Conditions for Quantum Network Teleportation

We provide conditions for establishing the teleportation protocol between any two nodes within a quantum network. The conditions include: (i) the unitary action on the quantum state after measurement during teleportation must be the inverse of the unitary operator corresponding to the classical outcome from the composition of encoding functions through the intermediate nodes, and (ii) a network connection must exist between any two nodes. These conditions are verified in a butterfly network and over a general version of the butterfly network, termed the G
$k$
network, with
$k$
sender/receiver pairs.