Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 2

Entanglement monogamy says that strong entanglement between $A$ and $B$ limits the entanglement of either with an independent $C$. For qubits, the Coffman--Kundu--Wootters inequality isIn quantum cryptography, nearly maximal correlations between legitimate partners imply that an eavesdropper cannot hold a purification with equally strong correlations. Measuring an error rate therefore bounds the eavesdropper's information and permits privacy amplification.

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Interpret the sum as one term per unordered pair. Withthe all-to-all XY Hamiltonian isFor fixed total spin $S$, choose $|m|=S$, giving $E=2S-N$. The smallest allowed total spin is $S=0$ for even $N$ and $S=1/2$ for odd $N$, soDividing by $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ givesFor three qubits, $E_0=-2$ and there are three pairs:The finite system therefore has a smaller pair-energy density; monogamy prevents every pair from independently attaining the two-qubit minimum.

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In a local gapped ground state, correlations and the entanglement needed to lower a local interaction energy are concentrated over a finite distance. Degrees of freedom deep inside a region use most of their entanglement with nearby degrees of freedom that are also inside; entanglement monogamy limits their simultaneous entanglement with the exterior. Only degrees of freedom within a correlation length of the boundary can contribute extensively across the cut, giving an entanglement area lawMonogamy supplies the physical mechanism, while locality and suitable ground-state assumptions are essential; monogamy alone is not a general mathematical proof of an area law.

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Cutting an open-boundary matrix product state crosses one virtual bond of dimension $\chi$. Its Schmidt rank is at most $\chi$, soA periodic interval crosses two bonds and obeys $S(A)\le 2\log \chi$. The one-dimensional boundary has a constant number of points, so this is an area law.

For a projected entangled pair state, cut every virtual bond crossing the boundary of a region $A$. If $n_{\partial }$ bonds are cut, the Schmidt rank is at most ${\chi }^{n_{\partial }}$, and thereforeSince $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.

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