Timeline for Does this derivation of the reduced density matrix really prove it containing "all measurement statistics"?
Current License: CC BY-SA 4.0
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| Jun 23, 2022 at 12:28 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 14:07 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 13:52 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 13:51 | vote | accept | manuel459 | ||
| Jun 21, 2022 at 13:50 | comment | added | manuel459 | Forget my last comment. I can fully follow your thoughts now! Now it totally makes sense to me. Thank you for the dialogue! | |
| Jun 21, 2022 at 13:50 | comment | added | Tobias Fünke | As a side note: Every density matrix of a system can be seen as a reduced density matrix of a composite system (see purification) | |
| Jun 21, 2022 at 13:48 | comment | added | manuel459 | In other words: The postulates work with densitry matrices that are descriptions of systems (which is based on physical thoughts). But reduced density matrices (by now) only formal identical objects that (in my opinion) not necessarily are a descritpion of the subsystem - this has to be proven using postulates and the densitry matrix $\rho$. | |
| Jun 21, 2022 at 13:48 | comment | added | Tobias Fünke | @manuel459 But this is what I did, at least tried to. I've added a few sentences... Let me know if something is still unclear. I've started with the rules of quantum mechanics (for bipartite systems) and then, using the properties of the reduced density matrix, derived the equations $(3)$ and $(5)$. | |
| Jun 21, 2022 at 13:47 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 13:44 | comment | added | manuel459 | I think this hits the nail on his head: Why do I know I can calculate it this way? this foralisms are used for density matrices and I wonder (while reduced density matrices are mathematically identical to density matrices) if this is true for reduced density matrices. It seems to me one might need to prove that using (3), (4) and so on give the same solutions one gets when considering probabilities and states after measurement when considering $\rho$ and $O_{\mathcal{A}}\otimes\mathbb{I}$ using the respective formalisms. | |
| Jun 21, 2022 at 13:43 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 13:36 | comment | added | Tobias Fünke | To summarize: The right-hand sides of $(3)$ and $(5)$ are results, derived from the postulates of quantum mechanics, and not postulates themselves. | |
| Jun 21, 2022 at 13:35 | comment | added | Tobias Fünke | Well, in my opinion, the above considerations prove this: All predictions you can make regarding observable properties of $H_A$ require only knowledge of $\rho_A$. Put differently: Suppose you know $\rho$. Then you can calculate all this stuff (this is axiomatic). Now if you are interested in the statistics of the subsystem only, then the above calculations show that the knowledge of $\rho_A$ suffices. I did nowhere postulate that we compute the probabilities etc. from $\rho_A$; I derived that. Do I miss or misunderstand something? | |
| Jun 21, 2022 at 13:31 | comment | added | manuel459 | I understand what you are saying, but still have a little worry. Let me try to put it into words: Since the reduced density matrix is a density matrix and the above expected value can be calculated - how can I infer that one is allowed to calculate state after measurement and measure-probability as it is done when considering density matrices? This could already be done once you know the reduced density matrix is a density matrix - no need for proving that the expected value can be calculated? It seems there is a non mathematical, rather physical thought still to be seen by myself. | |
| Jun 21, 2022 at 13:27 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 13:17 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 12:57 | history | edited | Tobias Fünke | CC BY-SA 4.0 |
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| Jun 21, 2022 at 12:55 | history | undeleted | Tobias Fünke | ||
| Jun 21, 2022 at 12:40 | history | deleted | Tobias Fünke | via Vote | |
| Jun 21, 2022 at 12:35 | history | answered | Tobias Fünke | CC BY-SA 4.0 |