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Hi,
I recently posted a paper that gives, I think, the most general prescription possible for what experiments are. It essentially derives these notions from basic facts about spacetimes. In the paper I give a quasi classical experiment as an example. In this paper, we can actually see another example for quantum systems. Example 5.2 explains that any Frobenius structure on a monoidal category exists because the associated endofunctor $- \otimes B$ is a frobenius monad. Todd gave an answer to a question about the ambidextrous adjuction and pointed out that the category on the other end of the adjunction is the category of B-modules.
I am encouraged by Heunen’s paper. My theory suggests that the adjunciton binds the experiment you are doing to the theory of the system you are probing. In the question with Todd’s responds, I asked about having Hilbert spaces as the base category. $B$, is then an object in Hilb, and the Frobenius structure associated with $B$, I believe is a basis for a hilbert space, which is how we come to know the state of a quantum system, ie measure it. If we take my paper seriously, we expect that at the other end of the adjunction is a theory for the underlying system we are looking at. I am not sure what a B-module is, but I know that module are a generalization of vector spaces, and hence Hilbert spaces.
I would like to open a discussion about how the ambidextrous adjunction points to a theory of Hilbert spaces. Let me know if you agree or disagree.
If you are given a monoidal category $\mathbf{H}$ (e.g., $Hilb$) and a monoid object $B$ with respect to that monoidal product, then a (right) $B$-module is simply an object $M$ of $\mathbf{H}$ together with a morphism $\alpha: M \otimes B \to M$ that satisfies the usual “unit” and “associativity” axioms. It’s the same as an algebra over the monad $- \otimes B: \mathbf{H} \to \mathbf{H}$.
The correspondence between orthonormal bases and Frobenius monoid structures sounds pretty cool; I was not aware of it before. I’d like to put a pin through that.
The correspondence between orthonormal bases and Frobenius monoid structures sounds pretty cool
This is something that the school of finite quantum mechanics in terms of dagger-compact categories has been amplifying for a while. Review includes Chris Heunen’s book def. 3.3.4 and example 3.3.5.
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