I'm trying to understand a paper and have trouble seeing why the following can be written:
$Tr_E\{[ \rho,V] \} = \sigma Tr_E\{\rho_E V\} - Tr_E\{ V \rho_E \} \sigma$, when we know the following definitions
$\rho = \sigma \rho_E$ and $\sigma = Tr_E\{ \rho \}$.
Here $Tr_E$ is the partial trace. I think we have at least two spaces here, space $E$ and space $I$.
And.. $\rho \in I \otimes E$, whereas $\sigma \in I$ and $\rho_E \in E$.
Sorry I couldn't formulate my question in a clearer form. I'm not very familiar with the calculus involving traces. Partial traces - even less so.
I would tremendously appreciate a clarification.
P.S. This is from a physics paper, in case you were wondering. The only explanation I could come up with: $Tr \{ Tr{\rho} \} = Tr{\rho}$, because the partial trace is a sort of a "projection" to a space, so projecting again to the same space wouldn't change the result?