of $\theta$, where we parametrize $\bm\tau=(\tau \sin \theta,0,\tau \cos\theta)$. The curves correspond to different values of $\tau$ and therefore illustrate the behavior as one goes from a maximally mixed state ($\tau = 0$) to a pure state ($\tau = 1$). ](plot4.pdf){width="40.00000%"}

Wehrl entropy flux for a general spin
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In the Appendix, we compute the entropy flux given by Eq. (\[Phi\_damping\_angular\]) exactly for a general spin. We also show that the Wehrl entropy flux for $T\rightarrow 0$ simplifies to $$\Phi =  2  \gamma  J [  J +  \langle J_z \rangle ],$$ which is valid for any $J$.

Thus, as with the dephasing noise, with the Wehrl formalism, we obtain a well-behaved result even at zero temperature. The structure of this expression is also surprisingly similar to the structure found for bosonic systems in Ref. [@Santos2017b]. It shows that the flux is related to the difference between the instantaneous value of $\langle J_z \rangle$ and the bath-induced value $-J$ (which is the target state of the amplitude damping at $T\to0$).

Energy flux vs Entropy flux
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We can also relate the entropy flux with the energy flux, assuming a Hamiltonian $H=\omega J_z$. The energy flux is given by $\Phi_E=-\operatorname{tr}(HD(\rho))$, which may 