  We study a protocol in which many parties use quantum communication to
transfer a shared state to a receiver without communicating with each other.
This protocol is a multiparty version of the fully quantum Slepian-Wolf
protocol for two senders and arises through the repeated application of the
two-sender protocol. We describe bounds on the achievable rate region for the
distributed compression problem. The inner bound arises by expressing the
achievable rate region for our protocol in terms of its vertices and extreme
rays and, equivalently, in terms of facet inequalities. We also prove an outer
bound on all possible rates for distributed compression based on the multiparty
squashed entanglement, a measure of multiparty entanglement.
