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Paolo Perrone (Jun 04 2020 at 19:16):Hey all,
This is the discussion thread of Arthur Parzygnat's talk, "Categorical probability in the quantum realm".
The talk, besides being on Zoom, is livestreamed here: https://youtu.be/Ur6vxdRKr9g
Paolo Perrone (Jun 04 2020 at 19:23):Date and time: Sunday, 7 Jun, 15h UTC.
Arthur Parzygnat (Jun 07 2020 at 07:25):The slides to my talk are available here. And here's a version without slide-repetitions caused by the \pause command.
Javier Prieto (Jun 07 2020 at 13:40):@Arthur Parzygnat thanks for sharing them in advance! I think your copy and discard morphisms in the definition of a quantum Markov category are going the opposite way - or am I missing something?
Arthur Parzygnat (Jun 07 2020 at 13:41):They look like they're going in the opposite direction, but because I'm working with algebras, that's like the opposite of the category of spaces so morphisms get reversed.
Arthur Parzygnat (Jun 07 2020 at 13:43):Oh I see, I think you're referring to why I call them copy and discard perhaps. Yeah, maybe "copy" isn't the best word. Think multiplication.
Javier Prieto (Jun 07 2020 at 13:46):Yeah, I was confused by the name too. I see now in slide 23 that the relevant subcategory is equivalent to _the opposite_ of FinStoch, so everything's going in reverse.
Paolo Perrone (Jun 07 2020 at 14:40):Hi all! The talk will start in 20 minutes.
Manuel Bärenz (Jun 07 2020 at 15:17):Hi @Arthur Parzygnat thanks a lot! I think your categories are related to (special cases of?) "involutive monoidal categories". There are some references here:
Most of these works are in a more general, not necessarily symmetric setting, but maybe there is still some interesting overlap. In particular, I find this formulation the most enlightening way to think about antilinear maps.
Robert Furber (Jun 07 2020 at 15:50)::clap:
Arthur Parzygnat (Jun 07 2020 at 16:16):Manuel Bärenz said:
Hi Arthur Parzygnat thanks a lot! I think your categories are related to (special cases of?) "involutive monoidal categories". There are some references here:
- Bart Jacobs: http://www.cs.ox.ac.uk/people/bob.coecke/PDFS/11-Jacobs.pdf
- Jamie Vicary: https://arxiv.org/pdf/0805.0432.pdf
- Jeff Egger: http://www.tac.mta.ca/tac/volumes/25/14/25-14abs.html
- My thesis (second part): https://www.manuelbaerenz.de/files/Thesis.pdf
Most of these works are in a more general, not necessarily symmetric setting, but maybe there is still some interesting overlap. In particular, I find this formulation the most enlightening way to think about antilinear maps.
Thank you for the references! I'll be sure to check these out and compare the definitions!
Arthur Parzygnat (Jun 07 2020 at 16:17):@Juan Pablo Vigneaux I was curious about the question you asked me at the end. Do you happen to know any specific physical systems for which we would like to analyze? I've been working through a few examples but so far they're quite elementary and I'd be super interested in examining more! (Also, I just saw your thesis is about entropy. I'm excited to check it out!)
Arthur Parzygnat (Jun 07 2020 at 16:56):@Rob Spekkens Thanks for your question! I want to write it here for the record: How is the Bayesian inverse of a map and a convex combination of states related to a Bayesian inverse of that map together with the two states comprising that convex combination? (Similar questions could be asked about convex combinations of the maps as well.)
Tomáš Gonda (Jun 07 2020 at 16:59):Arthur Parzygnat said:
Rob Spekkens Thanks for your question! I want to write it here for the record: How is the Bayesian inverse of a map and a convex combination of states related to a Bayesian inverse of that map together with the two states comprising that convex combination? (Similar questions could be asked about convex combinations of the maps as well.)
I wonder if this property can be stated more categorically as a structure-preservation property for a Bayesian inversion (partial) functor?
Alex Simpson (Jun 07 2020 at 17:06):That was a very nice talk, Arthur. I wanted to ask the following question. Are there interesting models in which almost everywhere equality coincides with actual equality? Or is it at least consistent to have such coincidence? Can one, for example, build state into the objects by considering the co-slice category 1/C, whose objects are maps from the terminal object - I'm using the standard Markov category orientation - and quotient that by almost everywhere equality? Does this quotient retain the category theoretic structure?
Arthur Parzygnat (Jun 07 2020 at 18:35):Alex Simpson said:
That was a very nice talk, Arthur. I wanted to ask the following question. Are there interesting models in which almost everywhere equality coincides with actual equality? Or is it at least consistent to have such coincidence? Can one, for example, build state into the objects by considering the co-slice category 1/C, whose objects are maps from the terminal object - I'm using the standard Markov category orientation - and quotient that by almost everywhere equality? Does this quotient retain the category theoretic structure?
Thank you! Yes! Actually, I was secretly working with a co-slice category in some of the definitions without saying so. And yes, the quotient does remain a category! In fact, it works for all the notions of positivity I talked about. I can't remember if it works for all linear maps, but composition of a.e. equivalence classes of positive unital maps is well-defined (this was proved in the 2019 paper with Russo that I mentioned). There are special cases when a.e. equivalence coincides with equality. One case is when the state has full support (trivial null-space), which is not very surprising, but at least satisfying. Another situation that's interesting is the following: if a positive unital map is a.e. equivalent to the identity on a matrix algebra (not a direct sum), then it is a.e. equivalent to the identity. I suspect much more interesting phenomena occurs in the infinite-dimensional setting, though we are only beginning to study these things in that setting more recently.
Alex Simpson (Jun 07 2020 at 19:42):@Arthur Parzygnat Thank you for the answer. You say " And yes, the quotient does remain a category! ". I am hoping it additionally remains a category with all the relevant structure (i.e. the relevant version of the Markov category structure).
Juan Pablo Vigneaux (Jun 07 2020 at 21:17):Arthur Parzygnat said:
Juan Pablo Vigneaux I was curious about the question you asked me at the end. Do you happen to know any specific physical systems for which we would like to analyze? I've been working through a few examples but so far they're quite elementary and I'd be super interested in examining more! (Also, I just saw your thesis is about entropy. I'm excited to check it out!)
I'm not an expert in quantum probability, but I looked at the literature because I was trying to extend my cohomological computations to quantum probabilities. More precisely: in my thesis, I showed that the only real-valued functions of gaussian laws that satisfy the chain rule are
1) Shannon entropy, and
2) the dimension of the support.
(The precise formulation uses presheaves, because one also need the functions to be "local" in an appropriate sense: the entropy of X should depend only on the law of X.)
I wanted something similar for quantum gaussians. Immediately, a notion of conditional probability is needed (for quantum gaussians or more general quantum random variables); in fact, I was looking for disintegrations, which directly give regular versions of conditional probabilities. These conditional probabilities appear in the cohomological formalism through an action of random variables on probabilistic functionals (if you don't like cohomology, just consider an alternative question: what is conditional entropy for noncommuting observables?).
As I mentioned in the talk, some people claim that quantum conditional probabilities of noncommuting observables do not make sense. The main reference I found is: http://eprints.ucm.es/10636/ When you showed your last result (if F has a disintegration then F is deterministic a.e.), I wonder if it was a partial justification of that claim ("there is a conditional probability of X given Y only if Y commutes with X").
Anyway, some general definitions of quantum conditional expectation appear in the literature. For instance, https://arxiv.org/pdf/math/0009023.pdf introduces a conditional expectation that in particular gives a disintegration in your sense (but a priori the condition there is stronger). Then the author computes the conditional moments of certain standard operators defined on the q-Fock space (Corollary 2).
It would be also interesting to study the notion of quasi-Conditional expectation introduced by Accardi in the context of noncommutative Markov chains (see this attachment).
A big motivation for quantum conditional probabilities is the quantum Bayes rule and a quantum version of belief propagation. I guess @Rob Spekkens would be able to summarize the state of the art in that area.
Paolo Perrone (Jun 07 2020 at 23:05):Hi all! Here's the video.
https://youtu.be/vDrppYDUOHo
Sam Staton (Jun 08 2020 at 09:34):Hi Arthur Parzygnat, nice talk. I have a question about the first part. I know CPU maps have a physical motivation. Do you have any physical motivation for the supercategories of CPU maps that you considered?
Rob Spekkens (Jun 08 2020 at 14:21):Hi Arthur, sorry I couldn't respond earlier. First, let me say that I really enjoyed your talk. The background to my question is the analysis found in this article, which I mentioned briefly in my talk:
Dominic Horsman, Chris Heunen, Matthew F. Pusey, Jonathan Barrett, Robert W. Spekkens
Can a quantum state over time resemble a quantum state at a single time?
Proc. R. Soc. A 473(2205), p. 20170395 (2017)
[arXiv:1607.03637 (quant-ph)]
There, we are considering the problem of defining a joint state over systems that may be related as cause and effect, a problem that we formalize as follows: given a state rho_A and a quantum channel from A to B, represented by the Jamiolkowski isomorphic operator on the composite AB, denoted rho_{B|A}, and constituting a quantum analogue of a conditional B|A, what is the quantum analogue of the joint state on AB, denoted rho_{AB}? We stipulate that rho_AB= rho_B|A * rho_A for some operator product * and ask what properties this operator product should have. In other words, we are trying to constrain the quantum analogue of the relation that holds between joints and conditionals, which is at the heart of defining the condition for Bayesian inversion. We argue that the operator product should be bilinear. For instance, the argument for linearity in the second argument is that the joint state on AB associated to a convex mixture of states on A should be the convex mixture of the joint states on AB obtained from each state on A. As we note in the article, this fails for the type of proposal that Leifer and I investigated. We also argue that the operator product should be associative, as follows. Consider a state on A going through a channel to B and then another channel to C. One can imagine computing the joint state on ABC in two ways: (i) combine rho_A and rho_B|A into a joint rho_AB and then apply the conditional rho_C|B to this, or (ii) combine rho_{C|B} and rho_{B|A} into a conditional rho_{BC|A} and then combine with rho_A. Associativity of the operator product is then motivated by the assumption that one should get the same joint state regardless of which way one computes it. Such associativity also fails for the type of proposal that Leifer and I investigated. Let me finally note that in our article we also derive a no-go theorem---a set of conditions on the operator product that cannot all be jointly satisfied.
My question is essentially what your approach might imply for the properties of the relationship between joints and conditionals in a quantum analogue of probability theory.
By the way, you mentioned in your talk that you could recover the formula for Bayesian inversion that Leifer and I studied by looking into a particular off-diagonal element of the quantum conditional operator (or something like that). Do you have anything written up about that? Thanks!
Arthur Parzygnat (Jun 08 2020 at 20:29):Alex Simpson said:
Arthur Parzygnat Thank you for the answer. You say " And yes, the quotient does remain a category! ". I am hoping it additionally remains a category with all the relevant structure (i.e. the relevant version of the Markov category structure).
Ah, yes that's an important question, @Alex Simpson. I didn't think about this question yet, but I believe it's still a monoidal category, at least in the finite-dimensional case---I have to think for a bit about the von Neumann algebra version. The reason is because the nullspace of the tensor product of states equals the tensor product of their nullspaces (or equivalently, the support of the tensor product of two positive semi-definite matrices is the tensor product of their supports). Whether this works much more generally for -algebras using some appropriate tensor product is currently beyond my level of understanding.
As for other structure, could you say in more detail what other things you were thinking about? I'm guessing your question is motivated by some theorems regarding the structures that descend to slice categories? If so, is there a word for this (I haven't yet learned the abstract results of this kind, if they exist)?
Update (Jun 9): after sleeping on it, the monoidal structure on the slice category is actually immediate from the definition of the Markov category axioms and the definition of a.e. equivalence (proof attached---primes denote CPU maps that are a.e. equivalent to the ones without primes with respect to the states and drawn). In terms of the algebraic structure, the statement I said about supports is true, but the statement about nullspaces is false since (as opposed to equal) for the associated supports and of and , respectively.
aemonoidal.JPG
Arthur Parzygnat (Jun 08 2020 at 21:10):Rob Spekkens said:
...
My question is essentially what your approach might imply for the properties of the relationship between joints and conditionals in a quantum analogue of probability theory.
By the way, you mentioned in your talk that you could recover the formula for Bayesian inversion that Leifer and I studied by looking into a particular off-diagonal element of the quantum conditional operator (or something like that). Do you have anything written up about that? Thanks!
Thank you very much for taking the time to explain all this, @Rob Spekkens. It will definitely help me when I read the paper! At some point in the future, I hope I'll have a better response then what I can offer at the moment:
For now, I can answer your last question in the affirmative and the reference is https://arxiv.org/pdf/2005.03886.pdf. Apologies for the length of this paper (it includes a decent amount of background material), but I can point you to more specific locations. Corollary 5.32 reproduces the recovery map formula (the additional term is there to indicate that we are only looking at one of four component of a Bayesian inverse). Remark 5.37 explains some of the similarities and differences between what we call a Bayesian inverse and what Leifer's formula gives (this is not the same as your update rule, as I now understand it thanks to your clarification). The other three components come from taking different parts of the projection and its complement. One of those extra three parts is a consequence of Definition 3.26, and this fact is quantified by Proposition 5.1. In expression (5.2) the fact that appears only on the left of tells you that you know both and . Demanding a completely positive extension allows one to figure out the remaining parts (sometimes uniquely), and this is described in our main theorem, which is Theorem 5.62. You can see why I didn't mention this result in my talk... we don't yet have a simpler description.
As for your first question, the relationship between joints and conditionals does not look very promising in our setup (see Remark 5.96 and Proposition 5.99---the prime notation denotes commutants). The main issue is the non-positivity of the copy map, which is not as serious a problem in formulating inversion. Nevertheless, with other collaborators, I'm trying to work out some ideas to see if any of them bypass this result. In the end, we may just recover the Choi--Jamiolkowski isomorphism or its variant, but that's not yet obvious to us at the moment.
Arthur Parzygnat (Jun 09 2020 at 07:09):Sam Staton said:
Hi Arthur Parzygnat, nice talk. I have a question about the first part. I know CPU maps have a physical motivation. Do you have any physical motivation for the supercategories of CPU maps that you considered?
Hi @Sam Staton, thanks for your question! Maybe calling them S-positive wasn't the best idea since S stands for "Schwarz" haha. I should have emphasized this crucial point during my talk, but the copy map in the quantum setting is not positive and is therefore not in your S-positive subcategory (for otherwise, it would enforce that the multiplication map is commutative as described by the version of the no-cloning theorem I discussed). My point of introducing the category SPU of Schwarz-positive unital maps was initially purely mathematical since I was able to prove that this structure was sufficient for the axiom Tobias provided for positivity in the (classical) Markov setting. Whether it is necessary is one of the open questions I posed. If it is indeed necessary, then calling this an S-positivity condition is, in my opinion, justified, and provides a very interesting characterization for S-positivity. The physical motivation is that this would give a different way to understand the necessity of CPU maps, since CPU maps form the largest monoidal subcategory of Schwarz-positive unital maps (in general, the tensor product of two SPU maps is not SPU).
Sam Staton (Jun 09 2020 at 17:56):Thanks @Arthur Parzygnat ! I understand and it is interesting that you can discuss positivity at the Schwarz level.
It would indeed be interesting to see whether CPU maps have some canonical status in this way. But what is wrong with the following argument:
Suppose there is a monoidal-supercategory of CPU maps that is a subcategory of SPU maps. Then for any in that category, must be in the category too, and so is completely positive?
Maybe I am missing something?
Sam Staton (Jun 09 2020 at 18:03):Also, I have been interested in another universal property for CPU maps (shorter paper, longer). For simplicity I state it for the dual case of CPTP maps:
The monoidal category of CPTP maps is the semi-cartesian reflection of the monoidal category of isometries.
In other words, we arrive at CPTP maps by supposing over pure quantum theory that it is possible to hide things.
This is close to the sort of Markov category axiomatization, so I wonder if there is anything to investigate there.
Oscar Cunningham (Jun 09 2020 at 18:21):@Sam Staton Is there an analogous classical version of that result, starting with the category of injections?
Sam Staton (Jun 09 2020 at 18:29):Hi @Oscar Cunningham, That's one of the strange things about it! The semi-cartesian reflection of the category of injections was explored by Hermida and Tennent for very different reasons, and has nothing to do with probability, as far as I can see! So it seems there's something very special about isometries and how pure quantum gives rise to probability via hiding.
Arthur Parzygnat (Jun 10 2020 at 12:39):Sam Staton said:
Thanks Arthur Parzygnat ! I understand and it is interesting that you can discuss positivity at the Schwarz level.
It would indeed be interesting to see whether CPU maps have some canonical status in this way. But what is wrong with the following argument:
Suppose there is a monoidal-supercategory of CPU maps that is a subcategory of SPU maps. Then for any in that category, must be in the category too, and so is completely positive?
Maybe I am missing something?
@Sam Staton , oh, yes, once you know that the largest S-positive subcategory of all unital maps is SPU, then it immediately follows that CPU is the largest monoidal closed subcategory by your argument. The question is: "Is SPU the largest S-positive subcategory of all unital maps?" As far as I am aware, it could be possible that one can enlarge this category by throwing in some linear map (not necessarily positive) and then taking the smallest category containing that additional morphism with the possibility that it still satisfies the (S-)positivity condition Tobias introduced. (Also, I don't know what a supercategory is, but I'm assuming that's what I was calling a -graded category?---I try to avoid the word "super" unless I'm specifically talking about fermions and supersymmetry.)
Sam Staton (Jun 10 2020 at 13:15):Thanks Arthur Parzygnat, I think I misunderstood what you wrote, sorry, my fault. I understand your question now and it is interesting. (When I said "supercategory", I just meant the opposite of "subcategory", like superset versus subset, nothing very "super" about it at all!)
Arthur Parzygnat (Jun 10 2020 at 15:36):Thanks! Now I know the terminology haha.