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''': :$\backslash ;\; C\_\{AB\}^2\; +\; C\_\{AC\}^2\; \backslash leq\; C\_\{A(BC)\}^2,$ where $\backslash ;\; C\_\{AB\},\; C\_\{AC\}$ are the concurrences between A and B respectively between A and C, while $\backslash ;\; C\_\{A(BC)\}$ is the [[concurrence]] between subsystems A and BC. It was proved that the above inequality can be extended to the case of $\backslash ;\; n$ qubits. More generally, the '''monogamy inequality''' can be expressed in terms of [[entanglement measures]] $\backslash ;\; E$, as follows: '''For any tripartite state of systems $\backslash ;\; A$, $\backslash ;\; B\_1$, $\backslash ;\; B\_2$ we have :$\backslash ;\; E(A\; |\; B\_1)\; +\; E(A\; |\; B\_2)\; \backslash leq\; E(A\; |\; B\_1B\_2).$''' 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: :$\backslash ;\; E(A\; |\; B\_1)\; +\; E(A\; |\; B\_2)\; +\; \backslash ldots\; +\; E(A\; |\; B\_N)\; \backslash leq\; E(A\; |\; B\_1B\_2\backslash ldots\; B\_N)\; .$ Notice that the [[entanglement measures]] $\backslash ;\; E\_C$ and $\backslash ;\; E\_F$ 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 $\backslash ;\; \backslash varrho$ 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 :$\backslash ;\; |Tr(\backslash mathcal\{B\}\_\{CHSH\}^\{AB\}\backslash varrho)|\; +\; |Tr(\backslash mathcal\{B\}\_\{CHSH\}^\{AC\}\backslash varrho)|\; \backslash leq\; 4\; .$ This means that if AB violate the CHSH inequality then AC cannot. == Related papers == *[http://pra.aps.org/abstract/PRA/v61/i5/e052306 V. Coffman ''et al.'', ''Phys. Rev. A'' '''61''', 052306 (2000)] *[http://dx.doi.org/10.1016/S0024-3795(00)00251-2 B. M. Terhal, ''Linear Algebra Appl.'' '''323''', 61 (2001)] *[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5388928 B. M. Terhal, ''IBM J. Res. Dev.'' '''48'', 71 (2004).] Available at [http://arxiv.org/abs/quant-ph/0307120 arXiv:quant-ph/0307120] *[http://pra.aps.org/abstract/PRA/v69/i2/e022309 M. Koashi, A. Winter, ''Phys. Rev. A'' '''69''', 022309 (2004)] *[http://prl.aps.org/abstract/PRL/v96/i22/e220503 T. J. Osborne, F. Verstraete, ''Phys. Rev. Lett.'' '''96''', 220503 (2006)] *[http://rspa.royalsocietypublishing.org/content/465/2101/59 B. Toner, ''Proc. R. Soc. A'' '''465''', 59-69 (2009)] {{stub}} [[Category:Entanglement]]