  Subsystems of composite quantum systems are described by reduced density
matrices, or quantum marginals. Important physical properties often do not
depend on the whole wave function but rather only on the marginals. Not every
collection of reduced density matrices can arise as the marginals of a quantum
state. Instead, there are profound compatibility conditions -- such as Pauli's
exclusion principle or the monogamy of quantum entanglement -- which
fundamentally influence the physics of many-body quantum systems and the
structure of quantum information. The aim of this thesis is a systematic and
rigorous study of the general relation between multipartite quantum states,
i.e., states of quantum systems that are composed of several subsystems, and
their marginals. In the first part, we focus on the one-body marginals of
multipartite quantum states; in the second part, we study general quantum
marginals from the perspective of entropy.
