Entanglement monogamy describes limits on sharing quantum entanglement among several partners. If two qubits form a pure Bell state, their joint state cannot also be correlated with a third system: the Bell pair already has a pure joint density operator, so the full state factorizes from the third system.
For partially entangled states, a quantitative sharing rule depends on the measure and subsystem dimensions. The Coffman--Kundu--Wootters inequality bounds the sum of squared pairwise concurrences for three qubits by the entanglement of one qubit with the other two together. Squaring and the chosen measure matter; one cannot assume an identical linear inequality for every entanglement measure. Classical correlations can be shared much more freely, distinguishing monogamy from a general ban on correlated triples.
For a pure state of three qubits, the squared concurrence obeys . In particular, if and form a maximally entangled pair, then has no entanglement with .
An entanglement area law bounds a region's entanglement entropy by its boundary size rather than its volume, in a specified family of states and with a stated short-distance convention. For a lattice region , a typical form is with a constant independent of the region's linear size.
The physical intuition is that correlations contributing most strongly across the cut can be concentrated near its boundary. A generic highly entangled state need not satisfy this: independent Bell pairs joining many interior sites to the exterior can produce volume-scaling entropy. Tensor-network area-law bounds make the boundary counting explicit through the number of cut bonds. Area laws are therefore useful structural constraints, not automatic consequences of calling a state quantum or a Hamiltonian local.
If a bipartition cuts tensor-network bonds of dimension at most , its Schmidt rank is at most and its entanglement entropy is at most .
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