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September 19, 2021

Axioms for the Category of Hilbert Spaces

Posted by Tom Leinster

Chris Heunen and Andre Kornell have found an axiomatic characterization of the category of Hilbert spaces!

Chris Heunen and Andre Kornell, Axioms for the category of Hilbert spaces. arXiv:2109.07418, 2021.

A bit more precisely, they axiomatize the monoidal dagger category of Hilbert spaces: Hilb\mathbf{Hilb} equipped with the operations of tensor product and taking the adjoint.

Their paper appeared on the arXiv this week, and uses Solèr’s theorem, which John wrote about here in 2010.

I don’t have time to write more, but I wanted to make sure that Café readers don’t miss this treat.

Edit: Chris has now written a more substantial post on his and Kornell’s paper.

Posted at September 19, 2021 11:20 AM UTC

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Re: Axioms for the Category of Hilbert Spaces

There is a typo in the article. When defining the dagger on page 1, the condition should be id H =id Hid_H^\dagger = id_H, not id H =id Aid_H^\dagger = id_A.

Posted by: Madeleine Birchfield on September 19, 2021 3:20 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

Is that a dagger category I see before me?

Posted by: Swill Stroganoff on September 19, 2021 5:40 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

Chris Heunen nicely summarized the paper on Twitter, and said:

The axioms are purely categorical. Physically, you could say they are in terms of systems and how they interact only. The million dollar question: do the axioms have a physical interpretation? Or can you rephrase them so that they do? I’d love to hear your thoughts!

By the way, the axioms characterize the monoidal dagger-categories of complex or real Hilbert spaces and bounded linear operators. You need an extra axiom to pick out one or the other of these categories. For the real case you can require f =ff^\dagger = f for every endomorphism of the tensor unit. For the complex case you can require that this is false — or if you prefer, that there’s an endomorphism ff of the tensor unit with f 2=1f^2 = -1.

Posted by: John Baez on September 20, 2021 12:55 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

Yes, it took me a while to appreciate that they were actually characterizing two categories. (We need a picture of two amigos.) So the theorem is:

Every monoidal dagger category satisfying such-and-such conditions is equivalent to either Hilb \mathbf{Hilb}_\mathbb{C} or Hilb \mathbf{Hilb}_\mathbb{R}.

Posted by: Tom Leinster on September 20, 2021 10:20 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

I wonder if there’s another way of distinguishing Hilb \mathbf{Hilb}_\mathbb{C} from Hilb \mathbf{Hilb}_\mathbb{R}.

There’s a nontrivial involution of the category Hilb \mathbf{Hilb}_\mathbb{C}, right? Given a Hilbert space HH, we get a new one, H¯\overline{H}, which is the same as HH except for two things. First, scalar multiplication is twisted by complex conjugation: to scalar-multiply in H¯\overline{H}, you conjugate the scalar and then scalar-multiply in HH. Second, the order of the inner product is reversed. This should mean that the inner product on H¯\overline{H} is still sesquilinear.

But I don’t see how there could be a nontrivial involution of Hilb \mathbf{Hilb}_\mathbb{R}.

This is off the top of my head. I may be confused.

Posted by: Tom Leinster on September 20, 2021 10:30 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

I think that the authors briefly refer to this point, saying that one can distinguish between these two cases (R vs C) by postulating a nontrivial involution. But I am going from a vague memory of a brief skim!

Posted by: Yemon Choi on September 20, 2021 5:44 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

Oops, ignore my previous comment – I see that the quote upthread from Chris H. already addresses this, and that you are asking about something different (an involution on the category itself, not on the unit of the monoidal structure).

Posted by: Yemon Choi on September 20, 2021 5:46 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

I think you’re right, Tom, that Hilb \mathsf{Hilb}_{\mathbb{C}} has a covariant involution sending each complex Hilbert space to its ‘conjugate’ space, defined as you explained.

There’s also a contravariant involution on Hilb \mathsf{Hilb}_\mathbb{C} sending each Hilbert space to its dual, and each morphism f:HKf: H \to K to its adjoint f *:K *H *f^\ast: K^\ast \to H^\ast. This is an involution up to natural isomorphism.

The interaction between these two involutions creates endless fun and confusion.

Posted by: John Baez on September 21, 2021 12:08 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

It’s not just endless fun and confusion, but it’s what gives the dagger structure, isn’t it?

That is, if you apply the contravariant involution and then the covariant one, you can identify the conjugate of the dual with the original space (the Riesz-Frechet theorem), and an arrow f:HKf:H \to K gets sent to f :KHf^\dagger: K\to H, which is the “operator algebraists’ adjoint”.

Is there thus a “Klein-four” action on this category?

Posted by: Yemon Choi on September 21, 2021 1:09 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

All very interesting! And peculiarly unrelated-seeming to the other ways that the “reconstruction of quantum theory” genre has found to select \mathbb{C} instead of \mathbb{R}. The most common, I think, is to postulate what’s called “local tomography”; another is to posit that the generators of time evolution are conserved “observables”.

Posted by: Blake Stacey on September 22, 2021 10:54 PM | Permalink | Reply to this

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