Distinguishability and nonclassicality of one-mode Gaussian states

We consider the class of the one-mode Gaussian states of the quantum radiation field. The relative entropy of such a state with respect to a similar one is derived. We analyze the entropic amount of nonclassicality of a Gaussian state defined as the minimal relative entropy of any classical Gaussian state with respect to it. A similar quantity built with the Hilbert-Schmidt distance is also calculated. Both nonclassicality measures are then compared with the Bures-metric degree of nonclassicality evaluated previously. The properties of the closest classical Gaussian state, as well as the decrease of the nonclassicality under thermal noise mappings are carefully examined in each case. For mixed states we find that only the Bures-distance amount of nonclassicality is equivalent to the nonclassical depth