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

In this paper, we introduce a noiselessly amplified thermal state (ATS), by operating the noiseless amplification operator ($g^{\hat{n}}$) on the thermal state (TS) with corresponding mean photon number (MPN) $\bar{n}$. Actually, the ATS is an new TS with MPN $\bar{N}=g^{2}\bar{n}/[1-\bar{n}\left(g^{2}-1\right)]$. Furthermore, we introduce photon-added-ATS (PAATS) and photon-subtracted-ATS (PSATS) by operating $m$-photon addition ($\hat{a}^{\dag m}$) and $m$-photon subtraction ($\hat{a}^{m}$) on the ATS, respectively. We study photon number distributions (PNDs), purities, and Wigner functions (WFs) for all these states.