Monogamy is one of the most fundamental properties of entanglement and can, in its extremal form, be expressed as follows: If two qubits A and B are maximally quantumly correlated they cannot be correlated at all with a third qubit C. In general, there is a trade-off between the amount of entanglement between qubits A and B and the same qubit A and qubit C. This is mathematically expressed by the Coffman-Kundu-Wootters (CKW) monogamy inequality:
where 
are the concurrences between A and B respectively between A and C, while 
is the concurrence between subsystems AB and C.
It was proved that the above inequality can be extended to the case of 
qubits.
More generally, the monogamy inequality can be expressed in terms of entanglement measures 
, as follows:
For any tripartite state of systems 
, 
, 
we have
If the above inequality holds in general, i.e. not only for qubits, then it can be immediately generalized by induction to the multipartite case:
Notice that the entanglement measures 
and 
do not satisfy the monogamy inequality, whereas squashed entanglement does.
Moreover, is was proved that the Bell-CHSH inequality is monogamous: if three parties A, B and C share a quantum state 
and each chooses to measure one of two observables, then the trade-off between AB’s and AC’s violation of the CHSH inequality is given by
This means that if AB violate the CHSH inequality then AC cannot.
Related papers
- V. Coffman et al., Phys. Rev. A 61, 052306 (2000)
- B. M. Terhal, Linear Algebra Appl. 323, 61 (2000)
- T. J. Osborne, F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)
- M. Koashi, A. Winter, Phys. Rev. Lett. 69, 022309 (2004)
- B. F. Toner, Monogamy of nonlocal quantum correlations, e-print {{Arxiv number=quant-ph/0601172} (2006).
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