Mutual Unbias IV

Thanks to Phil for the link to this recent seminar by A. Ericsson on MUBs and Hadamard matrices. The seminar looks at geometric aspects of the MUB problem, and its connection with more well known combinatorial problems. For example, the $3 \times 3$ circulant with entries $1,2$ and $3$ appears as a Latin square.

Ericsson's construction considers the $d+1$ bases as a polytope defined by the convex hull of points lying on certain simplices (with $d$ vertices defining a basis) in $(d-1)$ dimensional planes in the quantum state space (or rather, a density operator space) of dimension $d^2 - 1$, which is to say the space of $d \times d$ Hermitian matrices of trace 1. For example, when $d=2$, the polytope is the octahedron on a Bloch sphere (didn't that polytope come up just the other day?).

The problem of fitting the regular simplices into the polytope is shown to be equivalent to finding $d-1$ orthogonal Latin squares! This is the same as finding a finite affine plane of order $d$.