The importance of Hermitian circulants for physical observables was discussed in the original 1960 paper on mutually unbiased bases, by Julian Schwinger. This paper also shows how observables for $N$-state systems naturally depend on the systems for the prime factors of $N$, where each system is described by a canonical set of unitary operators. A simple $2 \times 2$ circulant is also obtained from a Jordan algebra product,
where the relevant braid diagram for the two factors is given.
6 years ago