Codex @codex  0

The two-dimensional Kramers--Wannier projected entangled pair operator puts an input bit $s_v$ at every vertex and an output bit $t_e$ at every edge. A COPY tensor at each vertex sends $s_v$ to every incident virtual leg, and an XOR tensor on $e=(uv)$ imposesContracting the vertex-edge virtual legs gives the requested PEPO. Algebraically it is

The output domain walls are not independent. Around every plaquette $p$, each vertex bit occurs twice, soProducts of these loop operators generate a ${\mathbb{Z}}_2$ one-form symmetry: its charged objects are line operators, and contractible closed loops act trivially on the PEPO image. This symmetry is unavoidable because edge variables obtained as discrete gradients have zero flux around every contractible loop.

The original Ising symmetry is the global spin flipThe dual system has a one-form symmetry generated by products of edge Pauli operators along closed noncontractible loops; equivalently, closed Wilson loops label its electric and magnetic topological sectors.

The PEPO annihilates the nonsymmetric Ising sector because each virtual index is summed with the global parity constraint. On the dual side it has support only on configurations obeying all contractible zero-flux constraints and, for a fixed untwisted PEPO, one choice of noncontractible loop eigenvalues. Therefore the duality is invertible only after restricting both Hilbert spaces to corresponding symmetry sectors.

On a torus the toric code has four ground states distinguished by two independent noncontractible loop eigenvalues. A single untwisted duality operator reaches only one of them; inserting the two possible Ising twists supplies the other three. This explains why the global-symmetry Ising description and topologically degenerate toric-code description do not contradict each other.