Monogamy of Entanglement
Entanglement cannot be freely shared: the more one party is entangled with a second, the less it can be with a third.
A shared resource with limits
Classical correlations can be broadcast without penalty; one variable can be perfectly correlated with arbitrarily many others. Entanglement cannot. Monogamy is the principle that if qubit A is maximally entangled with B, it has no entanglement left to share with C. The tighter the bond between two parties, the weaker their bonds with everyone else.
The CKW inequality
Coffman, Kundu, and Wootters made this quantitative for three qubits: C(A:B)^2 + C(A:C)^2 <= C(A:BC)^2, where C denotes concurrence and C(A:BC) treats BC as a single system. The pairwise entanglements are bounded by the entanglement of A with the rest. The gap, the three-tangle, measures genuine tripartite entanglement not captured by any pair.
- A maximally entangled with B: zero entanglement A-C, A-anyone else
- W state: entanglement spread across pairs, three-tangle zero
- GHZ state: pairwise concurrence zero, three-tangle maximal
Why it holds
Monogamy is rooted in the no-cloning theorem and the structure of quantum states. If A could be maximally entangled with both B and C, one could effectively clone quantum information, which is forbidden. More formally, a maximally entangled pair is in a pure joint state, which by purity must be uncorrelated with anything else. Extreme entanglement forces the rest of the world out.
Consequences
Monogamy is the security backbone of quantum key distribution: an eavesdropper's entanglement with the transmitted qubits is bounded by how much the legitimate parties share, so their correlations limit any leak. It shapes the physics of many-body systems, where each particle's entanglement budget is finite, driving area laws and constraining ground-state structure. It also underlies debates about black-hole information, where monogamy tensions sharpen the puzzle.