  We present a new scheme for the compression of one-way quantum messages, in
the setting of coherent entanglement assisted quantum communication. For this,
we present a new technical tool that we call the convex split lemma, which is
inspired by the classical compression schemes that use rejection sampling
procedure. As a consequence, we show new bounds on the quantum communication
cost of single-shot entanglement-assisted one-way quantum state redistribution
task and for the sub-tasks quantum state splitting and quantum state merging.
Our upper and lower bounds are tight up to a constant and hence stronger than
previously known best bounds for above tasks. Our protocols use explicit
quantum operations on the sides of Alice and Bob, which are different from the
decoupling by random unitaries approach used in previous works. As another
application, we present a port-based teleportation scheme which works when the
set of input states is restricted to a known ensemble, hence potentially saving
the number of required ports. Furthermore, in case of no prior knowledge about
the set of input states, our average success fidelity matches the known average
success fidelity, providing a new port-based teleportation scheme with similar
performance as appears in literature.
