production rate, given by Eq. (\[Pi\_deph\_Wehrl\]). The curves were computed assuming $\tau_x^2 + \tau_y^2 = \tau^2$. The von Neumann measure diverges for a pure state ($\tau \to 1$), whereas the Wehrl measure remains finite. ](plot1.pdf)

Application: spin in a rotating magnetic field
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As an example, let us consider a spin-$1/2$ particle in the presence of a rotating magnetic field. We take the system Hamiltonian to be $$\label{H_example}
H(t) = -\frac{b_0}{2}\sigma_z - \frac{b_1}{2}(\sigma_x\cos(\omega t)+\sigma_y\sin(\omega t)),$$ and assume that the system is also subject to the dephasing dissipator (\[D\_dephasing\]).

The trajectory of the system in the Bloch sphere is shown in Fig. \[fig:bloch\_dephasing\], together with a comparison of the Wehrl and von Neumann entropy production rates. As can be appreciated, the Wehrl entropy production rate is capable of capturing the same features as its von Neumann counterpart, but remains finite throughout the motion.

![\[fig:bloch\_dephasing\] Evolution of a spin-$1/2$ particle under a time-dependent magnetic field \[Eq. (\[H\_example\])\] and a dephasing bath \[Eq. (\[D\_dephasing\])\]. Left: trajectory in the Bloch sphere. Right: Wehrl and von Neumann entropy production rates. We assume the system initially starts in the state $|x_-\rangle = (|z_+\rangle - |z_-\rangle)/\sqrt{2}$ (with $\sigma_z|z_{\pm}\rangle=\pm|z_{\pm}\rangle$). The chosen parameters were $b_0/\lambda=5$, $b_1/\lambda=1$, and $\omega/\lambda=1$. ](deph_ex_a.pdf){width="100.00000%"}

