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Stream: theory: categorical probability

Topic: Thesis of Nicholas Gauguin Houghton-Larsen

Tobias Fritz (Mar 20 2021 at 19:14):

To those of you who are interested in categorical probability, categorical quantum mechanics, and/or potential applications of these to information processing, cryptography or causal inference, I'd like to advertise the recent thesis of Nicholas Gauguin Houghton-Larsen at the University of Copenhagen. It's here:

https://arxiv.org/abs/2103.02302

This document is a treasure trove of ideas and results that may be of interest for years to come, and it's quite a pleasant read on top of that. Unfortunately, Nicholas may be about to leave the scientific community, and it is unclear whether any of this will appear in any other form than in the thesis. I'm therefore trying to advertise this material a bit in order to prevent it from going unnoticed, especially as Nicholas has not known about the ACT community until very recently.

As a teaser, here's a selection-biased summary of the things that I've personally found the most interesting:

• Chapter 2 shows that one can already do a surprising amount of categorical probability and categorical quantum mechanics in any semicartesian monoidal category satisfying a bunch of natural axioms. These axioms are then used to prove an entirely new generalization of the no-cloning theorem as Theorem 2.5.18, which in addition also specializes to the recent no-masking theorem and other results of this type. I'm already using some of the chapter's theory of dilations in my own work.
• Chapter 3 considers the extension to categories enriched in metric spaces and proposes axioms for how the metric ought to be compatible with dilations. It applies these definitions and results in order to develop a new notion of distance of quantum channels.
• Chapter 4 is the heart of the thesis. It develops a categorical framework for the joint analysis of causal structure and information leaking to and from the environment. Among other things, this involves a generalization of traced monoidal categories that seems to be new. It clearly provides a convincing categorical setting for causal inference. I also wonder if this framework could serve as a foundation for a categorical approach to cryptography, at least insofar as the analysis of information flow and information leakage is concerned.

So if you know of any others who may be interested in this sort of thing but may not read this forum, please spread the word!

Jules Hedges (Mar 20 2021 at 19:43):

By chapter 2, do you mean that it's possible to something similar to Markov categories but without the comultiplication/copy morphisms?

Tobias Fritz (Mar 20 2021 at 19:58):

I just mean that it's been surprising to me personally that one can do anything that vaguely resembles probability theory or a nontrivial theory of information flow with semicartesianness only. I can't say for sure that one can do as much with it as one can do with Markov categories, but I believe that the answer is yes, in the following (uninteresting) way.
My hypothesis is that suitable versions of Nicholas's axioms, in particular universal dilations, will imply that copy morphisms exist, with the idea being that copying and leaking one copy is the universal way of leaking information to the enviroment. If this conjecture is true, then being a Markov category is morally an extra property of a semicartesian category rather than extra structure. (Although it technically is extra structure, that extra structure is unique in typical examples and behaves more like extra property.) Does this sound plausible?

Jules Hedges (Mar 20 2021 at 20:09):

Interesting...

John van de Wetering (Mar 21 2021 at 12:48):

Thanks for pointing out this thesis! I haven't gotten very far yet, but I already feel like I'm learning new things about dilations

John van de Wetering (Mar 21 2021 at 13:05):

It does look like he's missing several references to work that I'd say should be of interest to him, like the symmetric purification of Selby et al. https://arxiv.org/abs/1802.00367 or the categorical reconstruction of Tull that uses dilations in a more abstract categorical section to derive something very close to quantum theory https://lmcs.episciences.org/6035, or the theory of infinite-dimensional dilations of Westerbaan^2 https://arxiv.org/abs/1603.04353v2

Tobias Fritz (Mar 21 2021 at 16:06):

Thanks @John van de Wetering for pointing those out! I've forwarded that to Nicholas and encouraged him to also drop by here and say hi. (And I had recently already pointed out the Westerbaan^2 paper to him as well as @Arthur Parzygnat's Stinespring's construction as an adjunction, which seem to be of a similar flavour as the thesis's Theorem 2.4.11. It's a bit unfortunate that he missed all of this literature which arguably should have been referenced and discussed, but also not very surprising given that he was embedded in a different community.)

Jules Hedges (Mar 21 2021 at 16:09):

This is a reminder of what the situation was like for everybody only a few years ago, before the amazing community building work by the organisers of ACT1

Fabrizio Genovese (Mar 21 2021 at 16:44):

Yeah, personally I'm legendarily bad at the reference game, so a community like this (and a place like Zulip) are fundamental for me to have a vague idea of what's going on in ACT. I'd have no hope otherwise

Nicholas Gauguin Houghton-Larsen (Mar 22 2021 at 10:53):

Hi all,

First of all many thanks @Tobias Fritz for promoting my thesis and introducing me to this community!

As Tobias wrote I will probably soon be leaving academia, but I am planning to first write a few papers based on the thesis. It has been very uplifting to me that this work may be of interest more broadly. If you wish to discuss elements of the thesis, or have other comments or corrections, please feel free to contact my by email (nicholas.gauguin@gmail.com).

Many thanks @John van de Wetering for pointing out those references, I'll consult them before future presentations of the material. It was never an explicit objective for me to reconstruct quantum theory axiomatically, but it is of course a natural problem to consider in this context.

Best regards,
Nicholas

Nicholas Gauguin Houghton-Larsen (Mar 22 2021 at 10:56):

Tobias Fritz said:

I just mean that it's been surprising to me personally that one can do anything that vaguely resembles probability theory or a nontrivial theory of information flow with semicartesianness only. I can't say for sure that one can do as much with it as one can do with Markov categories, but I believe that the answer is yes, in the following (uninteresting) way.
My hypothesis is that suitable versions of Nicholas's axioms, in particular universal dilations, will imply that copy morphisms exist, with the idea being that copying and leaking one copy is the universal way of leaking information to the enviroment. If this conjecture is true, then being a Markov category is morally an extra property of a semicartesian category rather than extra structure. (Although it technically is extra structure, that extra structure is unique in typical examples and behaves more like extra property.) Does this sound plausible?

@Tobias Fritz I think this sounds very plausible :)

John van de Wetering (Mar 22 2021 at 11:00):

@Nicholas Gauguin Houghton-Larsen My primary reason for mentioning those two reconstruction results is because the way they use dilations seems to me to be more closely aligned to how you do it. In Tull's paper, he defines purity as dilational purity, as opposed to convex purity in the Pavia reconstruction, and in Selby et al. they use a symmetric dilation that holds for all processes, and not just states, and which is also true for classical processes and not just the quantum ones.

John van de Wetering (Mar 22 2021 at 11:00):

Anyway, let me congratulate you on an excellent thesis. I'm finding it a joy to read :)

Nicholas Gauguin Houghton-Larsen (Mar 22 2021 at 11:48):

@John van de Wetering I'm pleased that you like it!
And I see, that is of course relevant. I think I only reference Chiribella (Def. 2.4 in https://arxiv.org/pdf/1411.3035.pdf) when comparing to other categorical notions of purity.
Note, incidentally, that his purity notion (which agrees with Tull's as far as I can see) is slightly different from the one I employ, due to their restriction to one-sided dilations. For example, in the category of surjective functions, a bijection has only trivial one-sided dilations (i.e. is pure in their terminology) but in general has non-trivial two-sided dilations (i.e. is not pure in my terminology).
This is briefly mentioned in footnote 7 on page 49 of the thesis.

Martti Karvonen (Mar 22 2021 at 14:42):

There's also this approach to purification: https://arxiv.org/abs/1705.07652

Oscar Cunningham (Mar 22 2021 at 14:50):

That paper is actually referenced. :smile:

Martti Karvonen (Mar 22 2021 at 19:55):

Oh oops

Nicholas Gauguin Houghton-Larsen (Mar 23 2021 at 09:17):

@Martti Karvonen Absolutely, my apologies. I guess I meant "when comparing to purity notions based on dilations".
@Oscar Cunningham Maybe I missed a journal reference, which I'll be happy to give in the future (I reference an arXiv-version) :)

Oscar Cunningham (Mar 23 2021 at 09:37):

I'm not fussed, but it's in the conference proceedings here http://eptcs.web.cse.unsw.edu.au/paper.cgi?QPL2017.20 if you want to cite the 'real' version.