# Monogamy of entanglement

'''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**:

*C**A**B*2 + *C**A**C*2 ≤ *C**A*(*B**C*)2, where *C**A**B*, *C**A**C* are the concurrences between A and B respectively between A and C, while *C**A*(*B**C*) is the concurrence between subsystems A and BC.

It was proved that the above inequality can be extended to the case of *n* qubits.

More generally, the **monogamy inequality** can be expressed in terms of entanglement measures *E*, as follows:

'''For any tripartite state of systems *A*, *B*1, *B*2 we have

*E*(*A*∣*B*1) + *E*(*A*∣*B*2) ≤ *E*(*A*∣*B*1*B*2).''' 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:

*E*(*A*∣*B*1) + *E*(*A*∣*B*2) + … + *E*(*A*∣*B**N*) ≤ *E*(*A*∣*B*1*B*2…*B**N*).

Notice that the entanglement measures *E**C* and *E**F* 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 $\; \varrho$ 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

$$\; |Tr(\mathcal{B}_{CHSH}^{AB}\varrho)| + |Tr(\mathcal{B}_{CHSH}^{AC}\varrho)| \leq 4 .$$ 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 (2001) - B. M. Terhal,
*IBM J. Res. Dev.**'48*, 71 (2004). Available at arXiv:quant-ph/0307120 - M. Koashi, A. Winter,
*Phys. Rev. A***69**, 022309 (2004) - T. J. Osborne, F. Verstraete,
*Phys. Rev. Lett.***96**, 220503 (2006) - B. Toner,
*Proc. R. Soc. A***465**, 59-69 (2009)