Rank-one local systems on the elliptic curve

The operational use below concerns degree-zero Hilbert coefficient spaces. A chain complex is not treated as a density matrix.

The character tensor product satisfies

This follows by multiplying their monodromies. It is not a reversible quantum map $|k,l⟩\mapsto |k+l⟩$ on a full coherent label register. If a classical fusion channel is desired, it is separately the complete probability pushforward under addition modulo $d$.