  Continuous variable quantum teleportation provides a path to the
long-distance transmission of quantum states. Photon-varying non-Gaussian
operations have been shown to improve the fidelity of quantum teleportation
when integrated into the protocol. However, given a fixed non-Gaussian
operation, the achievable fidelity varies with different input states. An
operation that increases the fidelity for teleporting one class of states might
do the contrary for other classes of states. A performance metric suitable for
different input states is missing. For a given type of non-Gaussian operation,
the achievable fidelity also varies with parameters associated with the
operation. Previous work only focuses on particular settings of the parameters.
Optimization over the parameters is also missing. In this work, we build a
framework for photon-varying non-Gaussian operations for multi-mode states,
upon which we propose a performance metric suitable for arbitrary teleportation
input states. We then apply the new metric to evaluate different types of
non-Gaussian operations. Starting from simple multi-photon photon subtraction
and photon addition, we find that increasing the number of ancillary photons
involved in the operation does not guarantee performance improvement. We then
investigate combinations of the operations mentioned above, finding that
operations that approximate a particular form provide the best improvement. The
results provided here will be valuable for real-world implementations of
quantum teleportation networks and applications that harness the
non-Gaussianity of quantum states.
