this is equivalent to using a distribution of single-mode Gaussian states for the probe, which, under the masking of strong environmental noise, gives an approximately equal knowledge about a weakly reflecting object as using a single-mode coherent state probe.

Relating Quantum Advantage to Discord Consumed {#sec_results_discord}
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To calculate the consumed discord $\delta_{\rm con}({\rm A}|{\rm B})$, we need to compute the discord of states $\rho^{(0)}$ and $\bar\rho$ when the entangled state $\rho_{\rm EPR}$ is used as probe and idler. $\rho^{(0)}$, the resulting state when Alice does nothing, is a Gaussian state whose discord is equal to the Gaussian discord, and additionally this discord is obtained when the measurement is a heterodyne measurement [@PhysRevLett.113.140405]. The state after encoding $\bar\rho$, however, is not Gaussian, and thus the same rule does not apply. Unfortunately, calculating the discord of a general state is an NP-hard problem [@1367-2630-16-3-033027], so there is no method to calculate it efficiently. Here, we simplify the problem by restricting ourselves to Gaussian discord and calculate the consumed Gaussian discord $\delta^{\rm G}_{\rm con}({\rm A}|{\rm B})$ instead. This is just Eq.  with the discords replaced with Gaussian discords.

The Gaussian discord of state $\bar\rho$ was obtained by numerically optimizing Eq.  over Gaussian 