Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what conditions are these compatible with the same global state of $ABC.$
In other words when does a state $\tau_{ABC}$ exist such that $Tr_{C}(\tau_{ABC}) = \rho_{AB}$ and $Tr_A(\tau_{ABC}) = \rho_{BC}.$
A necessary condition for the existence of such a state would be $Tr_A(\rho_{AB}) = Tr_C(\rho_{BC}).$
Is this condition sufficient? If not, are sufficient conditions known?
