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We investigate some basic scenarios in which a given set of bipartite quantumstates may consistently arise as the set of reduced states of a globalN-partite quantum state. Intuitively, we say that the multipartite state"joins" the underlying correlations. Determining whether, for a given set ofstates and a given joining structure, a compatible N-partite quantum stateexists is known as the quantum marginal problem. We restrict to bipartitereduced states that belong to the paradigmatic classes of Werner and isotropicstates in d dimensions, and focus on two specific versions of the quantummarginal problem which we find to be tractable. The first is Alice-Bob,Alice-Charlie joining, with both pairs being in a Werner or isotropic state.The second is m-n sharability of a Werner state across N subsystems, which maybe seen as a variant of the N-representability problem to the case wheresubsystems are partitioned into two groupings of m and n parties, respectively.By exploiting the symmetry properties that each class of states enjoys, wedetermine necessary and sufficient conditions for three-party joinability and1-n sharability for arbitrary d. Our results explicitly show that althoughentanglement is required for sharing limitations to emerge, correlations beyondentanglement generally suffice to restrict joinability, and not all unentangledstates necessarily obey the same limitations. The relationship betweenjoinability and quantum cloning as well as implications for the joinability ofarbitrary bipartite states are discussed.

Compatible quantum correlations: on extension problems for Werner and isotropic states