The employ of a heralded noiseless linear amplifier has been proven as a useful tool for mitigating imperfections in quantum channels. Its analysis is usually conducted within specific frameworks, for which the set of input states for a given protocol is fixed. Here we obtain a more general description by showing that a noisy and lossy Gaussian channel followed by a heralded noiseless linear amplifier has a general description in terms of effective channels. This has the advantage of offering a simpler mathematical description, best suited for mixed states, both Gaussian and non-Gaussian. We investigate the main properties of this effective system, and illustrate its potential by applying it to loss compensation and reduction of phase uncertainty.
Blandino, R., Barbieri, M., Grangier, P., & Tualle-Brouri, R. (2015). Heralded noiseless linear amplification and quantum channels. PHYSICAL REVIEW A, 91(6) [10.1103/PhysRevA.91.062305].