### 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.

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.

## 3 Comments:

Have you seen the paper by John Baez and Michael Stay at

http://math.ucr.edu/home/baez/rosetta.pdf

entitled

"Physics, Topology, Logic and Computation:

A Rosetta Stone"

in which they say

"... Quantum physics has long made extensive use of `symmetric monoidal categories with duals', whether physicists knew it or not.

Topology - especially knot theory - uses `braided monoidal categories with duals', which are slightly more general.

However, it became clear by the 1990s that these

more general gadgets are useful in physics too.

Logic and computer science used to focus on `cartesian closed categories' - where `cartesian' can be seen, roughly, as an antonym of `quantum'. However, thanks to more recent work on linear logic and quantum computation, some logicians and computer scientists have now dropped their insistence on cartesianness: now they study more general sorts of `symmetric monoidal closed categories'.

... we shall focus our attention on `braided monoidal closed categories'.

These are just general enough to include all the aforementioned gadgets as special cases. ...".

As John Baez indicated in his diary entry for February 9, 2008, the paper is a work in progress, but it already seems to be nice and useful.

Tony Smith

Haven't had a look at it yet, Tony, but thanks for the link.

Well I'm currently exploring extensions of MUBs to the set of all possible bases. This allows cancelation of bilinears, which is what you need if you want to treat different bilinears differently (as for symmetry breaking).

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