A random unitary channel (RUC) acting on a quantum system $S$ is any completely positive and trace preserving (CPTP) linear map $\mathcal{E}$ that can be expressed as \begin{align} \mathcal{E}(\rho_S) = \frac{1}{n} \sum_i^n U_i \rho_S U_i^{\dagger}, \end{align} where $\{U_i\}_{i=1}^n$ is a collection of $n$ unitary operators. All RUCs can be implemented using a single bipartite unitary $U_{SB}$ and a source of randomness $B$, i.e. a maximally-mixed state of a sufficiently large dimension: \begin{align} \mathcal{N}(\rho_S, U_{SB}) = \text{Tr}_B\left[ U_{SB}\left(\frac{\mathbb{1}_B}{d_B} \otimes \rho_S\right)U_{SB}^{\dagger}\right], \end{align} with $d_B = n$. The common implementation uses a controlled unitary \begin{align} U_{SB}^{\text{controlled}} = \sum_{i=1}^n |{i}\rangle\langle i|_B \otimes U_i, \end{align} so that $\mathcal{E}(\rho_S) = \mathcal{N}(\rho_S, U_{SB}^{\text{controlled}})$ for all density operators $\rho_S$. The unitary $U_{SB}^{\text{controlled}}$, however, leaves the final state of $SB$ highly correlated and this can be very problematic in certain applications.
My question: Is it possible to implement a fixed RUC with vanishing correlations with the randomness source?
My initial approach: Let us consider the simpler case when $\rho_S$ and $\sigma_{SB}$ are both diagonal density operators. Correlations can be quantified using quantum mutual information which in the present case reads \begin{align} I(S:B) = \left[S(\sigma_S) - S(\rho_S)\right] + \left[S(\sigma_B) - S(\rho_B)\right], \end{align} where $S(\rho) = -\text{Tr} \rho \log \rho $ is the von Neuman entropy. Since the first square bracket is fixed, the only way to reduce $I(S:B)$ is by changing the state of $B$. A naive approach is to consider unitaries of the form $U_{SB} = V_{SB} U_{SB}^{\text{controlled}}$, where $V_{SB}$ is a unitary. The unitary $U_{SB}^{\text{controlled}}$ implements the desired map on $S$, whereas $V_{SB}$ partially decorrelates the two systems by changing only the state of $B$. For example we can choose $V_{SB}$ to be a unitary that acts in the following way: \begin{align} |{i}\rangle_S |{r}\rangle_B \leftrightarrow |{i}\rangle_S |{r'}\rangle_B \qquad \text{for all} \,\, i \in \{1, \ldots, d_S\} \,\,\text{and} \,\, r \neq r' \in \{1, \ldots, d_B\}, \end{align} where for each $r$ we choose $r'$ so that $\left[S(\sigma_B) - S(\rho_B)\right]$ (which is non-positive) is minimal. Although this allows to remove part of correlations, in general it still leaves $SB$ correlated. I'm wondering if there is a way to achieve arbitrarily small correlations, e.g. by increasing the size of the source of randomness, even for the simpler diagonal case.