Suppose $$\mathcal{H}=\bigotimes_{i\in I} \mathcal{H}_i$$ is a tensor product of Hilbert spaces, where $I$ is some index set. Given a $J\subset I$, let $$\mathcal{H}_J=\bigotimes_{i\in J} \mathcal{H}_i,$$ and let $\bar{J}=I\setminus J$. Given a state $\rho$ on $\mathcal{H}$ (i.e. a linear operator mapping $\mathcal{H}$ to itself which satisfies $\operatorname{tr}(\rho)=1$), we define its reduced state in $J$ as $$\rho_J = \operatorname{tr}_{\bar{J}}(\rho);$$ in other words we just take the partial trace over $\mathcal{H}_{\bar{J}}$.
Given a collection of subsets $J_1,J_2,\dots\subset I$, and states $\sigma_{J_1},\sigma_{J_2},\dots$ on $\mathcal{H}_1,\mathcal{H}_2,\dots$, when does there exist a state $\rho$ on $\mathcal{H}$ whose reduced states on each of $J_1,J_2,\dots$ are given by $\sigma_{J_1},\sigma_{J_2},\dots$?
Clearly this is possible when the subsets $J_1,J_2,\dots$ are pairwise disjoint. Clearly also, this is only possible if the reduced states on the intersections of the subsets agree. However, I know that the answer is further constrained by various inequalities such as monogamy of entanglement.
I am also interested in the more general analogous question for states on Von Neumann algebras and the induced states on their subalgebras.
