denotes an operator-valued Wigner function (also known as the partially-Wigner transformed density matrix), which is both an operator in the Hilbert space of the $\hat q$’s and a function of the bath coordinates $X$. The partial Wigner transform of an arbitrary operator $\hat\chi$ is analogously given by $$\begin{aligned}
\hat\chi_{\rm W}(X) &=& \int dZe^{iP\cdot Z/\hbar}
\langle Q-\frac{Z}{2}\arrowvert
\hat\chi\arrowvert Q+\frac{Z}{2}\rangle \;.
\label{eq:partWtchi}\end{aligned}$$

Taking the partial Wigner transform of Equation (\[eq:Trq\]) leads to the expression for the average of $\hat\chi$ given in Equation (\[eq:TrW\]). The partial Wigner transform of the Hamiltonian in Equation (\[eq:H\]) is given in Equation (\[eq:H\_W\]).

Upon taking the partial Wigner transform of the QLE, Equation (\[eq:qle\]), and truncating the resulting equation after first order in $\hbar$, one arrives at the QCLE $$\begin{aligned}
\frac{\partial}{\partial t}\hat W(X,t)
&=&-\frac{i}{\hbar}
\left[\hat H_{\rm W},\hat W(X,t)\right]
+ 
\frac{1}{2} \hat H_{\rm W} \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}\hat W(X,t)
-
\frac{1}{2} \hat W(X,t)
\overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}\hat H_{\rm W}
\nonumber\\
&=&
-i{\cal L}\hat W(X,t) \;,
\label{eq:qcle}\end{aligned}$$ where the last equality defines the quantum Liouville operator $i{\cal L}=(i/\hbar)[\hat H_{\rm W},\cdot] - (1/2)
(\hat H_{\rm W} \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}} \cdot)
+ (1/2)
( \cdot \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}} \hat H_{\rm W})$. To arrive at Equation (\[eq:qcle\]), we have used 