Timeline for Proof that the reduced density matrix of free fermions is thermal?
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Feb 27, 2021 at 17:37 | history | bounty ended | FriendlyLagrangian | ||
| S Feb 27, 2021 at 17:37 | history | notice removed | FriendlyLagrangian | ||
| Feb 27, 2021 at 17:37 | vote | accept | FriendlyLagrangian | ||
| Feb 26, 2021 at 21:00 | history | tweeted | twitter.com/StackPhysics/status/1365406322234183686 | ||
| Feb 26, 2021 at 20:38 | history | edited | Norbert Schuch |
edited tags
|
|
| Feb 26, 2021 at 18:10 | answer | added | Norbert Schuch | timeline score: 7 | |
| Feb 26, 2021 at 15:49 | comment | added | Norbert Schuch | Happy to help. I guess I'll have to write an answer. | |
| Feb 26, 2021 at 14:31 | comment | added | FriendlyLagrangian | @NorbertSchuch I would offer you the bounty but I don't know how :( | |
| Feb 26, 2021 at 14:15 | comment | added | FriendlyLagrangian | That's so true... I'm jealous of your knowledge! Thank you so much for your help and patience Norbert :) | |
| Feb 26, 2021 at 13:56 | comment | added | Norbert Schuch | Because a state is equivalent to all possible expectation values. States associate an expectation to every possible expectation value - nothing more, and nothing less. Thus, they are fully characterized by all possible expectation values. Anything else which is not specified by those would be entirely unphysical. | |
| Feb 26, 2021 at 12:40 | comment | added | FriendlyLagrangian | @NorbertSchuch I agree with (1) & (2). If I understood you correctly, your argument boils down to: if I find a form for $\rho_b$ that reproduces all the n-point correlators correctly then this is valid (I agree) and unique (?). I don't understand why one can't find another different form of $\rho_B$ such that it also reproduces the correct n-point functions. | |
| Feb 25, 2021 at 19:07 | comment | added | Norbert Schuch | I'm saying: (1) If I know all n-point correlators, for any n, this uniquely determined the state. Is that part ok? If yes, then (2) Gaussian states satisfy Wick's theorem. Is that part ok? Then, from (2): Gaussian states satisfy Wick's theorem, which fixes all n-point correlators, and from (1): A state where I know all n-point correlators is determined uniquely, so => it must be the Gaussian states (as it satisfies the desired properties, and such a state is unique). | |
| Feb 25, 2021 at 10:38 | comment | added | FriendlyLagrangian | @NorbertSchuch so are you saying: n-point correlators can be determined from 2-point correlators $\implies \rho_B \sim \exp (-\sum_{ij} M_i^{\dagger} C_{ij} M_j)$ (with $dim(\rho_B)=dim(M_i)$). But $\exp (-\sum_{ij} M_i^{\dagger} C_{ij} M_j)$ satisfies Wick’s theorem $\implies \rho_B = \exp (-\sum_{ij} c_i^{\dagger} h_{ij} c_j)$ (with $c_i$ being the original fermionic operators)? I might need a little bit more help with this, I still dont see why this last form is unique. | |
| Feb 24, 2021 at 20:32 | comment | added | Norbert Schuch | If you have Wick's theorem for arbitrary (many-operator) correlations, the "iff" follows trivially since the state is fully specified by all such correlations; if they can be determined from the two-point correlations then this must be the Gaussian state, since the Gaussian state satisfies Wick's theorem and, as stated, the state is determined uniquely by all correlations. Does this answer the unclear direction? | |
| S Feb 22, 2021 at 0:54 | history | bounty started | FriendlyLagrangian | ||
| S Feb 22, 2021 at 0:54 | history | notice added | FriendlyLagrangian | Draw attention | |
| Feb 19, 2021 at 12:00 | history | edited | FriendlyLagrangian | CC BY-SA 4.0 |
added 123 characters in body
|
| Feb 19, 2021 at 11:54 | comment | added | FriendlyLagrangian | Right, my problem is that this might not be the unique form of $\rho_B$. The above to me is a bit meaningless without the iff. | |
| Feb 19, 2021 at 11:53 | comment | added | OON | The iff part of the argument is what really is dubious. The "if" works according to their reference [8] that proves the variant of the Wick theorem (that represents the products as sum of two-particle correlation functions and smaller product) to compute $\mathrm{Tr}(\rho abc\ldots)$ for the thermal state | |
| Feb 19, 2021 at 11:37 | history | edited | FriendlyLagrangian | CC BY-SA 4.0 |
added 12 characters in body
|
| Feb 19, 2021 at 11:26 | history | edited | FriendlyLagrangian | CC BY-SA 4.0 |
added 127 characters in body
|
| Feb 19, 2021 at 11:20 | comment | added | FriendlyLagrangian | @Vadim Im adding this, thanks for the feedback! Well, once you trace over the degrees of freedom of $A$, the subsystem left (described by $\rho_B$) “looks” thermal, or that’s what they claim. It is not really surprising, one can view the rest of the system as a thermal bath to the degrees of freedom at $B$. I will also include that $\langle \cdot \rangle_{\psi}=\langle \psi | \cdot | \psi \rangle.$ | |
| Feb 19, 2021 at 11:13 | comment | added | Roger V. | I think you need to put all this in the question, since these are essential details. I still don't see though how a term thermal can be applied to a single state. By the way, what is the averaging in your proof - the thermal averaging? | |
| Feb 19, 2021 at 11:11 | review | Close votes | |||
| Feb 22, 2021 at 1:04 | |||||
| Feb 19, 2021 at 11:08 | comment | added | FriendlyLagrangian | @Vadim The subscript $B$ indicates subsystem $B$, indeed those two equations are not describing the same thing. Given an eigenstate $|\psi\rangle$ and its density matrix $\rho=|\psi\rangle\langle\psi|$ the reduced density matrix $\rho_B$ in a say, bi-partite system $A\cup B$ is obtained by tracing out $A$: $\rho_B := \text{tr}_A(\rho)$. Does this make more sense? | |
| Feb 19, 2021 at 10:57 | history | edited | Urb | CC BY-SA 4.0 |
deleted 66 characters in body
|
| Feb 19, 2021 at 10:56 | comment | added | Roger V. | There is a discrepancy in notation between the first and the second equation. Also, the question seems to lack essential information about how the reduced density matrix is defined, what is its relation to $\psi$ etc. The question just doesn't make sense in its current form. | |
| Feb 19, 2021 at 10:52 | history | edited | FriendlyLagrangian | CC BY-SA 4.0 |
added 2 characters in body
|
| Feb 19, 2021 at 9:30 | history | asked | FriendlyLagrangian | CC BY-SA 4.0 |