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The von Neumann entropy $-\mathrm{Tr}(\rho\ \log\rho)$ of a quantum thermal state with $\rho=\frac{1}{Z}e^{-\beta H}$ gives the thermal entropy, see e.g. this question.

The von Neumann entropy is a fine-grained entropy, does that mean thermal entropy is also a fine-grained entropy? To my understanding, the thermal entropy (in classical sense) is a coarse-grained entropy. Is there a difference between "classical" and "quantum" thermal entropy? Do the thermal and von Neumann entropies follow the second law?

Moreover, we could purify a thermal state to a thermal field double (TFD). The thermal entropy is given by the entanglement entropy of the TFD. Is this only a trick, or is there any relationship between thermal entropy and entanglement entropy?

Edit: For coarse-grained entropy, I use the definition in section 4 of The entropy of Hawking radiation (2020). When we could only observe some simple observables $A_i$, the coarse-grained entropy is given by the maximum von Neumann entropy of all possible $\tilde{\rho}$ with $-\mathrm{Tr}(\rho A_i)=\mathrm{Tr}(\tilde{\rho} A_i)$.

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    $\begingroup$ Consider to make your question more readable. Also, consider to put some references or links and define what you mean e.g. with "fine-grained" etc. $\endgroup$
    – Tobias Fünke
    Commented Oct 7, 2023 at 7:41
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    $\begingroup$ @Tobias Fünke I think it looks messy because I am asking three questions in one post. I will separate them $\endgroup$
    – gshxd
    Commented Oct 8, 2023 at 3:39
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    $\begingroup$ Yes, that is a good idea. But you still should define what you mean with fine and coarse grained, exactly. Do you know the work of Jaynes? In both the classical and quantum case you get the equilibrium density (operator) from the MaxEnt principle. Regarding your last point: You can quite generally ask whether or not purification is (always) meaningful and to this end, see here. $\endgroup$
    – Tobias Fünke
    Commented Oct 8, 2023 at 5:44
  • $\begingroup$ @Tobias Fünke Yes, I know the work of Jaynes, but I have not read the paper. Can you elaborate a little bit more on the MaxEnt principle? $\endgroup$
    – gshxd
    Commented Oct 10, 2023 at 5:01
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    $\begingroup$ How to Build the Thermofield Double State seems quite relevant for the second part of the question. See also Is purification physically meaningful? $\endgroup$
    – Quillo
    Commented Oct 10, 2023 at 7:05

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