  Hybrid entanglement between discrete-variable (DV) and continuous-variable
(CV) quantum systems is an essential resource for heterogeneous quantum
networks. Our previous work showed that in lossy channels the teleportation of
DV qubits, via CV-entangled states, can be significantly improved by a new
protocol defined by a modified Bell state measurement at the sender. This work
explores whether a new, similarly modified, CV-based teleportation protocol can
lead to improvement in the transfer of hybrid entangled states. To set the
scene, we first determine the performance of such a modified protocol in
teleporting CV-only qubits, showing that significant improvement over
traditional CV-based teleportation is obtained. We then explore similar
modifications in the teleportation of a specific hybrid entangled state showing
that significant improvement over traditional CV-based teleportation is again
found. For a given channel loss, we find teleporting the DV qubit of the hybrid
entangled state can always achieve higher fidelity than teleporting the CV
qubit. We then explore the use of various non-Gaussian operations in our
modified teleportation protocol, finding that, at a cost of lower success
probability, quantum scissors provides the most improvement in the loss
tolerance. Our new results emphasize that in lossy conditions, the quantum
measurements undertaken at the sender can have a surprising and dramatic impact
on CV-based teleportation.
