Purification of Gaussian states by photon subtraction or addition

Photon subtraction (PS) can enhance entanglement, which, for pure states, induces a decrease in the purity of reduced states. In contrast, by analyzing the purities of Gaussian states before and after subtracting or adding a single photon, we prove that the purity of a Gaussian state can be increased by less than 20%. On the one hand, it reveals that PS and photon addition (PA) can reduce entanglement, and on the other hand, it shows that PS or PA can achieve a limited amount of Gaussian state purification. Through the analysis of several examples, we demonstrate the inherent mechanism and applicable scope of photon-subtraction-based purification. We further compare the performance between PS and PA for purifying single-mode Gaussian states, and analyze their capabilities in generating Wigner negativity within purification regimes. In a multimode system, we find that PS can simultaneously increase entanglement and purify some of the reduced states. These findings demonstrate the new application potential of PS and PA, specifically their capability to directly suppress Gaussian noise.