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Stream: theory: categorical probability

Topic: Tensor product of convex spaces

Nathaniel Virgo (Oct 26 2024 at 15:31):

This is a question I'm coming back to that I was stuck on a few years ago. A convex space is an algebra of the [[distribution monad]], and the intuition is that it's a set where you can form "mixtures" or of its points, so it behaves like a convex subset of $\mathbb{R}^n$, although there are plenty of examples that aren't of that form. Let's write $\bf Conv$ for the category of convex spaces.

Since the distribution monad is a commutative monad we can define the [[tensor product of algebras over a commutative monad]], and I'm pretty sure this exists in this case (because lots of people have said so), giving $\bf Conv$ a monoidal product.

What I'm stuck on is characterising this more concretely. I can unpack the definition, but it feels like it should simplify quite a lot in this particular case. I'd ideally like to be able to get a nice geometric picture in my head of what the tensor product looks like, and to be able to do basic calculations with them. (For example: what's the tensor product of the unit disk in $\R^2$ with itself?)

Does anyone know a good way to think about this, or know somewhere where it's written down?

Benedikt Peterseim (Oct 26 2024 at 16:15):

Is the usual construction of taking the free algebra on the cartesian product and modding out the “obvious relations” not concrete enough? Are you aiming for an intuition or a gadget to do calculations with (whatever they may be)?

Tobias Fritz (Oct 26 2024 at 16:16):

There's quite a bit of literature on tensor products like this, mostly either on tensor products of polytopes, which are an actual special case of the tensor product of convex spaces, or the more analytically flavoured tensor products of compact convex sets. Often this goes under the name of "minimal tensor product", since one can think of it as the smallest convex space containing all elementary tensors $x \otimes y$, and (at least for convex sets in vector spaces) there's also a dual notion of maximal tensor product. In terms of quantum mechanics, the minimal tensor product of two sets of quantum states gives you the set of separable states, so this may help somewhat with gaining intuition about it.

Tobias Fritz (Oct 26 2024 at 16:19):

More widely studied used is the analogously defined and intimately related minimal tensor product for convex cones. Somewhat in the same spirit is also the projective tensor product of Banach spaces, where the unit ball is spanned by all elementary tensors and which has the same universal property. Up to technical differences like taking the completion, this is also a tensor product of algebras for a commutative monad, namely that of linear combinations $\sum_i c_i x_i$ with coefficients such that $\sum_i |c_i| \le 1$, see totally convex spaces.

Tobias Fritz (Oct 26 2024 at 16:23):

In terms of mental visualization, I personally find that working with cones is easier than with convex sets/spaces, because the dimensions multiply as usual for tensor products. Working with cones rather than convex sets is often a good idea anyway.

Tobias Fritz (Oct 26 2024 at 16:38):

The tensor product of the unit disk in $\mathbb{R}^2$ with itself is not so easy to understand at all. It's going to be an 8-dimensional convex set with an intricate facial structure that should be similar to that of the convex set of mixed quantum states (density matrices) on a complex 4-dimensional Hilbert space, meaning the convex set of $4\times 4$ positive semidefinite matrices of trace 1.

Tobias Fritz (Oct 26 2024 at 16:41):

The tensor product of the unit square with itself is easier: it's an 8-dimensional polytope sometimes called the "CHSH polytope".

Ned Summers (Oct 29 2024 at 16:59):

Are there known convex spaces for which all tensor products agree, a la nuclear C*-algebras?

Tobias Fritz (Oct 29 2024 at 17:08):

I'm not sure what a maximal tensor product for convex spaces in general would be. But if we restrict to compact convex sets $C$ in finite-dimensional vector spaces, then the answer is known:

Theorem: Such $C$ is nuclear, in the sense that $C \otimes_{\min} D = C \otimes_{\max} D$ for all $D$ of the same kind, if and only if $C$ is a simplex.

Tobias Fritz (Oct 29 2024 at 17:09):

The "if" part is quite straightfoward. The "only if" direction was open for a long time and proven only quite recently by Aubrun, Lami, Palazuelos and Plávala in Entangleability of cones, using the more convenient language of convex cones.

Tobias Fritz (Oct 29 2024 at 17:11):

[fixed the reference]

Ned Summers (Oct 30 2024 at 12:07):

Oh interesting, thank you!

Benedikt Peterseim (Oct 30 2024 at 21:45):

I think finite-dimensional compact convex sets form a *-autonomous category (via polar duality / the “bi-polar theorem”). Wild guess, but could it be that the maximal tensor product from above is is the “dual tensor product” $((-)^*\otimes(-)^*)^*$? @Tobias Fritz

Benedikt Peterseim (Oct 30 2024 at 21:48):

That would be quite nice because then the result you cite above would mean that simple lives are exactly the “dualisable objects” in that symmetric monoidal category, I think. (Not sure what to make of that other than “curious fact”, though.)

Benedikt Peterseim (Oct 30 2024 at 21:50):

*simplices, not “simple lives” (autocorrect, sorry)

Tobias Fritz (Oct 30 2024 at 22:01):

Yes, that sounds right -- nice observation about the dualizable objects!

Tobias Fritz (Oct 30 2024 at 22:04):

I haven't seen this done anywhere for convex sets as such, but for convex cones there is some literature on the *-autonomy. I'm not sure what the best reference is, but it appears for example in the recent Beyond Operator Systems by my local Innsbruck colleagues.