  We generalize the concept of optical state truncation and noiseless linear
amplification to enable truncation of the Fock-state expansion of an optical
state to higher order and to simultaneously amplify it using linear optics. The
resulting generalized quantum scissors are more efficient for noiseless linear
amplification than employing multiple scissors in parallel and are
experimentally practical. As a particular example, we focus on a third-order
scissor device and demonstrate advantages in terms of fidelity with the target
state, probability of success, distillable entanglement, and the amount of
non-Gaussianity introduced.
