Noiselessly amplified thermal states and after multi-photon addition or subtraction

In this paper, we study the combined effects of the ideal amplification operator ([Formula: see text]) and m-photon added operator [Formula: see text] (or m-photon subtracted operator [Formula: see text]) on the thermal state ([Formula: see text]), with mean photon number (MPN [Formula: see text]). By only operating [Formula: see text] on [Formula: see text], we introduce a noiselessly amplified thermal state (ATS [Formula: see text]) with larger MPN [Formula: see text]. The selection of [Formula: see text] and g must ensure that the process is physical ([Formula: see text]). By operating [Formula: see text] and [Formula: see text] (or [Formula: see text]) on [Formula: see text] successively, we further introduce photon-added-ATS (ρ m+ ) or photon-subtracted-ATS (ρ m− ). Then, we analyze their photon number distributions and prove that the purity of ρ m+ is equal to that of ρ m− . Meanwhile, we study their nonclassicality by examing Mandel parameter and second-order correlation function. Finally, we study their three quasiprobability distributions, whose negativities are other signatures of the nonclassicality. Our study demonstrates that photon addition has more pronounced effects on nonclassicality than photon subtraction. Our study will provide useful theoretical references for experiments and applications.