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We introduce and analyze graph-associated entanglement cost, a generalizationof the entanglement cost of quantum states to multipartite settings. Weidentify a necessary and sufficient condition for any multipartite entangledstate to be constructible when quantum communication between the multipleparties is restricted to a quantum network represented by a tree. The conditionfor exact state construction is expressed in terms of the Schmidt ranks of thestate defined with respect to edges of the tree. We also study approximatestate construction and provide a second-order asymptotic analysis.

Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions