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1. • CommentRowNumber1.
   • CommentAuthorBen_Sprott
   • CommentTimeJun 11th 2018
   • PermaLink
   Author: Ben_Sprott
   Format: MarkdownItexHi, I have been studying polynomial monads for a little bit now, especially in terms of their application to data structures under the title containers. I have found fun things like List and Bag, whose categories of algebras are Monoids and commutative Monoids. There isn't much work on applying this work to the Quantum realm, despite the level of sophistication we see in the subject of quantum computing. I would think we need to apply this to monoidal categories so that we can talk about quantum containers. I think [this paper](https://arxiv.org/abs/1106.1983) states that if you have a category with pullbacks, you can host the subject of polynomial functors and thus containers. Hilb has pullbacks, so we can begin the subject. It would amount to a study of quantum data structures. This should become a well trodden subject in the coming years. An important data structure in quantum computing and quantum information is the Frobenius Algebra which captures the presence of classical data within a quantum context. [Heunen and Karvonen](https://arxiv.org/pdf/1602.04324.pdf) pointed out that the internal Frobenius structure is generated by the monad $- \otimes B$, where $B$ is any object in your symmetric monoidal dagger category which supports a frobenius algebra (this may be putting poorly). The first question, for which the answer is likely "yes", is whether $- \otimes B$ is polynomial. I am interested in a larger question, and that concerns quantum theory itself. There are two papers, [Tull](https://arxiv.org/abs/1804.02265) and [Selby, Scandolo, Coecke](https://arxiv.org/abs/1802.00367), that give diagrammatic presentations of quantum theory. I am working on a very basic intuition that all string diagrams will be generated by some monad which is polynomial. The reasoning from that point is whether or not we can see these diagrammatic presentations of all of quantum mechanics as a polynomial monad. I am guessing that the host category for the monad would be Hilb. To begin this program of research, we need to ask the following: given that the Frobenius algebra is generated by a polynomial monad, what kinds of string diagrams can be given by a polynomial monad? T hen we can ask, what is the monad that generates the axioms in Tull, Coecke et al.?

   Hi,

   I have been studying polynomial monads for a little bit now, especially in terms of their application to data structures under the title containers. I have found fun things like List and Bag, whose categories of algebras are Monoids and commutative Monoids. There isn’t much work on applying this work to the Quantum realm, despite the level of sophistication we see in the subject of quantum computing. I would think we need to apply this to monoidal categories so that we can talk about quantum containers. I think this paper states that if you have a category with pullbacks, you can host the subject of polynomial functors and thus containers. Hilb has pullbacks, so we can begin the subject. It would amount to a study of quantum data structures. This should become a well trodden subject in the coming years.

   An important data structure in quantum computing and quantum information is the Frobenius Algebra which captures the presence of classical data within a quantum context. Heunen and Karvonen pointed out that the internal Frobenius structure is generated by the monad $- \otimes B$, where $B$ is any object in your symmetric monoidal dagger category which supports a frobenius algebra (this may be putting poorly). The first question, for which the answer is likely “yes”, is whether $- \otimes B$ is polynomial.

   I am interested in a larger question, and that concerns quantum theory itself. There are two papers, Tull and Selby, Scandolo, Coecke, that give diagrammatic presentations of quantum theory. I am working on a very basic intuition that all string diagrams will be generated by some monad which is polynomial. The reasoning from that point is whether or not we can see these diagrammatic presentations of all of quantum mechanics as a polynomial monad. I am guessing that the host category for the monad would be Hilb.

   To begin this program of research, we need to ask the following: given that the Frobenius algebra is generated by a polynomial monad, what kinds of string diagrams can be given by a polynomial monad? T hen we can ask, what is the monad that generates the axioms in Tull, Coecke et al.?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJun 11th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt seems unlikely to me that $(-\otimes B)$ would be a polynomial functor; how would you encode information about the tensor product *structure* of the category in terms of a polynomial which uses only pullbacks and their adjoints?

   It seems unlikely to me that $(-\otimes B)$ would be a polynomial functor; how would you encode information about the tensor product structure of the category in terms of a polynomial which uses only pullbacks and their adjoints?

3. • CommentRowNumber3.
   • CommentAuthorBen_Sprott
   • CommentTimeJun 12th 2018
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   Author: Ben_Sprott
   Format: MarkdownItexI guess it wouldn't be a post by me if I didn't state my conjecture as though it were true (with no proof). Would anyone like to suggest a polynomial functor on FHilb using, as Mike has suggested, only pullbacks and their adjoints? Mike, are you saying that we cannot have a polynomial functor on FHilb?

   I guess it wouldn’t be a post by me if I didn’t state my conjecture as though it were true (with no proof).

   Would anyone like to suggest a polynomial functor on FHilb using, as Mike has suggested, only pullbacks and their adjoints?

   Mike, are you saying that we cannot have a polynomial functor on FHilb?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 12th 2018
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   Author: Mike Shulman
   Format: MarkdownItexA polynomial endofunctor of a category $C$ is determined by an [[exponentiable morphism]] $p:B\to A$. I don't know whether there are any nontrivial exponentiable morphisms in $Hilb$, but identity morphisms are always exponentiable, so at least there are those polynomials, which are the cartesian products (= coproducts) with a fixed object $A$.

   A polynomial endofunctor of a category $C$ is determined by an exponentiable morphism $p:B\to A$. I don’t know whether there are any nontrivial exponentiable morphisms in $Hilb$, but identity morphisms are always exponentiable, so at least there are those polynomials, which are the cartesian products (= coproducts) with a fixed object $A$.

5. • CommentRowNumber5.
   • CommentAuthorBen_Sprott
   • CommentTimeJun 13th 2018
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   Author: Ben_Sprott
   Format: MarkdownItexThanks Mike!

   Thanks Mike!