Timeline for Dependence of trace norm on matrix size for smooth vs. random matrices.
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Aug 15, 2010 at 0:14 | history | edited | Jess Riedel | CC BY-SA 2.5 |
added 681 characters in body
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| S Jul 21, 2010 at 19:03 | vote | accept | Jess Riedel | ||
| S Jul 21, 2010 at 19:02 | vote | accept | Jess Riedel | ||
| S Jul 21, 2010 at 19:03 | |||||
| Jul 21, 2010 at 19:01 | vote | accept | Jess Riedel | ||
| S Jul 21, 2010 at 19:02 | |||||
| Jul 21, 2010 at 1:39 | answer | added | Willie Wong | timeline score: 5 | |
| Jul 20, 2010 at 22:44 | comment | added | Daniel Litt | Ah yup that's what I meant. | |
| Jul 20, 2010 at 22:37 | answer | added | Helge | timeline score: 2 | |
| Jul 20, 2010 at 22:32 | comment | added | Jess Riedel | @Wille: whoops, you're right. | |
| Jul 20, 2010 at 22:22 | comment | added | Jess Riedel | @Helge: Yes, I figured this was a standard result from random matrix theory. Unfortunately, I'm really interested in matrices constructed like $S$. I was including $R$ for contrast. | |
| Jul 20, 2010 at 22:17 | comment | added | Willie Wong | @Jesse, I suspect Daniel meant $[0,1/2]\times (1/2,1]$ and vice versa. | |
| Jul 20, 2010 at 22:10 | history | edited | Jess Riedel |
Removed bad tag
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| Jul 20, 2010 at 22:09 | comment | added | Jess Riedel | The tag has been removed. Thanks for the advice. | |
| Jul 20, 2010 at 22:08 | comment | added | Jess Riedel | But Daniel, isn't true that if $f(x,y) = 1$ on $[0, 1/2]\times [0, 1/2]$ and vanishes otherwise, then $\mathrm{Tr} [\sqrt{S^\dagger S}] = \mathrm{Tr} [S] = d/2$ ? In any case, I agree that the linear behavior of $||S||$ might not survive for carefully chosen $f(x,y)$ (and, of course, for $f=0$), but I think it should be preserved for "most" $f$. | |
| Jul 20, 2010 at 22:00 | comment | added | Helge | I guess the first statement should follow from well-known principles. We know where the density of states for i.i.d. matrices converges to, and then the claim follows by integrating $|x|$ against it. The second statement seems more tricky. | |
| Jul 20, 2010 at 21:54 | comment | added | Jess Riedel | Maybe not. This is the mathematical part of a quantum information calculation, but I don't really know what the "quantum-algebra" tag is. If someone else agrees that it's not appropriate, I'll happily remove the tag. | |
| Jul 20, 2010 at 21:53 | comment | added | Daniel Litt | Note that $||S||=0$ for all $d$ if the support of $f$ is contained in $[0, 1/2]\times [0, 1/2]$ or $[1/2, 1]\times [1/2, 1]$. | |
| Jul 20, 2010 at 21:45 | history | edited | Yemon Choi |
added tag
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| Jul 20, 2010 at 21:45 | comment | added | Yemon Choi | Not sure if the "quantum-algebra" tag is appropriate, but I'm not competent to judge... | |
| Jul 20, 2010 at 21:24 | history | asked | Jess Riedel | CC BY-SA 2.5 |