Timeline for Why can every $|X\rangle\in H_1\otimes H_0$ be written as $|X\rangle=(X\otimes I_{H_0})|\Omega \rangle$ for some $X\in\mathcal L(H_0,H_1)$?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 19, 2020 at 9:31 | comment | added | JSdJ | You're welcome! Daftwullie's answer is indeed really nice :) | |
| Aug 19, 2020 at 9:27 | comment | added | Marco Fellous-Asiani | Thanks for your answer/comment. It solved a part of my problem but DaftWullie gave me the exact step I was looking for so I accepted his answer instead. Thanks anyway for your instructive answer ! | |
| Aug 18, 2020 at 19:06 | comment | added | JSdJ | You might be able to use the first identity on this wikipedia page about the vectorization operator. (Coincidentally, this is also identity you can use to arrive at a 'natural representation' of a channel as a big matrix, which you asked about in one of your other questions today) | |
| Aug 18, 2020 at 19:05 | comment | added | JSdJ | You're welcome! The $d$ is indeed because of normalization - I've been very sloppy with normalization in the above derivation (I also swept under the rug that the $a$ and $b$ vectors aren't properly normalized Hilbert-space vectors, but you can check that it works out in the end (if you still don't agree, I would be interested in a discussion :) )). I'm not $100 \%$ sure about the second question though - the partial trace is a vital part of the derivation so I don't think that the mapping works always. I'll have to think about it more. | |
| Aug 18, 2020 at 18:54 | comment | added | Marco Fellous-Asiani | Thank you for your answer. Two questions: why do you have a $d$ that multiplies $tr_2$ in your definition of the isomorphism ? Is it because in your convention $| \Omega \rangle$ is normalized (whereas in mine it is not) ? Second question: does that mean in an indirect way that indeed any vector belonging in $H_1 \otimes H_0$ can be written under the form $A \otimes \mathcal{I} | \Omega \rangle$ ? | |
| Aug 18, 2020 at 18:39 | history | answered | JSdJ | CC BY-SA 4.0 |