  We consider a network of three spatially separated labs of Alice, Bob, and
Charlie, with a two-qubit state shared between Alice-Bob and Bob-Charlie, and
all of them can collaborate through LOCC. We focus on the problem of optimal
and deterministic distribution of a quantum teleportation channel (QTC) between
Alice and Charlie. This involves distributing a two-qubit entangled state
between Alice and Charlie with an optimized fully entangled fraction (FEF) over
all three-party trace-preserving (TP) LOCC, exceeding the classical bound.
However, we find that the optimal distribution of QTC generally has no
one-to-one correspondence with the optimal distribution of entanglement. For
some specific class of pre-shared two-qubit states, we identify the set of
sufficient TP LOCC strategies that optimally distribute QTC. In this context,
the mentioned set is restricted, with Bob initiating operations and
subsequently sharing the outcomes with Alice and Charlie. Following Bob's
contribution and after it is discarded, Alice and Charlie have the freedom of
local post-processing. It seems that if one of the pre-shared entangled states
is noisy, the optimal distribution may not necessarily require the other one to
be most resourceful, i.e., a maximally entangled state (MES). Furthermore, when
both of the pre-shared entangled states are noisy, there are instances where an
efficient Bob-assisted protocol (generally a suboptimal protocol distributing a
channel with FEF larger than the classical bound) necessarily requires Bob's
joint measurement to be either performing projective measurement (PVM) in
partially entangled pure states or performing POVM. In this regard, our study
also reveals that the RPBES protocol introduced in Ref. [Phys. Rev. Lett. 93.
260501] for efficient entanglement distribution (even optimally for some
cases), is not an efficient protocol in general.
