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Tim Hosgood (Jan 09 2021 at 18:41):here's a very vague and barely-thought-out question that i'm hoping somebody will reply to with either an answer or a better version of my question: when dealing with quantum channels (or super-operators, or CPTP maps, or whatever you want to call them), it is a nice fact that we can always decompose them into three parts:
Now, i'm very much not a quantum information scientist, nor am i really a category theorist (in fact, what am i?), but this sort of decomposition looks very categorical to me. I know that categorical quantum mechanics is a hot topic, but i've only ever seen the categories show up in terms of string diagrams and such like. What's happening here looks a bit more like some sort of "model category" thing, or maybe something closer to an exact sequence...
Tim Hosgood (Jan 09 2021 at 18:41):i guess there's not really a question in the above (so i've become one of those people who, during the question time after a talk, says "this is really more of a comment...", sorry!), but i guess i'm just looking for comments rather than answers then :-)
Cole Comfort (Jan 09 2021 at 18:45):Given any -compact closed category, there is a notion of category of quantum channels, called the CPM construction. This was first discovered in the following paper of Selinger https://ncatlab.org/nlab/files/SelingerPositiveMaps.pdf
Tim Hosgood (Jan 09 2021 at 18:50):that's a really interesting read, thanks!
Sam Staton (Jan 09 2021 at 20:13):In addition to the various CP constructions that Cole mentioned, the factorization that you mentioned has been studied in terms of different universal properties by A & B Westerbaan and Arthur Parzygnat. Arthur gave a talk about it in the MIT series not long ago.
My own simple position, with Mathieu Huot, is this. Say a monoidal category is "affine" if its unit is a terminal object. For any monoidal category C, there is an "affine reflection" LC, which is the universal category with a terminal unit and a monoidal functor C->LC. Now here's a theorem: the category of CPTP maps (your parts 1, 2 and 3) is the affine reflection of the category of isometries (just parts 1 and 2). I quite like this because it matches my intuitions from physics: a terminal unit means that you can hide things, and the passage from pure quantum (isometries) to quantum channels (CPTP) arises precisely from hiding things in a free way. And it turns out that the affine reflection of _any_ monoidal category has factorizations like your 1,2,3.
There has been some work on connecting all these different views, but I think there is more to be done.
Tim Hosgood (Jan 09 2021 at 21:44):seeing universal properties turn up is also exactly the sort of thing that i was looking for, so thank you!
Tim Hosgood (Jan 09 2021 at 21:46):i still feel that there is something hidden in the fact that 1. looks a bit like a cofibration, 2. like a weak equivalence (since unitary "=" time invertible), and 3. looks a bit like a fibration, but i have nothing more concrete than a sneaking suspicion...
John van de Wetering (Jan 10 2021 at 13:59):Cole and Sam's comments show how you can go from a category to a bigger one where the maps have this sort of decomposition. However, it is also possible to "recognize" categories that arise in these ways via environment structures. When you know even more about the category this is even sufficient to characterise it as a "category of matrices": https://arxiv.org/abs/1804.02265
Oscar Cunningham (Jan 10 2021 at 21:25):Also related might be my paper with Chris Heunen where we show that deterministic and random channels form a weak factorisation system. https://arxiv.org/abs/1705.07652
Tim Hosgood (Jan 10 2021 at 22:20):Oscar Cunningham said:
Also related might be my paper with Chris Heunen where we show that deterministic and random channels form a weak factorisation system. https://arxiv.org/abs/1705.07652
oh this is very much an answer to my vague question, thank you!