12:18 AM
@glS would u mind explaining why u removed the qubyte tag from my qubyte post?

6 hours later…
6:45 AM
@meowzz because "qubyte" is not a thing.. Also it was used only for that question, making it pretty useless imo

4 hours later…
11:07 AM
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I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes. It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended. May I ...

11:28 AM
@Blue Could you merge the tags "phase-estimation" and "quantum-phase-estimation"?

11:56 AM
@Nelimee Do we need to merge? Currently, there's just one question with "phase-estimation" and another question with "quantum-phase-estimation". Might we as well use just one tag? (say just "phase-estimation")

I think they both convey the same meaning

@Nelimee I know. I'm saying that I could delete the "quantum-phase-estimation" tag instead
And replace it with "phase-estimation"

As you want :)

Okay, cool! Actually there are just too many tags beginning with "quantum"
It becomes a mess when you're searching for a particular tag
So I wanted to minimize the number of tags beginning with "quantum"
The quantum part is obvious since the site is about quantum computing ;)

Yup you are right :) I also added 3 tags excerpt, waiting for review =)

12:00 PM

@Blue 'merging', if I'm getting the terms right, is a specific single action that does exactly that and is generally preferable to editing tags on questions. Having said that, if it's just one question, it doesn't really matter although performing a proper merge is still probably preferable

@Mithrandir24601 Well, merging does create permanent tag synonyms I guess? Do we really need more synonyms beginning with "quantum"? :P
And yeah, it was simply a matter of 1 question, and that too, my question
@Nelimee In the review queue I approved all of them. Just approval from one more user is needed I suppose
Or @Mithrandir24601 could do it

Ha ok! :) Thanks =)

@Blue ah no, that's a different thing :P
You can do that as part of the merge process but it's optional

@Mithrandir24601 Do what?

12:09 PM
@Blue create synonyms

@Mithrandir24601 Oh, I didn't know that
@Mithrandir24601 Are synonymous tags on a question automatically replaced by the master tag? If so, you could add "quantum-phase-estimation" as a synonym for "phase-estimation"

Merging is taking all the questions with a specific tag and replacing that tag with a different one, on all those questions, on a tag level, without permanently changing anything about the underlying tags
@Blue yeah, you could do that. It generally requires votes, so it's probably not worth bothering when only one question has that tag

12:35 PM
@Blue You have a typo in quantumcomputing.stackexchange.com/questions/2564/… : the last Rzz is $R_zz$ and should be $R_{zz}$ :p

@Nelimee Corrected!
@glS That paper does have a few mistakes. We were discussing that. But on making some corrections, it does give the correct solution for the system of linear equations
The basic idea in the paper seems to be correct though

@Blue I'm puzzled about the removing of details in the latest version though. Also, they published the work in a weird journal.

I was puzzled too :P

Yep, they removed nearly all the interesting material x)

But well, if we can make it work, we're done ;)

12:42 PM
And their justification is not really clear

@Nelimee is it included in the published version (if you checked that)?

But after a few corrections, it worked so it's fine x)
What is included?

@Nelimee the details removed in the third arxiv version

@glS "Every hermitian matrix satisfy this property: more specifically, all and only Hermitian matrices have this property" ha? I though it was only a subset of the set of valid matrices ^^ Thanks for the precision :)

Based on version 2 it seems

12:45 PM
I'm not sure I understood your question... Are the removed parts published with the new paper? <- this is your question?
@Blue "Server is busy"

@Nelimee Try scihub
It's behind a paywall
@glS The journal published version of the paper seems to be exactly same as version 2 on arXiv
They didn't publish version 3 in a journal it seems

@Nelimee if you think about it it's quite easy to see. Unitary matrices are the ones with phases as eigenvalues, while Hermitians have real eigenvalues. Therefore, if a matrix is not Hermitian (does not have real eigenvalues), then its exponential will not have eigenvalues of the form $e^{i\phi}$ with $\phi\in\mathbb R$. Although I'm not sure whether there could be exceptions for non diagonalizable matrices (if $A$ is not diagonalizable, then the above argument doesn't work)

A matrix is not Hermitian does not mean that it does not have real eigenvalues. Or I completly missed something
Hermitian => real eigenvalues
real eigenvalues =/> Hermitian

Right, the converse isn't true

@Nelimee if a matrix $A$ is unitarily diagonalizable with real eigenvalues, then $A=U\Lambda U^\dagger$ with $\Lambda$ diagonal and real. Then, $A^\dagger = U\Lambda^\dagger U^\dagger = A$

12:49 PM
And... Every matrix is diagonalisable in $\mathbb{C}$ doesn't it?

Not every
There's some condition
But "most" are
Yes
8

This is an elementary question, but a little subtle so I hope it is suitable for MO. Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$. The characteristic polynomial $T - \lambda I$ splits into linear factors like $T - \lambda_iI$, and we have the Jordan canonical form: $$J = \begin... Linear algebra courses were 3 years ago, I may need a refresh x) @Nelimee no! unitarily diagonalizable matrices are all and only the normal ones (satisfying AA^\dagger =A^\dagger A). For general diagonalizability if I'm not mistaken onecharacterization is that the sum of the dimensions of the eigenspaces has to match the total dimension Eh, defining eigenspaces will be tricky if you're allowing for complex eigenvalues$$A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 2 & -4 & 2\end{pmatrix}$$has real eigenvalues \{1, 2, 3\} from en.wikipedia.org/wiki/… but is not Hermitian I don't know if e^{iAt} is unitary though 12:54 PM @Blue how so? given \lambda\in\mathbb C then the associated eigenspace is \{v: \,\, (A-\lambda)v=0\} @Nelimee indeed, my proof used unitarily diagonalizable matrices. You are right that the argument fails for generally diagonalizable ones The matrix e^{iAt} is not unitary Do you have any proof of your "only hermitian matrix works"? ok, let's rephrase. If A and e^{iA} are normal, then e^{iA} is unitary iff A is Hermitian I still don't see why "iff", but this is what you wrote in the comment yes! Wait Nope, still don't see it @glS Ah, you're right indeed @Nelimee If U=e^{iAt}, then what would U^{\dagger} be if A is not Hermitian? U^\dagger = e^{(iA)^\dagger t} 1:04 PM We could expand e^{iAt}=I+it A + (it)^2A^2/2! + ... if I'm not wrong So, all the A's get conjugate transposed @Nelimee Looks right Then UU^{\dagger}= ? U^{\dagger} = e^{-iA^{\dagger}t} @Blue I actually agree with Nelimee here that it's not that easy. You get UU^\dagger = e^{iA} e^{-iA^\dagger}, but if A and A^\dagger do not commute it's not straightforward that this doesn't give you an identity A and A^dagger don’t commute in general tho Sniped And applying an operator on an infinite sum needs some checks Infinites sums are bad boys, you don't want to manipulates them too much :p @Nelimee the exponential map converges with all the nice properties so usually that's not a problem @glS I was seeing how far we can go with that :P 1:08 PM Yup, right for the exponential, I forgot it was this one ^^ So it's not obvious that only hermitian matrices works You also get a fairly nice product if for instance [A,A^\dagger]=id I'm getting confused. I remember there being some theorem about one-to-one mappings between unitaries and hermitians provided by the exponential, but it was some time ago and may be confusing things in my head with the raising/lowering operators of a QHO being the easiest example @Semiclassical what does the notation with the square brackets means? Matrix product? Commutator [A,B]=AB-BA 1:11 PM Nice! Thanks :) Hummm, it should be 0 instead of Id no? I checked, it's OK, I need to re-do some calculus x) There's a theorem that every unitary matrix U can be written in the form U=e^{iA} where A is Hermitian though. But A isn't unique Got to invoke the Spectral theorem it seems @Nelimee if there is a 0 there then it becomes the normality condition. Otherwise it means that the matrix is not normal, therefore not unitarily diagonalizable, but still the product of exponentials is relatively easy to write But our question is precisely: Can exponential of non-Hermitian matrices be unitary? 1:19 PM @Blue you are right indeed. If U is unitary then for sure you can write it as exponential of an Hermitian (time i). This is easily proven because U is ensured to be unitarily diagonalizable, so you can simply compute it's logarithm through the eigenvalues. However, logarithms are tricky and multivalued, and there may be logarithms which are not diagonalizable at all. I've actually recently asked some questions on math.SE on related topics I will need to go (I need to work a little before France-Uruguay :p), but I follow the discussion here, even if I do not speak :p for example, Are all square roots of diagonalizable matrices diagonalizable?. I've yet to fully understand the last answer though, which I believe is very interesting given that the square root in the complex plane is essentially defined through the logarithm, there probably are similarities between the two questions I would argue that if the logarithm of a unitary U is diagonalizable, then it is ensured to be (anti-)Hermitian @glS not quite - non-Hermitian matrices can still have real eigenvalues @Blue presumably you mean that in the sense of e^{i A} with A non-hermitian. (A skew-hermitian \implies e^A is unitary) Ah, that's already been mentioned 1:31 PM @Mithrandir24601 indeed, that was also what @Nelimee showed with an example above. I believe my argument holds for unitarily diagonalizable matrices. If a matrix is only generally diagonalizable (so it's not normal) then it's not true fair enough @Blue nope hmm, suppose U(t)=e^{i A t} and U^\dagger(t)U(t)=id. Then \frac{d}{dt}U^\dagger(t)U(t) = e^{-i A^\dagger t}(A-A^\dagger)e^{i A t}=\frac{d}{dt}id=0 and in particular for t=0 one has A=A^\dagger so if that differentiation is legal then I'm not sure one can avoid it @Semiclassical nice argument, that seems to work. Isn't this the same as proving that e^{iA}e^{iB}=1 implies A=-B? quite probably i mean, if you modify that to e^{i A t}e^{i B t}=1 then the same argument should apply 1:43 PM also probably even more generally without i factors so, in conclusion, it does indeed seem that e^{iA} unitary implies A Hermitian. It therefore also seems that e^{iA} unitary implies A normal, so that also my argument passing through the spectra works (though one has to show that A is ensured to be normal) though, my argument only works if e^{i A t}e^{i B t} for all t (or at least all t within a nbhd including t=0) @Semiclassical isn't that the same? If e^{iA} is unitary then e^{iAt} is unitary for all t or isn't it? hmm that should be right, yes ...hmm something feels suspicious there does (U^t)^\dagger = (U^\dagger)^t hold for all t\in\mathbb R? For integer a, definitely. For non-integers...hmmmm 2:23 PM @glS yeah, no. see Emilio's example in h bar @Mithrandir24601 For real matrices A, it is true that if e^{iA} is unitary, it implies A is Hermitian. But I don't think we can say the same for complex matrices @Semiclassical Yup, I meant that the fact that e^{2\pi i}=1 doesn't imply 2\pi i=0 made me wonder if there was an escape clause and EP's example bears that out in The h Bar, 44 mins ago, by Emilio Pisanty$$\exp\left(i\begin{pmatrix} 2\pi & 1 \\ 0 & 0 \end{pmatrix}\right) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$according to Mathematica Nice example, hmm yeah it's a good one note, though, that if that's e^{i A} then e^{i A t}\neq 1 in general 0 What are the prominent visualisations used to depict large, entangled states and in what context are they most commonly applied? 2:38 PM 8 mins ago, by Blue @Mithrandir24601 For real matrices A, it is true that if e^{iA} is unitary, it implies A is Hermitian. But I don't think we can say the same for complex matrices Ah, so this is wrong @QuantumComputing my snarky response to this would probably be "Does sufficiently-complicated algebra count as visualization?" What does hold is: If e^A is unitary, then A is Hermitian (assuming A is real) But that i messes it up @Blue right. This also reminds me a lot of the other example I was given in the math.SE post Interesting well, if A is real then A being hermitian just means it's symmetric and e^A is orthogonal I guess 2:41 PM @Semiclassical Umm, but the requirement is skew-symmetry? I'm messing it up again derp yeah @Blue but then in this case the statement should be that A is skew-hermitian Right, makes sense Now what we need to look for is 1) The exact set of conditions for which the matrix exponential e^A of a complex matrix A, is unitary 2) The exact set of conditions for which the matrix exponential e^{iA} of a real matrix A is unitary Of course (1) will be a stronger version 11 mins ago, by Semiclassical note, though, that if that's e^{i A} then e^{i A t}\neq 1 in general This is again an interesting point What should be the constraint on t here? ^@Semiclassical dunno. my story was in the opposite direction: If it's true that e^{i At} is unitary for all t, then A is hermitian. @Semiclassical Proof? (sorry I'm a bit hazy here) 1 hour ago, by Semiclassical hmm, suppose U(t)=e^{i A t} and U^\dagger(t)U(t)=id. Then \frac{d}{dt}U^\dagger(t)U(t) = e^{-i A^\dagger t}(A-A^\dagger)e^{i A t}=\frac{d}{dt}id=0 yeah, that 3:07 PM @Semiclassical Shouldn't that be e^{-i A^\dagger t}(iA-iA^\dagger)e^{i A t}? I think there's an i missing there yeah sloppy Got it, no prob 3:25 PM @Blue fair enough - as with @Semiclassical I was thinking about it with the t parameter, as that's what we care about in physics :P I can possibly come up with a number of non-Hermitian matrices that gives unitary evolution for a specific t Or rather, the exponential of which is unitary for t+n\tau, although I'd need to check Unitary time evolution ftw @Semiclassical exactly 3:40 PM Unless your Hamiltonian is time-dependent. Then I just get a headache Although there's something to be said for quasi-unitary evolution as well :P (H at time 1 not commuting with H at time 2 is really a hassle) @Semiclassical my PhD is dealing with non-Hermitian stuff, so you'll have to try harder if you want any sympathy :P Lol My first project was semiclassical quantization / level-splitting of periodic pt-symmetric potentials Which somehow worked very well with numerics 3:55 PM @Semiclassical part of mine is simulating them, so fun times... (PT-symmetric Hamiltonians, that is) Oof The nice thing about the periodic case is that numerics are tractable (Fourier basis ftw) The moment you first discover that Green's functions break down is somewhat alarming, yeah 4:21 PM Oof Yeah, sounds right Non-hermitian spectral problems are generically pretty goofy 4:51 PM X is skew-Hermitian iff all \exp(sX) are unitary Nice! I believe this is the result I was remembering Is there any proof? @Blue @Semiclassical basically it holds exactly in the case as per @Semiclassical proof @Nelimee the excerpt is from pag. 7 of pi.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf @Nelimee they say it's because \exp\operatorname{tr}(X)=\det \exp X, but they don't prove this. This fact is easily shown for diagonalizable matrices but I'm not sure how to prove it in general actually on second thought, maybe the proof is exactly the one @Semiclassical gave above? 9 If you're afraid of the density of diagonalizable matrices, simply triangularize A. You get$$A=P^{-1}UP,$$with U upper triangular and the eigenvalues \{\lambda_j\} of A on the diagonal. Then$$ \mbox{det}\;e^A=\mbox{det}(P^{-1}e^UP)=\mbox{det}\;e^U.  Now observe that $e^U$ is upper ...

Seems to be covered here ^

4:58 PM
@Blue ah, great find!

@glS I was sort of expecting that after seeing SemiC's proof. I'll see it in more detail later, but thanks :)

@Blue yes it's basically what was found above, but it's good to see it mentioned in an authoritative reference I guess

7 hours later…
11:51 PM
0

There's 15 hours left on a bountied question, but the person who offered the bounty is suspended and his suspension doesn't expire until about 2 days, meaning he may not be able to award the bounty himself? That's not fair: It's a 300 point bounty. The largest bounty ever offered on QCSE. Let h...