Category theory and quantum mechanics

My last link to something categorical turned out to be pointing to a less-than-splended online resource. I hope this is rather better!

I’d heard tell of people interested in quantum foundations and quantum information getting entangled (see what I did there?) with category theory. And by chance, I  stumbled a few days ago across details of a course currently being run in Oxford. The course materials are a late draft of Categorical Quantum Mechanics by Chris Heunen and Jamie Vicary which has recently in fact been published as a book by OUP. This strikes me (in contrast I fear to that book I mentioned by Fong and Spivak) as extremely lucid and well-organized; and you don’t in fact have to read very far to see why quantum theorists might indeed be interested in monoidal categories as a mathematical tool. My QM is very very rusty; but if you have a smidgin of knowledge, this does seem worth dipping into, if only to get a glimpse from the sidelines about what the cool kids are up to …

2 thoughts on “Category theory and quantum mechanics”

  1. Why would quantum theorists be interested in monoidal categories? I mean, what can they do with the aid of monoidal categories that (a) they want to do and (b) they can’t do (or can’t do anywhere near so easily) without monoidal categories?

    And if the graphical language — described as “perhaps one of the most compelling features that monoidal categories have to offer” — is an important part of it, what shows that the graphical language can only (or best) be put on a rigorous footing using monoidal categories?

    I saw the Categorical QM book in a bookshop a while back and thought it might be category theory book that could get me past the great “so what?”, and reasoning with the diagrams seemed kind of cool, but the more I looked at it, the more the great “so what?” came on and, however useful the diagrams might be, the category-theoretic underpinnings they were being given did not seem worth the trouble it would take to learn them.

    The authors also seem to have the ‘ideological’ category theorist’s view that category theory doesn’t just provide an interesting or useful way to look at something but is instead the key to real understanding, because they write on p 4:

    Thus we can investigate exactly what it is that makes quantum theory ‘tick’, and what features set it apart from other compositional theories.

    I’m not a quantum theorist, so far all I know quantum theorists have an epiphany when they see this stuff and fall down crying out “How did I get by without this for so long? I have wasted years!” But if so, I’d like to see some solid evidence.

  2. What I’d really like to understand is why so many diverse subsets of the cool kids are talking about category theory right now. I’m expecting a book or conference on Category Theory and Machine Learning to pop up on my radar any day now. Or even Category Theory and Craft Beers.

    I doubt this is merely a fad. But I have little intuitive grasp on why CT would be such a tip-top tool for such varied interests. To be fair, I have a pretty limited grasp on the contents of CT!

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