Timeline for Why is the standard definition of fidelity unnecessarily complicated?
Current License: CC BY-SA 4.0
21 events
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| Feb 26, 2020 at 6:00 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Feb 26, 2020 at 3:16 | comment | added | tparker | @NorbertSchuch Done. | |
| Feb 26, 2020 at 3:16 | history | edited | tparker | CC BY-SA 4.0 |
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| Feb 25, 2020 at 16:28 | comment | added | Norbert Schuch | @tparker Why don't you just edit the answer to make more clear that there are two choices how to generalize this concept? | |
| Feb 25, 2020 at 16:21 | comment | added | tparker | @NorbertSchuch Respectfully, I think you're getting way too hung up on the question of whether PD2 is actually used in the existing literature. That's completely beside my point; I'm just considering the consequences of that choice of generalization of the standard definition and not claiming that it's a common or even good choice of definition. Feel free to mentally replace it with a different term if you'd prefer. | |
| Feb 25, 2020 at 16:18 | comment | added | tparker | @NorbertSchuch That's true for complex matrices but not for real ones. math.stackexchange.com/a/1555781/268333 states that PD2 is sometimes given as a definition of positive definite in the context of real matrices. | |
| Feb 25, 2020 at 15:37 | comment | added | Norbert Schuch | [...] but calling it PSD is simply contradicting the common use of the term. | |
| Feb 25, 2020 at 15:37 | comment | added | Norbert Schuch | @tparker The point is that PD2 implies hermiticity. That is, the generalization of "PD2" to non-hermitian matrices is "they are not positive semi-definite", which is the way it is supposed to be. An alternative definition is "a PSD matrix is a hermitian matrix with positive/non-negative eigenvalues" (cf. your link). The "generalization" of ither definition to non-hermitian matrices should not be called "positive semi-definite", as this concept is restricted to hermitian matrices. You can call it "my personal way of generalizing PSD" or whatever - [...] | |
| Feb 25, 2020 at 15:07 | comment | added | tparker | @NorbertSchuch Also, positive definite is not a well-defined concept for non-symmetric matrices. The whole point of my answer is that you need to choose which property to generalize when extending a definition to a larger set. And some sources do define positive definiteness in terms of the eigenvalues: math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/…. | |
| Feb 25, 2020 at 15:01 | comment | added | tparker | @NorbertSchuch I disagree. I state in my answer that it is a definition, while immediately following up with an alternate definition and then stating that the second definition is usually more useful, but in this particular case the less-common definition is the more useful one. I don’t think there’s any ambiguity here. | |
| Feb 25, 2020 at 8:32 | comment | added | Norbert Schuch | @tparker That is exactly the point. It is your private definition. The point about answers here is that they should not use private definitions of well-defined concepts. Next, you will use your private definition of mass, velocity, and temperature. For the very least, you have to make clear that this is your private definition (or private extension of the definition) - otherwise people are misled into believing that PD1 is the actual definition (and will tail their exam, or the like ... ). | |
| Feb 24, 2020 at 13:01 | comment | added | tparker | @glS Sure it's a definition - just not a standard or natural one. And for must purposes it's not a useful one, but for some purposes (e.g. determining whether a matrix power series converges, as in this case), it can be useful. It all just depends on the application. I'm not claiming that PD1 has appeared in any of the previous literature, but you're always free to generalize existing definitions in whatever way you want, although the result may or may not be useful. | |
| Feb 24, 2020 at 12:01 | comment | added | glS | PD1 is not a definition of "positive semidefinite" for non-normal matrices. Apparently, it's also not "natural" to talk about "positive semitedefinite" in such contexts, see discussion here | |
| Feb 24, 2020 at 9:18 | comment | added | user245141 | again note that this is true for finite spaces. For example: $\exp(\kappa x)$ and $\exp(\xi p)$ both have positive eigenvalues (though unbounded, so cannot be density operators). However, $\exp(\kappa x)\exp(\xi p) = \exp(\kappa x + \xi p + i \kappa \xi /2)$ which has complex eigenvalues. In contrast, $\exp(\kappa x/2)\exp(\xi p)\exp(\kappa x /2)$ is hermitian and will have real eigenvalues. | |
| Feb 23, 2020 at 18:30 | vote | accept | tparker | ||
| Feb 23, 2020 at 18:29 | comment | added | tparker | @NorbertSchuch (2) You're probably right that this answer is more detailed than necessary, but since a lot of confusion has been raised in the comments and answers to this question, I thought I'd clarify the situation as much as possible. (Also, the possibility of extra zero eigenvalues coming in from cyclic permutations actually only comes up for rectangular matrices. For square matrices, no additional zeroes come into the spectrum.) | |
| Feb 23, 2020 at 18:27 | comment | added | tparker | @NorbertSchuch (1) I've never seen definition PD1 in the literature, but I thought it was worth stating it separately (a) to show that it's distinct from the more usual definition PD2, and (b) because in this particular case (convergence of power series expansions), property PD1 is actually the more relevant one. | |
| Feb 22, 2020 at 14:36 | comment | added | Norbert Schuch | (P.S.: Note that even if there were branches, if you define them in some specific way you are fine, since tr(f(X)) is defined on the eigenvalues, and those do not change. | |
| Feb 22, 2020 at 14:34 | comment | added | Norbert Schuch | To add to this, I'm not sure about all this fuzz (not only in this answer) about subtleties. Spectra or products (except for zeros) are invariant under cyclic permutations, so as you correctly observed functions like square root etc. are cyclic as well, as long as there are no issues with branches - which is clearly the case here. I think if there is an answer to your question, it is that hermitian objects are more familiar and "nice", and Uhlmann and Josza certainly didn't think about computing those numerically for large systems. | |
| Feb 22, 2020 at 14:29 | comment | added | Norbert Schuch | Do you have a trustworthy source for your definition PD1? I have always only ever seen PD2 (which has nice features such that A>0, B>0 => A+B>0 etc. - certain "normal" math still works), which I strongly doubt holds for PD1. | |
| Feb 22, 2020 at 6:32 | history | answered | tparker | CC BY-SA 4.0 |