  Encoding and decoding quantum information in a multipartite quantum system
are indispensable for quantum error correction and also play crucial roles in
multiparty tasks in distributed quantum information processing such as quantum
secret sharing. To quantitatively characterize nonlocal properties of
multipartite quantum transformations for encoding and decoding, we analyze
entanglement costs of encoding and decoding quantum information in a
multipartite quantum system distributed among spatially separated parties
connected by a network. This analysis generalizes previous studies of
entanglement costs for preparing bipartite and multipartite quantum states and
implementing bipartite quantum transformations by entanglement-assisted local
operations and classical communication (LOCC). We identify conditions for the
parties being able to encode or decode quantum information in the distributed
quantum system deterministically and exactly, when inter-party quantum
communication is restricted to a tree-topology network. In our analysis, we
reduce the multiparty tasks of implementing the encoding and decoding to
sequential applications of one-shot zero-error quantum state splitting and
merging for two parties. While encoding and decoding are inverse tasks of each
other, our results suggest that a quantitative difference in entanglement cost
between encoding and decoding arises due to the difference between quantum
state merging and splitting.
