gaussian discord $\delta^{\rm G}({\rm A}|{\rm B})$ with a superscript G.

We now consider the evolution of the discord when quantum illumination is described by Figs. \[fig\_swapdiagram\](c) and \[fig\_swapdiagram\](d). After the noise-injection step, Alice is left with state $\rho$ with which she can encode information to send to Bob. We note that this state may have no entanglement due to the noise injection [@PhysRevLett.101.253601]. Alice encodes the value of 