To take advantage of this, these two-body ket states can be organized into "channels" of shared quantum numbers. In the case of the pairing model, the interaction preserves the total spin projection of a two-body state, \( S_{z}=s_{z1}+s_{z2} \). The single particle states can have spin of +1/2 or -1/2, so there can be three two-body channels with \( S_{z}=-1,0,+1 \). These channels can then be indexed with a unique label in a similar way to the single particle index scheme. In more complicated systems, there will be many more channels involving multiple symmetries, so it is useful to create a data structure that stores the relevant two-body quantum numbers to keep track of the labeling scheme.