### 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.

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.

## 2 Comments:

Nice to see the importance of the Democratic matrix showing up here.

Regarding cosmology, the modification of GR that I believe we have to reach is nicely described in the new PI lecture: Towards the end of the cosmological constant problem! : Niayesh Afshordi

This is a modification of GR. There are two important points of compatibility with the gravity I've been working on from the quantum side: (a) Assumes preferred reference frame, and (b) the added material (which he called aether) propagates faster than light.

There is a related arXiv paper, 0807.2639

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