  We investigate some basic scenarios in which a given set of bipartite quantum
states may consistently arise as the set of reduced states of a global
N-partite quantum state. Intuitively, we say that the multipartite state
"joins" the underlying correlations. Determining whether, for a given set of
states and a given joining structure, a compatible N-partite quantum state
exists is known as the quantum marginal problem. We restrict to bipartite
reduced states that belong to the paradigmatic classes of Werner and isotropic
states in d dimensions, and focus on two specific versions of the quantum
marginal 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 may
be seen as a variant of the N-representability problem to the case where
subsystems are partitioned into two groupings of m and n parties, respectively.
By exploiting the symmetry properties that each class of states enjoys, we
determine necessary and sufficient conditions for three-party joinability and
1-n sharability for arbitrary d. Our results explicitly show that although
entanglement is required for sharing limitations to emerge, correlations beyond
entanglement generally suffice to restrict joinability, and not all unentangled
states necessarily obey the same limitations. The relationship between
joinability and quantum cloning as well as implications for the joinability of
arbitrary bipartite states are discussed.
