$, where $$\begin{aligned}
\label{eqn_epr}
\ket{\psi_{\rm EPR}}&=\sqrt{1-\lambda^2}\sum_{n=0}^\infty(-\lambda)^n\ket{n}_\mathrm{A}\ket{n}_\mathrm{B} \ .\end{aligned}$$ where $\lambda=\tanh(r)$, and $r$ is the squeezing parameter.

Illumination can also be recast as a communication protocol. Let us suppose that Alice is in control of the object and she would like to communicate with Bob. She can do so by encoding a binary alphabet via the control of the object, such as in the Morse code. The message she sends to Bob can be described by realizations of a random variable $X$, where if $X=0$, Alice places the object in the noisy environment, and if $X=1$, Alice removes the object. Let $p_x$ be the prior probability that $X=x$, and let $p_0=p_1$, i.e., let both hypotheses be equally likely to occur. Let $\rho^{(x)}$ denote the state received by Bob when $X=x$. Noise is injected into the probe state before Alice encodes the value of $X$. This is shown diagrammatically in Figs. \[fig\_swapdiagram\](c) and \[fig\_swapdiagram\](d). We model the object as a beam splitter with reflectivity $\epsilon$. The environment-noise state $\rho_{\rm env}$ is a thermal state with mean photon number $\bar n_{\rm env}$, where $\rho_{\rm env}(\bar{n})=\sum_{n=0}^\infty\frac{\bar{n}^n}{(\bar{n}+1)^{n+1}}\ket{n}\bra{n}$. When the object is present, the environment noise is multiplied by a factor of $1/(1-\epsilon)$ such that the 