Timeline for Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2019 at 16:36 | comment | added | Paul B. Slater | Thanks for the comment--will consider what you suggest! | |
| Aug 24, 2019 at 12:57 | comment | added | J. E. Pascoe | @PaulB.Slater, it might be useful to consider the fact that operator monotone functions are exactly self maps of the upper half plane, and therefore have nice integral representations. In Peter Lax "Functional Analysis" book, I think these are called "Nevanlinna representations." To make a long story short, this would make your function $f$ depend on a real number $a,$ a nonnegative $b$ and a positive measure on the real line $\mu.$ | |
| Aug 21, 2019 at 0:33 | comment | added | Paul B. Slater | Thanks for the question J. E. Pascoe! I was conceiving of a (I guess) "functional" that would map the operator monotone functions into the values (25/341,...) of the "separability probabilities"--that is the ratio of the volume of the separable "two-qubit" states to the total (separable and entangled) states. In this conception, the states would be endowed with the metric corresponding (by the work of Petz-Sudar in J. Math. Phys.) to the specific operator monotone function. Happy to try to clarify further. But probably a "very tall order" for a "functional" to fulfill. "Pie-in-the-sky"? | |
| Aug 20, 2019 at 15:27 | comment | added | J. E. Pascoe | @PaulB.Slater, what was the goal here? That is, the original question. | |
| Jul 30, 2019 at 11:34 | comment | added | Paul B. Slater | J. E. Pascoe--I guess I didn't answer your question. $\log{t}$ clearly doesn't fit the titular relation employed by Denes Petz in his several papers pertaining to the matter. I'll have to re-examine them somewhat more closely, to see the origin of this specific definition. | |
| Jul 29, 2019 at 15:11 | comment | added | Paul B. Slater | J. E. Pascoe--if $t>0$ and $g(t)=\frac{(t-1)}{\log{t}}$, then $g(t)=t g(t^{-1})$, as per the operator monotone relation indicated in the title. | |
| Jul 29, 2019 at 13:23 | comment | added | J. E. Pascoe | Isn't $\log t$ operator monotone? That doesn't seem to satisfy that relation on $g$. | |
| Jul 29, 2019 at 12:58 | answer | added | Paul B. Slater | timeline score: 0 | |
| Jul 28, 2019 at 20:25 | comment | added | Paul B. Slater | Well, the class of operator monotone functions is nondenumerably infinite--and I know at most three function values exactly. I was thinking of a function that would be able to give the values/probabilities for any member of the class. | |
| Jul 28, 2019 at 19:59 | comment | added | მამუკა ჯიბლაძე | Is it acceptable, for example, to declare those values of $f$ that you want to be as you want and declare all other values of $f$ to be zero? | |
| Jul 28, 2019 at 18:22 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
modified title
|
| Jul 28, 2019 at 18:01 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
deleted 10 characters in body; edited title
|
| Jul 28, 2019 at 17:47 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
some rewriting
|
| Jul 28, 2019 at 17:42 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
added 1 character in body
|
| Jul 28, 2019 at 17:29 | comment | added | Paul B. Slater | Thanks, Pietro Majer! Well, certaintly the domain is the set of operator monotone functions--that is those for which $g(t)=t g(t^{-1})$--and the range is $[0,\frac{25}{341}]$. Otherwise, I certainly don't have definite specifications. What are the "possible interpretations"? (I am clearly "fishing" here with no idea, really at all, what the nature of the sought function would be, if it even exists in some sense or other.) | |
| Jul 28, 2019 at 17:21 | comment | added | Paul B. Slater | Theorem 7 of sciencedirect.com/science/article/pii/0024379594002118 tells us (changing $f$ there to $g$ here) that operator monotone functions $g(t)$ satisfy the relation $g(t)=t g(t^{-1})$--as can be checked with $\frac{1+t}{2}$, and the other examples. (So, maybe I should have set up the whole problem using $g$, not $f$.) | |
| Jul 28, 2019 at 17:18 | comment | added | Pietro Majer | Actually it is only clear that you are looking for a function $f$. You should kindly add a definition of its domain, a definition of its co-domain, and a list of properties you want it to have. (Each point in unambiguous way, otherwise the there will be a bunch of possible interpretations!) | |
| Jul 28, 2019 at 17:14 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
second paragraph expanded
|
| Jul 28, 2019 at 17:07 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
f(t) changed to f in title--per comment
|
| Jul 28, 2019 at 17:05 | review | Close votes | |||
| Jul 30, 2019 at 7:42 | |||||
| Jul 28, 2019 at 16:16 | comment | added | Iosif Pinelis | It's still unclear what $t$ is. It is not specified in your post by quantifiers "for all" or "there exist(s)" or in any other way. | |
| Jul 28, 2019 at 16:00 | comment | added | Paul B. Slater | Per comment of Iosif Pinelis, changed $f(t)$ to $f$ at outset of question. Hopefully, the intent of the question is clear. | |
| Jul 28, 2019 at 15:57 | history | edited | Paul B. Slater | CC BY-SA 4.0 |
deleted 3 characters in body
|
| Jul 28, 2019 at 15:30 | comment | added | Iosif Pinelis | What is $t$ in (say) $f(\frac{1+t}{2})=\frac{25}{341}$? Also, what do you mean by "function/functions $f(t)$"? Is $t$ the argument of a function $f$? Then the function is $f$, not $f(t)$. | |
| Jul 28, 2019 at 15:19 | history | edited | Paul B. Slater |
edited tags
|
|
| Jul 28, 2019 at 15:13 | history | asked | Paul B. Slater | CC BY-SA 4.0 |