  A minimally-invasive way to detect the presence of a stealth target is to
probe it with a single photon and analyze the reflected signals. The efficiency
of such a conventional detection scheme can potentially be enhanced by the
method of quantum illumination, where entanglement is exploited to break the
classical limits. The question is, what is the optimal quantum state that
allows us to achieve the detection limit with a minimal error? Here we address
this question for discrete signals, by deriving a complete and general set of
analytic solutions for the whole parameter space, which can be classified into
three distinct regions, in the form of phase diagrams for both conventional and
quantum illumination. Interestingly, whenever the reflectivity of the target is
less than some critical values, all received signals become useless, which is
true even if entangled resources are employed. However, there does exist a
region where quantum illumination can provide advantages over conventional
illumination; there, the optimal signal state is an entangled state with an
entanglement spectrum inversely proportional to the spectrum of the
environmental state. These results not only impose fundamental limits in
applications such as quantum radars, but also suggest how to become immune
against the attack of minimally-invasive detection.
