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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

Re: Axioms for the Category of Hilbert Spaces

We recently found a new way to exclud real number quantum theory which doesn’t need the local tomography hypothesis. It is based on concrete observation of some correlations in a Bell type experiement:

Posted by: Marc-Olivier Renou on November 4, 2021 10:50 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

Thanks for pointing this out! I see that a paper covering an actual experiment has been published already (covered here), but I cannot find the published version of your paper.

Posted by: David Roberts on November 4, 2021 3:27 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces

There’s an assumption discussed in the conclusion of arXiv:2101.10873 that I found of interest, that “the independence of two or more quantum systems is captured by the tensor product structure”. To write a complex matrix as a real matrix, Renou et al. introduce a little extra dimensionality, but it’s not really a degree of freedom — it’s not dynamical, but rather only notational. They then assume that these embiggened matrices then compose by the tensor product. As they note, it’s possible to drop that assumption; what I wonder is whether there might be a positive argument to do so. In other words, does the tensor-product assumption properly reflect the physical criteria for compositing systems, like the impossibility of instantaneous signalling? Or do those physical requirements have a different implication for a theory set up using nondynamical auxiliary qubits? Is Eq. (3) of Renou et al. the right way to go even for a bipartite system, never mind three or more portions?

Suppose that Alice has two systems on opposite sides of her laboratory. Each system she treats individually using quantum theory, describing her preparation of the left-hand system by a density matrix ρ L\rho_L and that of the right-hand system by ρ R\rho_R, modeling the measurements she might perform by POVMs, etc. Then physical constraints on signalling imply that effects combine by the tensor product while the possible joint states for the LRL R pair are “positive on pure tensor” operators: The probability of obtaining the pair of outcomes E,FE, F on the left and right systems is tr[(EF)W LR]\mathrm{tr}[(E \otimes F)W_{L R}] for some Hermitian operator W LRW_{L R} that is unit-trace but not necessarily positive semidefinite. Moreover, we can find some other effects E˜,F˜\tilde{E}, \tilde{F} and some joint density matrix ρ LR\rho_{L R} such that the probability of getting that pair of outcomes is tr[(E˜F˜)ρ LR]\mathrm{tr}[(\tilde{E} \otimes \tilde{F}) \rho_{L R}]. In other words, correlations that respect no-signalling and “local quantumness” can be emulated quantum-mechanically.

But what if we change our mathematical expression of being “locally quantum” to use embiggened real matrices? It seems to me that, from more primitive assumptions like these, one might conclude that the way to compose embiggened matrices is to extract the complex matrices, tensor-product those and embiggen the result. In other words, the way to have the theory be physically local in the way it should be is to have one generally ambient notational qubit floating about the equations. A “gambit” is somewhat like a ubit, but it doesn’t interact or have its own rotation rate; it just sits there like a cat in a sunbeam.

On the one hand, this sounds like a lengthy gloss on Renou et al.’s penultimate paragraph (“If we drop this assumption…”). On the other, there might be a little room here to explore (paraphrasing Pitowsky) the distinction between mathematical artifact and physical content.

My favorite reason to exclude real-number quantum theory is that a complete set of equiangular lines exists in 4\mathbb{C}^4 but not in 4\mathbb{R}^4. There’s some interesting number theory in that.

Posted by: Blake Stacey on November 8, 2021 3:21 PM | Permalink | Reply to this

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