![ **Diagram of illumination setup.** (a) With probability $p_0$, there is an object located in a noisy environment. The object is partially reflective (modeled as a beam splitter with reflectivity $\epsilon$). A probe is sent towards the object. The probe is mixed with the noisy environment, and reflected to the detector. (b) With probability $p_1$, an object is not present, in which case there is nothing to reflect the probe to the detector. Hence, only noise is detected. (c) An equivalent description of illumination whereby first noise is injected. Then, encoding is performed on the probe, whereby with a probability $p_0$, an identity operation is performed on the probe (after noise injection) and the environment noise, and (d) with probability $p_1$, a swap operation is performed on the probe and environment. In quantum illumination, we also have an idler initially entangled with the probe which is used to perform a joint measurement. Single-mode illumination is when there is no idler. $\rho_{\rm env}$ is the noisy environment and $\tilde{\rho}_{\rm env}$ is the environment with the mean photon number scaled by $1/(1-\epsilon)$. []{data-label="fig_swapdiagram"}](swapdiagram.pdf){width="7.4cm"}

Setup
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The illumination framework is described as follows: Bob wishes to determine whether an object is located in 