entropy production rate $\Pi$ and an entropy flux rate $\Phi$.

Information-theoretic aspects of the Wehrl entropy
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The Wehrl entropy has long been used as an information-theoretic tool when dealing with coherent states. Perhaps the most well-developed approach is that of Refs. [@General1995; @Buek1995], where the authors presented an operational interpretation of $S$ in terms of phase-space measurements subject to an additional filtering device (called quantum ruler) that coarse grains the knowledge acquired in the measurement. This then leads to the so-called sampling entropies, with the Wehrl entropy representing a special example for the case where the quantum ruler is a coherent state. Other operational uses of the Wehrl entropy include nontrivial measures of uncertainty [@Anderson1993; @Zyczkowski2001], measures of localization [@Gnutzmann2001], and its relation to quantum chaos [@Altland2012; @DelReal2013a].

\[sec:dephasing\_channel\] Dephasing channel
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General formulation
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As a first example, we consider the dephasing channel with Lindblad operator $$\label{D_dephasing}
D(\rho) = - \frac{\lambda}{2} [J_z,[J_z,\rho]].$$ This channel does not induce any population changes in the $J_z$ basis, but only causes a loss of coherence. The corresponding phase-space dissipator is simply $$\mathcal{D}(\mathcal{Q}) = - \frac{\lambda}{2}  \mathcal{J}_z (\mathcal{J}_z(\mathcal{Q})),$$ where $\mathcal{J}_z(\mathcal{Q})$ is given in Eq. (\[Jz\_orbital\]).

By replacing this in Eq. (\[contribution\_D\]) and integrating 