## Sunday, August 21, 2011

### Theory Update 105

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.