is that for continuous variable quantum illumination, the consumed discord is approximately equal to the Holevo information quantum advantage.

![image](plot1a.pdf){width="15.5cm"} ![image](plot1b.pdf){width="15.5cm"}

Analytic Result {#sec_results_analytic}
---------------

We prove the following theorem.

\[theorem1\] Let $\rho_{{\rm AB}}^{(0)}$ and $\rho_{{\rm AB}}^{(1)}$ be two arbitrary two mode states. If the following conditions are met:

1.  mode B is the same for both states, i.e., $\rho_{\rm B}^{(0)}=\rho_B^{(1)}$ where $\rho_{\rm B}^{(x)}=\Tr_A(\rho_{{\rm AB}}^{(x)})$ and where $\Tr_{\rm A}$ denotes the partial trace over subsystem A;

2.  $\rho_{{\rm AB}}^{(1)}$ is a product state, i.e., $\rho_{{\rm AB}}^{(1)}=\rho_A^{(1)}\otimes\rho_B^{(1)}$; and

3.  the Holevo information of local measurement $\chi_c$, the discord of $\bar \rho_{{\rm AB}}=p_0\rho_{{\rm AB}}^{(0)}+p_1\rho_{{\rm AB}}^{(1)}$, and the discord of $\rho_{{\rm AB}}^{(0)}$ are achieved by the same measurement,

then $\delta_{\rm con}({\rm A}|{\rm B})=\chi_q-\chi_c$, where $$\begin{aligned}
\chi_q&=\chi(\rho_{{\rm AB}}^{(0)},\rho_{{\rm AB}}^{(1)})\\
\chi_c&=\max_{\{\Pi_b\}}\sum_b p_b \chi(\rho_{{\rm A}|b}^{(0)},\rho_{{\rm A}|b}^{(1)}),\end{aligned}$$ where $p_b$ is the probability of measuring outcome $\Pi_b$ on subsystem B, and $\rho_{{\rm A}|b}^{(x)}$ are the states of subsystem A conditioned on that outcome.

Let $\{\Pi_b\}$ be the measurement in condition 3 that simultaneously optimizes $\chi_c$, as well as the discord of states $\bar \rho_{{\rm AB}}$ and $\rho_{{\rm AB}}^{(0)}$. The measurement outcome probability is $$p_b=\Tr((\Pi_b\otimes I)\rho_{{\rm AB}}^{(0)})=\Tr((\Pi_b\otimes I)\rho_{{\rm AB}}^{(1)}),$$ where we have used condition 1. The resulting conditional states 