M Theory Lesson 265

There is a nice series of papers by Godsil et al on combinatorics associated to MUBs. Consider the simple qubit Combescure matrix This is the only choice for the eigenvectors of the Pauli matrix $\sigma_{Y}$ that satisfies the following properties. The Schur multiplication of two matrices simply defines entries by the product of matching entries from the two components $A$ and $B$. That is $M_{ij} = A_{ij} B_{ij}$. Under this product, an inverse for $R_2$ is found relative to the democratic matrix (the Schur identity): An invertible matrix in this sense is type II if $M (M^{-1})^{T} = n I$, where $I$ is the ordinary identity matrix. This works for $R_{2}$, although only $R_{2}^{8} = I$.

A type II matrix is a spin model, in the sense of Jones, if all vectors of the form

$Me_{i} \circ M^{-1}e_{j}$

(for $e_{i}$ the standard basis vectors) are eigenvectors for $M$. One checks that this holds for $R_2$. Note that the Fourier (Hadamard) operator $F_{2}$, although a type II matrix, is not a spin model matrix.