  We introduce the resource marginal problems, which concern the possibility of
having a resource-free target subsystem compatible with a given collection of
marginal density matrices. By identifying an appropriate choice of resource R
and target subsystem T, our problems reduce, respectively, to the well-known
marginal problems for quantum states and the problem of determining if a given
quantum system is a resource. More generally, we say that a set of marginal
states is resource-free incompatible with a target subsystem T if all global
states compatible with this set must result in a resourceful state in T of type
R. We show that this incompatibility induces a resource theory that can be
quantified by a monotone and obtain necessary and sufficient conditions for
this monotone to be computable as a conic program with finite optimum. We
further show, via the corresponding witnesses, that (1) resource-free
incompatibility is equivalent to an operational advantage in some
channel-discrimination tasks, and (2) some specific cases of such tasks fully
characterize the convertibility between marginal density matrices exhibiting
resource-free incompatibility. Through our framework, one sees a clear
connection between any marginal problem -- which implicitly involves some
notion of incompatibility -- for quantum states and a resource theory for
quantum states. We also establish a close connection between the physical
relevance of resource marginal problems and the ground state properties of
certain many-body Hamiltonians. In terms of application, the universality of
our framework leads, for example, to a further quantitative understanding of
the incompatibility associated with the recently-proposed entanglement marginal
problems and entanglement transitivity problems.
