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Stream: event: Categorical Probability and Statistics 2020 workshop

Topic: Jun 8: Bart Jacobs' talk

Paolo Perrone (Jun 04 2020 at 19:18):

Hey all,
This is the discussion thread of Bart Jacobs' talk, "De Finetti's construction as a categorical limit".
The talk, besides being on Zoom, is livestreamed here: https://youtu.be/wsSpaIWqszQ

Paolo Perrone (Jun 04 2020 at 19:22):

Date and time: Monday, 8 Jun, 13h UTC.

Paolo Perrone (Jun 08 2020 at 13:05):

(the talk has started)

Robert Furber (Jun 08 2020 at 13:37):

:clap: (sorry Bart, I can't get on zoom)

Arthur Parzygnat (Jun 08 2020 at 13:57):

Ah, it doesn't look like Bart Jacobs is here on the zulip :( . I was definitely curious about the quantum case, even for qubits. Does anyone know if Bart said he knew what the objects and maps were but he was not sure about whether a universal property holds or not? It's difficult to use the Kleisli category in the quantum setting, but one might be able to work more directly with CPU maps instead? I wonder what all the data would be...

Sam Staton (Jun 08 2020 at 14:15):

Hi Arthur, Do you know the literature on quantum de Finetti? One option would be to consider a diagram $X: \mathit{Inj} \to \mathit{FdCStar}$ given by $X(n)=A^{\otimes n}$ for some algebra $A$, and see if it has a colimit in CPU? Here $\mathit{Inj}$ is natural numbers and injections.

Arthur Parzygnat (Jun 08 2020 at 14:22):

Thanks, Sam. I'm aware of the existence of a quantum de Finetti theorem, but have not learned it yet. I suspect it would encode states as well.

Rob Spekkens (Jun 08 2020 at 14:30):

A good reference on the quantum deFinetti theorem is this article from 2001
Carlton M. Caves, Christopher A. Fuchs, Ruediger Schack
Unknown Quantum States: The Quantum de Finetti Representation
J. Math. Phys. 43, 4537 (2002)
arXiv:quant-ph/0104088 (quant-ph)
It is more straightforward than the approach taken by earlier work on the subject by Hudson and Moody (Z. Wahrschein. verw. Geb. 33, 343 (1976))

Sam Staton (Jun 08 2020 at 15:56):

Thanks Arthur Parzygnat, Rob Spekkens, I think the Hudson/Moody set-up is indeed talking about a cone over this $\mathit{Inj}$-indexed diagram $A\rightrightarrows A\otimes A\rightrightarrows A^{\otimes 3}\rightrightarrows \dots$ in the category of CPU maps, with apex $\mathbb{C}$. Whether the theorem is saying that this diagram has a colimit in a category of C star/W star-algebras would be a bit of work to check, but it would be interesting to know.

Robert Furber (Jun 08 2020 at 16:07):

Sam, see here: https://www.sciencedirect.com/science/article/pii/0022123669900500

Robert Furber (Jun 08 2020 at 16:15):

Also Sam, taking colimits in W-star-algebras is very much unlike doing it for C*-algebras - see my post here: https://mathoverflow.net/a/284325

I can explain somewhere else what thing is analogous to taking the limit in measure spaces, if you want.

Sam Staton (Jun 08 2020 at 16:38):

Hi Robert Furber! Are you saying that the categorical fact is basically in Størmer's paper, or just that someone who wants to work it out should read that first?
I agree it might be best to use W star algebras, thanks for the pointer.

Robert Furber (Jun 08 2020 at 16:51):

Sam, I was trying to say the opposite- you don't want to take the colimit in W-star-algebras - the W-star-algebra you do want is not the colimit. In that specific example, the W*-algebra we want is L^\infty(2^\omega) with the coin-flipping measure, but we _don't_ get anything like that - we get a W*-algebra with no normal representation on a separable Hilbert space. I have tried to explain this to many people, but my words are always misinterpreted.

Robert Furber (Jun 08 2020 at 16:59):

@Sam Staton Regarding the Størmer paper - what he proves is that the compact convex set of states on an infinite tensor product of a unital C*-algebra A (e.g. M_2) that are invariant under finite permutations is isomorphic to the Radon measures on the state space of A (not easy to explain this in the chat). Of course, it is not just any old isomorphism, but the one that comes from mapping a state \rho to \rho \otimes \rho ... and extending by convexity and continuity.

Robert Furber (Jun 08 2020 at 17:30):

@Sam Staton The quantum de Finetti papers suggested by Rob Spekkens prove sharper statements because they actually deal with the finite-dimensional approximation part of the problem, and this is something that I think genuinely occurs in practice (applications of quantum de Finetti to quantum protocols). But the Størmer paper is closer to doing it by taking a colimit.

Paolo Perrone (Jun 08 2020 at 19:31):

Hi all! Here's the video.
https://youtu.be/OYtmXiRzAXo