The n-Category Café

Jonathan Vos Post wrote:

Similarly, the toy bit examples are individual points in a superspace of toy QM. Might it be useful to ask about them within *-algebras, as von Neumann had set up a playground for generalizations of QM?

It’s not a *-algebraic theory, or even in the set of convex theories investigated by Barnum and Co. It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.

If category theory has anything to say about this, particularly if it can significantly unify the different approaches to toy theories and/or generalizations of quantum theory, then I would definitely become a lot more interested in it.

Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.

I feel sure category theory would have interesting things to say about this theory, but only after I know what category Spekkens’ theory is talking about. I don’t yet know precisely what the state spaces are in his approach, and (more importantly) what the allowed maps between them are. He’s worked out the answers in some special cases, but I haven’t seen the general answer, and I’m not having much luck so far figuring it out. This is not a shortcoming of category theory: it’s shortcoming in my understanding of what his theory really says!

I should have tried hard to get him to explain this when I was in Les Treilles, but we were having too much talking about other things.

David wrote:

That’s an odd way of describing what de Finetti was doing with the concept of ‘exchangeability’. Remember he’s the one famous for ‘Probabilities do not exist’.

Interesting! But I can’t remember that, since I never knew it. I’d never heard of the de Finetti theorem until folks started telling me the quantum de Finetti theorem fails in quantum mechanics over the real numbers!

So, I must be foisting a wholly different interpretation on his theorem than he did. But hopefully I got the math of it right — though I only stated it for the quantum case.

In both the classical and quantum cases, it’s a characterization of mixed states that are mixtures of ‘independent identically distributed’ states.

Similarly, Fuchs and Caves are interested in exploring how much of the quantum formalism is describing one’s state of information rather than being about the world as it is in itself.

Yeah, I noticed that stuff when looking through their paper, but I tried to ignore it and dig out the mathematical ‘meat’: the theorem I stated here. I guess I was just in that kind of mood.

In fact, I am inclined to like that attitude.

While it is slightly bothersome that the Hestenes-Doran-Lasenby-et al. school keeps emphasizing how everything appears in a suggestive different light from their point of view, without (so far, as I am aware, but you will now immediately correct me) actually doing something with that which gets us to a point where we haven’t been before – while this is slightly bothersome, it doesn’t mean that they are not right!

It just means that somebody still may have to make that next step.

Something is going on, and nobody understands it yet. And it involves understanding what Clifford algebra really is and what supersymmetry really is. I claim. And I also claim that we know what anything really is once we know how it categorifies.

We have talked about that recently.

“Thin” refers to the fact that preorders are categories with hom-sets either singletons or empty. The order-structure, ie some special kind of orthomodular lattice, then captures the Hilbert space structure via Piron’s Thm. The “thick” varient has all linear maps in the Hom-set. Hilbert space is now captured by compact closure, which internalizes the Hom-sets as spaces (eg Deligne’s thm). Both approaches have a clear operational (or instrumentalist) connotation (eg Piron’s book for the case of lattices and my my notes for SMCs) but it’s not clear to me how they precisely relate, neither physically not mathematically. The obvious thing to do would be just to look at the category of those lattices but no notion of morphism seems to give the desired symmetric monoidal structure that allows to derive protocols such as quantum teleportation, provides trace structure etc.

Thanks for all the quaternionic QM references, Bob!

Bob wrote:

It is not clear to me how exactly this order-theoretic stuff relates to the *thick* categorical axiomatics for QM John mentioned above.

This is a fascinating puzzle! I ran a seminar on it once, but nobody took notes, and we didn’t get very far… Mike Stay got pulled over to working on the quantum lambda-calculus instead.

The basic idea went like this. Say we start from the ‘thick’ description in terms of a category $C$ of ‘state spaces’ and ‘processes’ — for example, the category $\mathrm{Set}$ of sets and linear operators, or $\mathrm{Hilb}$ of finite-dimensional Hilbert spaces and operators, or $n\mathrm{Cob}$ of compact oriented $(n-1)$-manifolds and cobordisms between them.

Next, say we want to get back the good old ‘thin’ description of logic in terms of posets, where the partial order is ‘implication’. We eventually want to get something like a lattice of propositions about the state for each ‘state space’ — that is, each object of $C$. But, we want all these lattices to fit together nicely!

Clearly, each object has a poset of subobject, and we can think of this partial order as ‘implication’. In some nice cases ($\mathrm{Set}$ or $\mathrm{Hilb}$) this poset will be a lattice. In some very nice cases of the ‘non-quantum’ or ‘cartesian’ flavor (e.g. $\mathrm{Set})$, it will be a Heyting algebra or even a Boolean algebra. In some very nice cases of the ‘quantum’ or ‘symmetric monoidal category with duals’ flavor (e.g. $\mathrm{Hilb}$), it will be an orthocomplemented lattice. And, in a bunch of these nice cases (e.g. both $\mathrm{Set}$ and $\mathrm{Hilb}$), we get a functor

$$F : C \to [lattices]$$

sending each object to its lattice of subobjects. Here the trick is making sure the morphisms in $C$ give lattice homomorphisms!

In nice cases, $F$ will be lax monoidal, so there will be a lattice homomorphism

$$F_{x,y}: F(x) \otimes F(y) \to F(x \otimes y)$$

This sends a pair $(P,Q)$ consisting of a proposition about the system $x$ and a proposition about the system $y$ to a proposition ‘$P$ and $Q$’ about the joint system $x \otimes y$. I guess a new thing about quantum systems is that $F_{x,y}$ need not be an isomorphism? That is, there are propositions about a joint quantum system that can’t be built from propositions of the form ‘$P$ (about the first system) and $Q$ (about the second)’ using the lattice operations?

One could have a lot of fun figuring out the precise conditions under which all these nice things happen. The ‘cartesian’ cases have been pretty much worked out already — a good place to start is Johnstone’s Elephant, with its discussion of regular categories and the like. So, the fun lies in a similar investigation of the ‘symmetric monoidal category with duals’ cases. Here one important trick is to focus, not on arbitrary subobjects, but on ‘isometric’ ones, i.e. monomorphisms $f: x \to y$ such that $f^* f = 1_x$. Clearly one needs to do something like this to get an orthocomplemented lattice.

If I could clone myself, I’d set one copy working on this project: the reconstruction of ‘thin’ quantum logic from the ‘thick’ categorical semantics of quantum systems. I do my best by having lots of grad students. But, I can’t clone my quantum state!

Maybe Isar Stubbe would be interested in this project. He’s one of the few people who knows all the relevant stuff.