Timeline for Is a probabilistic implementation of unitaries invertible?
Current License: CC BY-SA 4.0
8 events
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| Aug 26 at 2:42 | comment | added | Hans Schmuber | Sorry, I wasn't very clear! I had intended G to be a function of t in general. So where G appears above I really mean G(t). So the most general case I'm interested in would be to write $E_t = e^{tG(t)}$, provided $t$ is sufficiently small but finite. Naturally, the first thing one might try to do is take logs, this seems to run into problems as discussed above. Thank you again for your help! | |
| Aug 26 at 0:00 | comment | added | user196574 | (Small typo, $\alpha$ should run from $0$ to $3$; i.e. $\sigma^0 = I, \sigma^1 = \sigma^x, \sigma^2 = \sigma^y, \sigma^3 = \sigma^z$.) Let me know if the reasoning is clear; the $E_t$ above is smooth in $t$, and comparing its behavior near $t=0$ and $t=\pi/2$ should rule out a representation of the form $E_t = e^{tG}$. | |
| Aug 25 at 23:44 | comment | added | user196574 | @HansSchmuber For example, we can consider the case where you have $A \to E_t(A) = \frac{1}{4} \sum_{\alpha = 0}^4 e^{-i \sigma^\alpha t} A e^{i \sigma^\alpha t}$. When $t=0$, this returns $A$. This $E_t$ is not generically the depolarizing channel; however, when $t=\pi/2$, this recovers the fully depolarizing channel in my answer. The fully depolarizing channel causes traceless matrices to vanish, but $e^{tG}$ shouldn't be able to cause any nonzero matrix to vanish at finite $t$ if $G$ is well-defined. | |
| Aug 25 at 23:30 | comment | added | user196574 | @HansSchmuber Glad to hear it! I might be misinterpreting your case of interest; is it whether you can generically find such a $G$? I worry that your $E_t$ cannot generically be represented with such a simple time-dependence. | |
| Aug 25 at 22:12 | comment | added | Hans Schmuber | Ah, this answer is very helpful! Thank you! I'm particularly interested in the case where we can write $E_t = \sum_j p_j e^{-it ad_{h_j}}$ as a semi-group $E_t = e^{tG}$ for some generator $G$. I was hoping this could be proved by the existence of the matrix logarithm , but if $E_t$ is not generally invertible it seems this proof method must fail as the matrix log may not exist. | |
| Aug 25 at 22:08 | vote | accept | Hans Schmuber | ||
| Aug 25 at 21:43 | history | edited | user196574 | CC BY-SA 4.0 |
Added another detail about the action on generic
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| Aug 25 at 21:28 | history | answered | user196574 | CC BY-SA 4.0 |