  The physics of a many-particle system is determined by the correlations in
its quantum state. Therefore, analyzing these correlations is the foremost task
of many-body physics. Any 'a priori' constraint for the properties of the
global vs. the local states---the so-called marginals---would help in order to
narrow down the wealth of possible solutions for a given many-body problem,
however, little is known about such constraints. We derive an equality for
correlation-related quantities of any multipartite quantum system composed of
finite-dimensional local parties. This relation defines a necessary condition
for the compatibility of the marginal properties with those of the joint state.
While the equality holds both for pure and mixed states, the pure-state version
containing only entanglement measures represents a fully general monogamy
relation for entanglement. These findings have interesting implications in
terms of conservation laws for correlations, and also with respect to topology.
