Quantum illumination is to discern the presence or absence of a low
reflectivity target, where the error probability decays exponentially in the
number of copies used. When the target reflectivity is small so that it is hard
to distinguish target presence or absence, the exponential decay constant falls
into a class of objects called monotone metrics. We evaluate monotone metrics
restricted to Gaussian states in terms of first-order moments and covariance
matrix. Under the assumption of a low reflectivity target, we explicitly derive
analytic formulae for decay constant of an arbitrary Gaussian input state.
Especially, in the limit of large background noise and low reflectivity, there
is no need of symplectic diagonalization which usually complicates the
computation of decay constants. First, we show that two-mode squeezed vacuum
(TMSV) states are the optimal probe among pure Gaussian states with fixed
signal mean photon number. Second, as an alternative to preparing TMSV states
with high mean photon number, we show that preparing a TMSV state with low mean
photon number and displacing the signal mode is a more experimentally feasible
setup without degrading the performance that much. Third, we show that it is of
utmost importance to prepare an efficient idler memory to beat coherent states
and provide analytic bounds on the idler memory transmittivity in terms of
signal power, background noise, and idler memory noise. Finally, we identify
the region of physically possible correlations between the signal and idler
modes that can beat coherent states.Comment: 16 pages, 6 figure