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- Discussion Type
- discussion topicdistributive law
- Category Latest Changes
- Started by Mike Shulman
- Comments 22
- Last comment by Urs
- Last Active 2 minutes ago

I added some simpler motivation in terms of the basic example to the beginning of distributive law.

- Discussion Type
- discussion topicnonabelian Hodge theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by zskoda
- Last Active 1 hour ago

started a still highly stubby entry on nonabelian Hodge theory.

So far all I have is some references and the statement of "Corlette's theorem".

- Discussion Type
- discussion topicHodge structure
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by zskoda
- Last Active 1 hour ago

created

*Hodge structure*. Currently with nothing but a pointer to this nice book:- Chris Peters, Jozef Steenbrink,
*Mixed Hodge Structures*, Ergebisse der Mathematik (2007) (pdf)

Eventually I’d think we should move over Hodge-structure articles from

*Hodge theory*to here. But not tonight.- Chris Peters, Jozef Steenbrink,

- Discussion Type
- discussion topicCarlos Simpson
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 hours ago

- Discussion Type
- discussion topiclambda-connection
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 hours ago

- Discussion Type
- discussion topicKleisli category
- Category Latest Changes
- Started by Urs
- Comments 46
- Last comment by Urs
- Last Active 4 hours ago

I have tried to brush-up

*Kleisli category*; also made*Kleisli composition*redirect to it and cross-linked with*monad (in computer science)*

- Discussion Type
- discussion topicmodule category
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by mattecapu
- Last Active 4 hours ago

- Discussion Type
- discussion topicMauro Spera
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 4 hours ago

- Discussion Type
- discussion topicpositive-operator-valued measure
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active 4 hours ago

added monographs by Paul Busch to the references

and added quotes from Griffiths 2012 critiquing the idea of POVMs as “generalized measurements”

- Discussion Type
- discussion topicfunction type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active 5 hours ago

- Discussion Type
- discussion topicMatteo Spadetto
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 hours ago

- Discussion Type
- discussion topicDaniël Otten
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 hours ago

- Discussion Type
- discussion topicweak type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 19 hours ago

- Discussion Type
- discussion topicquantum reader monad
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 1 day ago

now creating this entry.

The technical material under “Details” (here) is copied over from what I had written at

*reader monad – Examples – quantum reader monad*. (There may still be room left to adjust the wording in order to reflect that this material moved to a new entry.)To this I have now added an Idea-section (here) which highlights the relation to (equivalence with) Bob Coecke’s “classical structures” (which term I made redirect to here now)

- Discussion Type
- discussion topicquantum state monad
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active 1 day ago

Starting an entry (to rhyme on “quantum reader monad”) on the linear version of the costate comonad and how its coKleisli category on the tensor unit alone exhibits the structure of quantum observables.

- Discussion Type
- discussion topiccategories of categories - contents
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicquantum circuits via dependent linear types
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by Urs
- Last Active 1 day ago

Am starting a write-up (here) of how (programming languages for) quantum circuits “with classical control and/by measurement” have a rather natural and elegant formulation within the linear homotopy type theory of Riley 2022.

Aspects of this have a resemblance to some constructions considered in/with “Quipper”, but maybe it helps clarify some issues there, such as that of “dynamic lifting”.

The entry is currently written without TOC and without Idea-section etc, but rather as a single top-level section that could be

`!include`

-ed into relevant entries (such as at*quantum circuit*and at*dependent linear type theory*). But for the moment I haven’t included it anywhere yet, and maybe I’ll eventually change my mind about it.

- Discussion Type
- discussion topicunistochastic quantum channel
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicJulian Sochocki
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 days ago

- Discussion Type
- discussion topicJosip Plemelj
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 2 days ago

Minimal person entry on a Slovenian mathematician, his formula plays major role in the entry principal value.

- Discussion Type
- discussion topicCauchy principal value
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active 2 days ago

I want to be adding some details to

*Cauchy principal value*. What’s a good reference? Say for the proof that up to addition of a delta-distribution, $f(x) = pv\left( \frac{1}{x}\right)$ is the unique distributional solution to $x f = 1$?

- Discussion Type
- discussion topictriple category
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by David_Corfield
- Last Active 3 days ago

Stub for triple category.

- Discussion Type
- discussion topicunital quantum channel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicmaximally mixed quantum state
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicmixed unitary quantum channel
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicbit-flip quantum channel
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicLex
- Category Latest Changes
- Started by SridharRamesh
- Comments 3
- Last comment by Tim_Porter
- Last Active 3 days ago

- Discussion Type
- discussion topictransformation of adjoints
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by nLab edit announcer
- Last Active 3 days ago

- Discussion Type
- discussion topicpostselection
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 3 days ago

- Discussion Type
- discussion topicScott Aaronson
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 3 days ago

Adding reference

- Scott Aaronson,
*Quantum computing, postselection, and probabilistic polynomial-time*, Proceedings of the Royal Society A, Volume 461, Issue 2063, pages 3473–3482, 08 November 2005. (doi:10.1098/rspa.2005.1546, arXiv:quant-ph/0412187)

Anonymouse

- Scott Aaronson,

- Discussion Type
- discussion topicPostBQP
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 3 days ago

- Discussion Type
- discussion topictensor linear category
- Category Latest Changes
- Started by mattecapu
- Comments 2
- Last comment by varkor
- Last Active 3 days ago

- Discussion Type
- discussion topicfunctional programming
- Category Latest Changes
- Started by AlexisHazell
- Comments 10
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicdualizable object
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active 4 days ago

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

- Discussion Type
- discussion topicsemisimple ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by JoseBrox
- Last Active 4 days ago

- Discussion Type
- discussion topicmonad (in computer science)
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicBQP
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active 4 days ago

- Discussion Type
- discussion topicQRAM
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicsemi-topological 4d Chern-Simons theory
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topictopological phases of matter via K-theory -- references
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topiccolimit
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 6
- Last comment by nLab edit announcer
- Last Active 4 days ago

- Discussion Type
- discussion topicJuan Pablo Paz
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicicon
- Category Latest Changes
- Started by Peter Heinig
- Comments 8
- Last comment by nilesjohnson
- Last Active 4 days ago

Added a literature reference to icon. Started some systematic notes on icons for monoidal-enriched bicategories, which I am currently using for something. Think the broken-off state of that section is not intolerable, in particular since I have seen similar work in progress on the nLab. Intend to continue them soon.

- Discussion Type
- discussion topicmodification
- Category Latest Changes
- Started by nilesjohnson
- Comments 1
- Last comment by nilesjohnson
- Last Active 4 days ago

- Discussion Type
- discussion topicpseudofunctor
- Category Latest Changes
- Started by Todd_Trimble
- Comments 7
- Last comment by nilesjohnson
- Last Active 4 days ago

I added a section on strictification of pseudofunctors $C \to Cat$ to pseudofunctor, after seeing Finn and Mike respond to Karol in the stable monoidal derivator thread at the Café. The discussion is fairly sketchy and pedestrian. Also added a few references.

- Discussion Type
- discussion topicJonathan Oppenheim
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicPawel Horodecki
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicMichal Horodecki
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicquantum channel
- Category Latest Changes
- Started by Eric
- Comments 101
- Last comment by Urs
- Last Active 5 days ago

Wikipedia has a nice article on quantum operations.

The nLab also had a page quantum operations and channels (cache bug?), but I’ve renamed this to simply quantum operation since a quantum channel seems to be nothing but a quantum operation when viewed from the perspective of quantum information theory. Eventually, this page might need some disambiguation since there may be several uses of the term, but for now I think it is “ok”.

I think this page can be cleaned up. I started, but don’t think I will be able to finish.

In particular, there is some background material that might be better on separate pages. I’ll continue trying to clean things up, but family might be calling soon and I’ll need to run quickly whatever state it is in.

I also made the simple statement

In quantum mechanics, a

*quantum operation*is a morphism in the category of density matricesat the beginning of the Idea section motivated by O’Loan’s comment

A quantum channel is a mapping which sends density matrices to density matrices.

This seems innocent enough, but someone might check the statement. For one, I’ve never seen a category of density matrices, but the idea seems obvious enough. Maybe a word on density matrix would be good.

- Discussion Type
- discussion topicquantum channels and decoherence -- section
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 5 days ago

this is a bare sub-section — meant to be

`!include`

ed into relevant entries such as at*quantum channel*and at*quantum decoherence*— towards the theorem that quantum channels are exactly the “bath-averages of bath-coupled unitary evolutions”

- Discussion Type
- discussion topicMOND
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by nLab edit announcer
- Last Active 5 days ago

recorded some recent surveys of the status of MOND at

*MOND*

- Discussion Type
- discussion topiclinear isometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicProf
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nonemenon
- Last Active 5 days ago

- Discussion Type
- discussion topiccompactum
- Category Latest Changes
- Started by David_Corfield
- Comments 6
- Last comment by Shamrock
- Last Active 6 days ago

Added the characterization of $Comp$ as the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object from

- Vincenzo Marra, Luca Reggio,
*A characterisation of the category of compact Hausdorff spaces*, (arXiv:1808.09738)

- Vincenzo Marra, Luca Reggio,

- Discussion Type
- discussion topicproto-exact category
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 6 days ago

- Discussion Type
- discussion topicpartial trace quantum channel
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 6 days ago

starting a bare minimum, to make links work.

(The single example included is copied over from revision 1 of

*partial trace*, where I had deleted it as announced there. It needs attention, if only for the typesetting).

- Discussion Type
- discussion topicenvironment rep of quantum measurement channel - section
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

a bare sub-section, to be

`!include`

-ed into relevant entries (such as at*quantum measurement channel*)

- Discussion Type
- discussion topicpartial trace
- Category Latest Changes
- Started by Ian_Durham
- Comments 5
- Last comment by Urs
- Last Active 7 days ago

- Just added a page on partial trace that is presently linked from quantum operations and channels which I also added to. However, note that the partial trace is not specific to physics so it needs embellishing by the mathematicians among us.

- Discussion Type
- discussion topicPaul Busch
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicPekka J. Lahti
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago