A deterministic quantum amplifier inevitably adds noise to an amplified signal due to the uncertainty principle in quantum physics. We here investigate how a quantum-noise-limited amplifier can be improved by additionally employing the photon subtrac
tion, the photon addition, and a coherent superposition of the two, thereby making a probabilistic, heralded, quantum amplifier. We show that these operations can enhance the performance in amplifying a coherent state in terms of intensity gain, fidelity, and phase uncertainty. In particular, the photon subtraction turns out to be optimal for the fidelity and the phase concentration among these elementary operations, while the photon addition also provides a significant reduction in the phase uncertainty with the largest gain effect.
