added (here) statement of the canonical Kraus form
]]>and this one:
Pointer to:
and
added more text (still just the very basics) to the Examples-section on quantum measurement channels (here)
will give this also its own entry quantum measurement channel now, for more informative cross-linking with POVM
]]>I have now (re-)written the first 3 subsections of the Definition-section (here).
]]>renaming entry from “quantum operation” to “quantum channel” which is a little more descriptive/evocative
]]>Have rewritten the (previously puny) Idea-section from scratch (here).
]]>added more references:
early discussion of “quantum operations”:
and dedicated discussion of “quantum channels” in the sense of Shannon theory:
Mark M. Wilde, Quantum Information Theory, Cambridge University Press (2013) [doi:10.1017/CBO9781139525343, arXiv:1106.1445]
Sumeet Khatri, Mark M. Wilde, §3.2 in: Principles of Quantum Communication Theory: A Modern Approach [arXiv:2011.04672]
added further references in the context of quantum computation:
John Preskill, §3.2 in: Measurement and Evolution, chapter 3 of: Quantum Information, lecture notes, since 2004 [pdf, web]
Peter Selinger, §6.3 in: Towards a quantum programming language, Mathematical Structures in Computer Science 14 4 (2004) 527–586 [doi:10.1017/S0960129504004256, pdf, web]
added pointer to:
added pointer to:
I spelled out the definition of a quantum operation, and made it clear that “quantum channel” is a synonym.
]]>I think that is a nice contribution.
Speaking as an outsider, though, it is somewhat confusing that the “Universal Property” section starts out by referring to “the category of natural numbers and quantum channels” when the page up until that point has not actually defined “a quantum channel”. The “Definition” section only defines “positive” and “completely positive” linear maps. The “Idea” section suggests that “quantum operations” should be, I guess, the trace-preserving and completely positive maps (although the connection is nowhere made to the Definition section). Is a “quantum operation” the same as a “quantum channel”? This is suggested by the fact that the “Properties” section defines a category (though it still never uses the word “quantum channel”) whose morphisms are the completely positive trace-preserving maps.
]]>I wrote about the universal property of the category of quantum channels. Here I am writing about my own recent work, because I think it is a good way to see this category from the nlab perspective. Of course, feel free to demote it if you like.
]]>I have added to quantum operation pointers to the historical articles by Stinespring and Kraus, predating Choi’s. See the new list of citations right below the theorem.
The entry as a whole still needs more improvement. Maybe later.
]]>Thanks, Urs – I should have figured it wasn’t you. I remembered you did a lot of work fixing things like quantum channel and quantum operation, and since the whole entry reads much more like something you would have written, I made an incorrect assumption.
]]>Thanks, Toby! I’ll go back and fix it.
]]>Todd, what you did wrong was to put HTML in a Markdown comment. (I don’t know why Markdown doesn’t fail gracefully, munging the link but rendering the rest.) Change
<a href="http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3158&page=1#Item_15">here</a>
to
[here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3158&page=1#Item_15)
to fix it.
]]>Okay, I have streamlined the whole paragraph after the statement of Choi’s theorem by making all pointers to the literature be actual pointers to the References-section and then adding the lines
]]>This is originally due to (Choi, theorem 1). A proof in terms of †-categories is given in (Selinger). A characterization of completely positive maps entirely in terms of †-categories is given in (Coecke).
Todd,
I didn’t write this. Instead, Ian Durham wrote this, in revision 48.
But I can fix it. I have fixed already many things in this entry, but seem to have missed this one.
]]>I hope my previous message is readable. I haven’t the foggiest idea what I did wrong.
]]>I’m looking at the page quantum operation, at a sentence written by Urs (I think):
“Note that Coecke has shown that FdHilb, the category of finite dimensional Hilbert spaces and linear maps, has the completely positive maps as morphisms.”
This sentence is confusing, since frequently the words ’maps’ and ’morphisms’ are used synonymously. I didn’t see a hint in the preceding text that gives warning about this. I am guessing, after looking briefly at Coecke’s paper that was linked to, that the ’morphisms’ here are arrows of a monoidal category that Coecke constructs out of a -symmetric monoidal category . For the present purposes, it looks like the -symmetric monoidal category is FDHilb.
I could try to insert clarifications myself, but this is not really my area. I was only led to that page while doing some preliminary looking around before I attempt a discussion with Ben over here.
]]>If one looks at wide pullbacks (look this up on the nlab - I’m being lazy and in a rush) then […]
That’s wide pullbacks. But that page doesn’t really explain things in the detail that Eric may need.
]]>To understand the concept of equalizer just realize that it’s the category theorist’s way of generalizing “the subset of on which two functions are equal”. We use equalizers all the time in math: that’s what we’re doing whenever we set two functions equal and look for the set of solutions!
Yet another crystal-clear explanation! Thanks John!
And having done that exercise doesn’t ensure the understanding of limits, but it hopefully helps :)
I’m probably deluding myself, but Jamie’s explanation of cones seemed to help me understand limits somehow. His description on p.7 makes me think of limits as kind of like a special “pathway” for getting to (I can visualize it in my head better than I seem to be able to put it into English, but I really do think I get it the more I look at it).
]]>