Are General Quantum Correlations Monogamous?

Quantum entanglement and quantum nonlocality 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.

Figure 1

Entanglement is monogamous: for a fixed amount of entanglement between $A$ and $BC$, the more entanglement exists between $A$ and $B$, the less can exist between $A$ and $C$. Quantitatively, this is expressed using the monogamy relation, see Eq. (1) in the main text. In particular, the latter implies—for a monogamous measure of entanglement $E$—that $E^{A|C}=0$ if $E^{A|BC}=E^{A|B}$. In this Letter we show that the monogamy relation does not hold, in general, for any quantum correlation measure beyond entanglement, i.e., for any measure that does not vanish on separable states.