  Quantum entanglement and quantum non-locality are known to exhibit monogamy,
that is, they obey strong constraints on how they can be distributed among
multipartite systems. Quantum correlations that comprise and go beyond
entanglement are quantified by, e.g., quantum discord. It was observed recently
that for some states quantum discord is not monogamous. We prove in general
that any measure of correlations that is monogamous for all states and
satisfies reasonable basic properties must vanish for all separable states:
only entanglement measures can be strictly monogamous. Monogamy of other than
entanglement measures can still be satisfied for special, restricted cases: we
prove that the geometric measure of discord satisfies the monogamy inequality
on all pure states of three qubits.
