Commutativity of $\rho_{AB}$ and $\mathbb{I}_A\otimes\rho_B$ Suppose the density operator of a composite system $AB$ is given by $\rho_{AB}$ and $\rho_B =\mathrm{Tr}_A(\rho_{AB})$ the marginal density operator of the sub-system $B$. I have some doubt whether $\rho_{AB}$ always commutes with $\mathbb{I}_A\otimes \rho_B$?
Is there a way to show this? Or is there some counter-example?
 A: It does not hold in general. Here is a class of counter examples:
Let $\rho^A_1, \rho^A_2$ and $\rho^B_1,\rho^B_2$ be density matrices and consider for $0 < p_1,p_2 < 1$ with $p_1+ p_2=1$ the following (separable, mixed) bipartite density matrix
$$\rho = p_1 \, \rho^A_1 \otimes \rho^B_1  + p_2\, \rho^A_2 \otimes \rho^B_2  \tag{1} \quad .$$
The reduced density matrix of the subsystem $B$ is
$$\rho_B= p_1\, \rho^B_1 + p_2\, \rho^B_2 \quad .\tag{2} $$
We compute:
\begin{align}
[\rho,\mathbb I_A\otimes\rho_B] &= \sum\limits_{k,q=1}^2 p_k\, p_q\, [\rho^A_k\otimes \rho^B_k, \mathbb I_A\otimes \rho^B_q] \tag{3} \\
&= \sum\limits_{k,q=1}^2 p_k\, p_q\, \rho^A_k \otimes [\rho^B_k,\rho^B_q] \tag{4}\\
&= p_1\, p_2\,\left( \rho^A_1 - \rho^A_2 \right)\otimes [\rho^B_1,\rho^B_2] \quad . \tag{5}
\end{align}
Note that if $\rho^A_1- \rho^A_2 \neq 0$ and $[\rho^B_1,\rho^B_2] \neq 0$, then $[\rho,\mathbb I_A\otimes\rho_B] \neq 0$. Indeed, both conditions mean that there exists at least one (non-zero) vector in each Hilbert space such that the respective action of these operators does not result in the zero vector. Let $|\psi_1\rangle$ and $|\psi_2\rangle$ denote these vectors; then $[\rho,\mathbb I_A\otimes\rho_B] |\psi_1\rangle \otimes |\psi_2\rangle \neq 0$, which means that the commutator cannot be zero.
So to construct an explicit counter example, simply take two non-equal density matrices for the subsystem $A$ and two non-commuting density matrices on the subsystem $B$ and construct the bipartite density matrix as in $(1)$. Finally, we note that the proposition however holds true for all density matrices (pure or mixed) of the form $\rho=\rho_A\otimes \rho_B$. The proof is trivial.
