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.

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