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. 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 resource-free incompatibility is equivalent
to an operational advantage in some subchannel discrimination task. Through our
framework, a clear connection can be established between any marginal problem
(that involves some notion of incompatibility) for quantum states and a
resource theory for quantum states. In addition, the universality of our
framework leads, for example, to further quantitative understanding of the
incompatibility associated with the recently-proposed entanglement marginal
problems as well as entanglement transitivity problems. As a byproduct of our
investigation, we obtain the first example showing a form of transitivity of
nonlocality as well as steerability for quantum states, thereby answering a
decade-old question to the positive.
