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Duyal Yolcu (Sep 11 2022 at 18:46):Hi, I have a question that hopefully has a short answer. I am interested in roughly the following categories to describe classical, pure-quantum and mixed-quantum "probability theories" respectively:
a) Objects are measurable spaces, morphisms are substochastic transition kernels (i.e. transitions are allowed to "lose probability mass"),
b) Objects are Hilbert spaces, morphisms are linear operators with Hilbert-Schmidt norm (i.e. operators are allowed to "lose amplitude"/get the system into a not-further-considered state with a certain probability),
c) Objects are -algebras with trace operators, morphisms are completely positive maps that don't increase the trace when acting on nonnegative operators.
My question: What are there literature name of these objects that I should use and (where) can I find a textbook-style discussion? I am particularly interested in whether/how one can define tensor products/direct sums. When I google around, I find scattered references to what seem to be things like that, but no unified treatment, and mostly in papers/blogposts.
In the next step, I'd like to do something with these categories (more specifically, construct the "states of knowledge" I am interested in in a more abstract way than I am currently doing). But that's separate from finding a definition of these categories in the first place.
To the extent that allowing infinite-dimensional spaces makes things more complicated, I am only interested in the finite dimensional case.
Simon Burton (Sep 12 2022 at 09:01):Are you familiar with the standard texts on categorical quantum mechanics?
Duyal Yolcu (Sep 12 2022 at 09:55):Simon Burton said:
Are you familiar with the standard texts on categorical quantum mechanics?
Thanks! This may be the short answer I was hoping for for now. I am looking through the first text I found now; I didn't yet find how to formalize the "morphisms must be norm-non-increasing" part, but it hopefully appears as well. Maybe it isn't even necessary for what I want to do.
For reference, ignoring the norm-non-increasing constraint and considering only finite-dimensional spaces, a) is , b) is , c) is according to 7.3.1 and 7.29 in that text.
(Update: and following the answer below and looking here and here, and are designed for classical probability distributions+transition matrices whose columns sum/integrate to exactly 1 (as opposed to at most 1; this is a long and apparently educative paper attempting to create a "Markov category" for quantum processes).
Simon Burton (Sep 12 2022 at 10:51):Ok great. I very much like the Heunen & Vicary text. There's also another book by Kissinger & Coecke, "Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning".
Plenty of other people here often talk about "Markov categories", which may also connect with what you are asking about.
John van de Wetering (Sep 13 2022 at 19:58):Hi Simon, definitely check out effectus theory. It was specifically designed to unify those three categories that you name. It also covers infinite-dimensional settings like the category of von Neumann algebras.
A good place to start would be Bas Westerbaan's PhD thesis (if you want a focus more on the quantum side) or Kenta Cho's PhD thesis (if you want a focus more on the logical side of things).
John van de Wetering (Sep 13 2022 at 20:00):The way you get from the categories of CQM to a setting where you have norm-non-increasing maps (I prefer the term sub-unital) is that you identity causal structure in the category in the form of discard maps. Then if you assume the category is enriched in semi-rings, so that you can add morphisms together, you can restrict to the category of maps that "lie below" the discard map (where the order relation is defined by iff there is an such that .
John van de Wetering (Sep 13 2022 at 20:01):While direct sums are native to effectus theory, although they appear in the form of coproducts in their particular setting, tensor products are less natural (although they can still be defined)