\Pi = \Phi + \frac{\gamma}{2}
\frac{2\bar{\tau}_z\tau_z-(\tau^2+\tau_z^2)}{2\bar{\tau}_z}
\;\bigg\{\frac{\tau-(1-\tau^2)\tanh^{-1}(\tau)}{\tau^3}\bigg\}$$ where $\Phi$ is given by Eq. (\[Phi\_damp\_W\]).

For comparison, the von Neumann entropy flux rate \[Eq. (\[Phi\_vN\])\] is $$\label{Phi_damp_vN}
\Phi_{\text{vN}}  =  \gamma\frac{\tanh^{-1}(\bar{\tau}_z)}{\bar{\tau}_z}(\tau_z-\bar{\tau}_z)$$ whereas the entropy production rate (\[Pi\_vN\]) reads $$\label{Pi_damp_vN}
\Pi_{\text{vN}}  =  \Phi_{\text{vN}} - 
\frac{\gamma}{2}\frac{\tanh^{-1}(\tau)}{\tau\bar{\tau}_z}
\bigg( \tau^2 + \tau_z(\tau_z - 2\bar{\tau}_z)\bigg)$$ Note how $\Phi_\text{vN}$ diverges in the limit $\bar{\tau}_z \to - 1$, in agreement with Eq. (\[Phi\_vN\]).

![\[fig:damping\] The entropy production rate \[von Neumann given by Eq. (\[Pi\_damp\_vN\]) and Wehrl given by Eq. (\[Pi\_damp\_spin12\])\] contribution of the amplitude bath for a spin-$1/2$ particle, as a function of $\tau = \tau_z$ for $\bar{\tau}_z = -1/2$. ](plot3.pdf){width="40.00000%"}

In fact, these measures diverge both when the bath is at zero temperature and when the state of the system is pure. A comparison between the von Neumann and Wehrl entropy production rates for the amplitude damping is shown in Fig. \[fig:damping\]. In the limit $T \to 0$ ($\bar{\tau}_z \to -1$), the Wehrl entropy production and fluxes become

[rCl]{} \[eq:Flux.damp.T0\] &=& (1+\_z)\
\[eq:Prod.damp.T0\] &=& + \[+(\^2-1)\^[-1]{}()\]

The behavior of the entropy production $\Pi$ in the limit $T\to 0 $ is shown in Fig. \[fig:damping\_theta\].

![\[fig:damping\_theta\] The Wehrl entropy production rate (\[eq:Prod.damp.T0\]) at $T = 0$ as a function 