Timeline for Does approximate equality of quantum states imply operator inequality in a large subspace?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 20 at 21:25 | comment | added | fedja | @Noel Sure. Whatever I post on MO is free for everybody to use in any way, no special permission required :-) | |
| Jul 20 at 20:01 | comment | added | Noel | Hello again @fedja, I am using this fact in a proof in my paper. I would like to include your proof. I think it's a nice technique and people in the quantum community would benefit from it. Let me know if this is okay. This is how it's currently written (attribution is red)- drive.proton.me/urls/KJXWX0T9QW#oV0DwK4fkGBA | |
| Sep 21, 2023 at 18:31 | comment | added | fedja | @Noel Each diagonal sum ($m=k-\ell$ with fixed $\ell=1,\dots,K$) is bounded by $\Sigma_1$ by Cauchy-Schwarz: $|\langle Bx_{k-\ell},x_k\rangle|\le \frac 12[\langle Bx_{k-\ell}, x_{k-\ell}\rangle+\langle Bx_k, x_k\rangle]$ ($B$ is positive definite). | |
| Sep 21, 2023 at 18:10 | comment | added | Noel | Hi again @fedja, what exactly did you have in mind when you said that you can bound $\Sigma_2 \leq 2K \Sigma_1$ because I was writing this out for myself and I can only do something $O(K^2)$ | |
| Aug 1, 2023 at 23:28 | history | edited | Michael | CC BY-SA 4.0 |
Removed an unnecessary comment.
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| Aug 1, 2023 at 15:24 | comment | added | fedja | @Noel 1) Just cite MO; they need citations more than I do :lol: 2) Yes, this technique of "geometric stairway decomposition" (the name is mine, I'm not sure what it is officially called, if anything) is pretty standard in analysis though I haven't seen it applied in linear algebra much. The key is that the "short support" functions/sequences cannot interact much with the "long support" ones (what that means depends on the context), so you can localize to same size entries, which often makes life simpler. I picked the idea from Bourgain's paper on $\Lambda(p)$-sets, but it is quite ubiquitous. | |
| S Aug 1, 2023 at 15:14 | history | edited | fedja | CC BY-SA 4.0 |
added brackets
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| S Aug 1, 2023 at 15:14 | history | suggested | Noel | CC BY-SA 4.0 |
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| Aug 1, 2023 at 15:00 | comment | added | Noel | Thank you, this is great. This answer is still pretty non-trivial to me and those who work with me. I do remember looking at this vector space decomposition but obviously could not go too far. I have two soft questions: 1. if I use this result in my research, how would you like to be cited (you can dm me). 2. if you do indeed think this is a simple problem, has a similar technique been used before and if so where? | |
| Aug 1, 2023 at 14:51 | vote | accept | Noel | ||
| Aug 1, 2023 at 14:50 | review | Suggested edits | |||
| S Aug 1, 2023 at 15:14 | |||||
| Jul 30, 2023 at 16:49 | history | edited | fedja | CC BY-SA 4.0 |
added 56 characters in body
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| Jul 30, 2023 at 16:34 | history | edited | fedja | CC BY-SA 4.0 |
added 4 characters in body
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| Jul 30, 2023 at 15:57 | history | answered | fedja | CC BY-SA 4.0 |