the special case where encodings were unitary, such that $\delta_{\rm loss}^{(x)} = 0$, $\delta_{\rm con}({\rm A}|{\rm B})$ was related to the advantage of using coherent interactions [@Gu:2012aa]. It is also interesting to note that $\delta_{\rm con}({\rm A}|{\rm B})$ also coincides with the the extra discord Bob sees between $A$ and $B$, should he learn the value of $X$.

In quantum illumination, when $X=0$, Alice performs an identity operation, and thus $\delta^{(0)}({\rm A}|{\rm B})=\delta({\rm A}|{\rm B})$ and $\delta_{\rm loss}^{(0)} = 0$. When $X=1$, Alice performs a swap operation between mode A of $\rho$ with the environment noise, destroying all correlations between the two modes. All discord is lost and $\delta_{\rm loss}^{(1)}=\delta({\rm A}|{\rm B})$. Putting this together, the average discord loss is thus $\delta_{\rm loss}=p_1\delta({\rm A}|{\rm B})$. Hence the consumed discord for quantum illumination is $$\label{eqn_denc}
\delta_{\rm con}(A|B)=p_0\delta^{(0)}({\rm A}|{\rm B})-\bar\delta({\rm A}|{\rm B}).$$

Method and Results {#sec_results}
==================

In Sec. \[sec\_results\_analytic\], we first derive a general result that if certain conditions are fulfilled, the discord consumed is equal to the Holevo information quantum advantage. In Sec. \[sec\_results\_info\], we numerically calculate the illumination information quantities. In Sec. \[sec\_results\_discord\], we numerically evaluate the consumed discord and compare it to the quantum advantages. Our main result 