Timeline for Simultaneous transformations on states and observables which leave predictions invariant
Current License: CC BY-SA 4.0
17 events
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| Nov 23, 2022 at 21:13 | comment | added | Norbert Schuch | Regarding the idea, it is not that simple -- I have to check the QI statement, but in particular, IIRC it says that any positive map can be written as a CP map and a CP composed with transposition -- but of course, CP maps are not of the form $U\rho U^\dagger$ (but $\sum K_i\rho K_i^\dagger$) unless they satisfy some additional (extremality?) property. | |
| Nov 23, 2022 at 21:12 | comment | added | Norbert Schuch | I understand that ACM's answer helped you settle your original confusion. Yet, it does not answer the question in your post (which I felt from the beginning was not really well-specified), which leaves me with the question whether the next time I should vote to close ... (I had this sentiment for a moment but then I felt you were someone who would clean this up in a timely fashion ... still, accepting this gives the impression you still don't know what your question is.) | |
| Nov 23, 2022 at 21:06 | comment | added | Tobias Fünke | Thanks for your updated answer, indeed interesting. I am of course interested in an answer to this question. Their answer helped me to understand my "original" confusion. Regarding your idea/ last comment in the answer. This is very interesting, and I think it is really necessary that both $\varphi_S$ and $\varphi_O$ should be positive here, no? Then, if your idea is correct, that would determine the general form, i.e. show that any such map, if existent, is made from conjugation + conjugation and transpose, if I understand all of this correctly. | |
| Nov 23, 2022 at 21:04 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Nov 23, 2022 at 20:56 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Nov 23, 2022 at 20:43 | comment | added | Norbert Schuch | @TobiasFünke Now with an improved counterexample. (Linear!) | |
| Nov 23, 2022 at 20:41 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Nov 23, 2022 at 20:12 | comment | added | Norbert Schuch | @TobiasFünke Sure. How much are you actually interested in the abovementioned answer to your question? (Given that you accepted ACuriousMind's answer, which does not answer the question, given that you don't give any information about $\varphi$.) (To be honest, I'd also appreciate to understand this in context, just to know for the future ... ) | |
| Nov 23, 2022 at 19:08 | comment | added | Tobias Fünke | Thanks Norbert, for your time and effort! It is much appreciated! I learned a lot through the discussions. | |
| Nov 23, 2022 at 18:55 | comment | added | Norbert Schuch | @TobiasFünke No -- not yet, at least ;) -- otherwise I would have written another answer. -- Mostly, I think it would be aesthetically pleasing, as it would remove the annoying feature of the complex conjugation that it is a basis dependent notion. -- But it seems like quite some work, basically one has to start to even derive linearity over R etc. from the property you demand. | |
| Nov 23, 2022 at 18:39 | comment | added | Norbert Schuch | @TobiasFünke Thinking about it, complex conjugation might be the only counterexample (up to unitary conjugation, of course). This would indeed be rather neat. | |
| Nov 23, 2022 at 18:28 | comment | added | Norbert Schuch | Let me note that my first example fails if you require tr[...]=tr[...] to hold over $\mathbb C$. On the other hand, if you don't restrict to the positive cone, $X\mapsto -X$ does the job. | |
| Nov 23, 2022 at 18:24 | comment | added | Norbert Schuch | It is all fine. The only thing which is not linear is the map from the observable algebra to their representation. But you did not ask about this (or for this) at all. (Did I mention I feel the question is underspecified for what you actually would like to know? It seems you would like to know under which conditions the unitary equivalence holds; then you should ask for that!) | |
| Nov 23, 2022 at 18:13 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Nov 23, 2022 at 18:06 | history | edited | Norbert Schuch | CC BY-SA 4.0 |
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| Nov 23, 2022 at 18:06 | comment | added | Norbert Schuch | Both. (This will conjugate the expectation value: But it is real.) -- Should I write this out? | |
| Nov 23, 2022 at 18:04 | history | answered | Norbert Schuch | CC BY-SA 4.0 |