M Theory Lesson 157

It's a while since we stressed the fact that there should be some category theory happening, so here is a diagram of a tensorator, which weakens a basic commutative square for maps involving tensor products. Observe that both sides of the square represent $f \otimes g$. These squares appear in the definition of a monoidal 2-category, where $\phi$ is essentially unique for any given $f: A \rightarrow A'$ and $g: B \rightarrow B'$. From a higher categorical point of view, we need to move away from Hilbert spaces when investigating mass matrices and MUBs. In particular, we are allowed to completely change the rules for tensor products of spaces.

What might this mean for mutually unbiased bases? Instead of bases we can consider $d$ change of basis maps for a prime $d$ dimensional space. Instead of being ordinary matrices, these operators are permitted to be less well defined under a tensor product, via the tensorator. For example, if $f \otimes g$ is only defined up to a scalar, depending on the order of composition, the map $\phi$ might correspond to multiplication by the scalar between the two alternative types of $f \otimes g$. Thus higher categories offer excellent ways of cheating to get just what one wants!

Aside: What is Category Theory? is a lovely book, which one can browse with Google. It contains a helpful article by Coecke: Introducing categories to the practicing physicist.