Including the vacuum in the post-selected model of quantum illumination

In the paper by Weedbrook et. al,(1) a post-selected model of Lloyd's original formulation of quantum illumination (QI)(2) was used to show that quantum discord explains the underlying advantage of QI over conventional illumination. The same model was used by Ray et. al(3) to show that the normalized Hilbert-Schmidt inner product (HSIP) is a valid distinguishability measure when analyzing quantum illumination. This post-selected model assumes that the detector always detects a photon whether it be from the signal or noise from the environment; thus, the vacuum is excluded from this model. In this paper, we include the vacuum back into these analyses. In the case of,(1) we show that the conclusion is unaffected by the inclusion of the vacuum. We then analyze the effects of including the vacuum when distinguishing between a noisy signal and noise. In this analysis, we found that the normalized HSIP is not monotonic with respect to the parameter that controls the brightness of the noise. Because of this, the normalized Hilbert-Schmidt inner product cannot be used as a distinguishability as was seen in the post-selected model. To reconcile this problem, we used the proposed alternative fidelity measure defined in,(4) and found that it is monotonic in the vacuum added model for all the examples considered.