Quantum marginal problem and incompatibility

One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in one-to-one correspondence with each other. Importantly, as incompatibility of measurements is a special case of channel incompatibility, it also forms an instance of the quantum marginal problem. The connection enables the translation of various results between the fields. As examples we solve the marginal problem for pairs of two-qubit Bell diagonal states, derive entropic criteria for channel incompatibility and give a task-oriented characterisation for a well-known semi-definite programming hierarchy of symmetric extendability of quantum states.