The two-dimensional <Kramers--Wannier duality> PEPO has one physical input leg at every original vertex and one physical output leg at every edge. A COPY tensor at each vertex sends its binary value to all incident virtual legs. An XOR tensor on each edge compares the two incident virtual bits and writes their sum modulo two to the edge output. Contracting every vertex--edge virtual leg gives the graphical network
$$
\text{vertex COPY tensors}\;-\;\text{edge XOR tensors},
$$
repeated over the graph. In the X-Fourier basis the same network exchanges the COPY and parity constraints. This local tensor description is a <projected entangled pair operator>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 4 / a / Solution

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Kramers--Wannier duality exchanges order variables with domain-wall or disorder variables. In multiple dimensions it naturally appears as a tensor-network map with a nontrivial symmetry-sector kernel.


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