a noisy environment. He sends a quantum state, referred to as the probe, to the location. If an object is present, part of the probe will be reflected back to Bob, along with some background noise. If the object is not present, Bob receives only the background noise. Bob may have another state called the idler, which was initially correlated with the probe.

If the probe and idler are quantum correlated (have a non-zero quantum discord), the scheme is called *quantum illumination*. If there is no idler, it is called *single-mode illumination*.

A diagram of illumination is shown in Figs. \[fig\_swapdiagram\](a) and  \[fig\_swapdiagram\](b). Bob performs a joint measurement on the idler and returning probe, and uses the results of the measurement to determine whether an object was present. For brevity in notation in the rest of the paper, modes A and B will label the probe and idler parts of the state, respectively.

We are interested in quantum illumination in the continuous-variable setting, where the probe and idler are Gaussian states. For single-mode illumination, Bob uses a coherent state $\rho_\alpha$, where $\alpha$ is its amplitude. For quantum illumination, Bob uses an Einstein-Podolsky-Rosen (EPR) state described by $\rho_{\rm EPR}=\ket{\psi_{\rm EPR}}\bra{\psi_{\rm EPR}}
