Of course I was wrong in #6 to say that quantum measurement channels give QuantumState (co)monad transformations. (They do preserve the (co)unit but not the (co)join.)

But partial trace quantum channels do give comonadic QuantumState transformations, have now added this here.

]]>added monographs by Paul Busch to the references

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

]]>(in fact I deleted the example now, it was wrong and its intended point not discernible)

]]>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).

I have tried to fix some of the oddities that this old entry inherited from its original revision 1. (It still leaves much room for improvement.)

In particular I deleted the lead-in paragraph (which suggested that the partial trace is a concept endemic to quantum physics) and the corresponding example (which looks incorrect to me).

Instead, I am creating now a separate entry *partial trace quantum channel* for such discussion.

brief `category:people`

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added (here) the example of the identity monad on $\mathbf{C}$ being the initial object in $Mnd(\mathbf{C})$

]]>added pointer to:

- Robert B. Griffiths,
*Consistent Quantum Theory*, Cambridge University Press (2002) [doi:10.1017/CBO9780511606052, webpage]

added pointer to what seems to be the original proof:

Lindblad 1975 (top of p. 149 and inside the proof of Lem. 5).

]]>brief `category:people`

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and this one:

- Göran Lindblad,
*Completely positive maps and entropy inequalities*, Commun. Math. Phys.**40**(1975) 147–151 [doi:10.1007/BF01609396]

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Pointer to:

- David E. Evans, John T. Lewis,
*Dilations of irreversible evolutions in algebraic quantum theory*, Communications of the Dublin Institute for Advanced Studies, Series A: Theoretical Physics**24**(1977) [eprint:34031, pdf]

and

- Karol Życzkowski, Ingemar Bengtsson,
*On Duality between Quantum Maps and Quantum States*, Open Systems & Information Dynamics**11**01 (2004) 3-42 [doi:10.1023/B:OPSY.0000024753.05661.c2]

added pointer to:

- Tom Schrijvers, Maciej Piróg, Nicolas Wu, Mauro Jaskelioff
*Monad transformers and modular algebraic effects: what binds them together*, in:*Haskell 2019: Proceedings of the 12th ACM SIGPLAN International Symposium on Haskell*(2019) 98–113 [doi:10.1145/3331545.3342595]

who cite Liang et al. broadly but then state the compatibility condition in terms of the join.

]]>expanded the proof (here) to derive the naturality clause

]]>Oh, I see now: Naturality of the transformation is already implied by its respect for `return`

and `bind`

. Will edit…

added a couple of lead-in paragraphs to the section “Interpretation of Deferred measurement” bringing out two points/claims more explicitly:

The folklore of quantum physics knows paradoxical-sounding stories under the title of

*Schrödinger’s cat*(1935)*Everett’s observers*(1957)*Wigner’s friend*(1961)

The author of these paragraphs asserts that:

These are all the same story, recast with different actors: Schrödinger’s cat plays the same role as Everett’s observer A and the same role as Wigner’s friend. The point in any case is that this first observer makes a quantum measurement and (only) ofterwards is himself observed by a second observer.

This is just what is formalized by the set-up of the deferred measurement principle:

The first observer (called “cat” or “A” or “friend”) is the controlled quantum gate denoted “$G$” above,

the quantum system observed by the first observer is $\mathrm{Q}W$ above,

the state space of the first observer is $\mathscr{H}$ (before) and $\mathscr{H}'$ (after the observation).

The second observer inspecting the scene at the end is the right hand side of the above setup, where the measurement is made at the end of the circuit execution. Before it is made, the first observer may have been in a superposition (in $\mathscr{H}'$).

But the deferred measurement principle says the outcome is indistinguishable from the situation where the first observer already collapses the original state in $\mathrm{Q}W$.

]]>

added a section (here)

making explicit the definition of monad transformers by Liang, Hudak & Jones 1995

and then proving that it is equivalent to monad morphisms in the sense of Maranda 1966.

That is, disregarding the naturality issue which Liang et al. seem to rather gloss over. What I am proving is that a natural transformation between monads satisfies their respect for the bind- operation iff it respects the join in the sense of Maranda.

(The statement/proof is evident/immediate, but I haven’t seen it mentioned anywhere before.)

]]>Thanks to a reference provided by Rod McGuire in another thread (here):

- F. William Lawvere:
*The legacy of Steve Schanuel!*(2015) [web]

we can settle the question of origin of the terminology ’rig’ – because Lawvere writes there, about his work with Schanuel, that:

We were amused when we finally revealed to each other that we had each independently come up with the term ’rig’.

Have added this to the entry.

]]>added

]]>14 July 1933 – 21 July 2014

Added Emily Riehl’s blog post on the n-Caategory Café about a construction of the Eudoxus real numbers

George Samson

]]>starting page on the n-Category Café

George Samson

]]>added (here) the statement that unitary channels are precisely the reversible quantum channels

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