appendixA.tex

\section{Tensor product and trace operator} 

A useful relation for the product of two tensor products is
\begin{align}
    (A \otimes B) (C \otimes D) = AC \otimes BD,
\end{align}
while the trace of a tensor product can be rewritten as
\begin{align}
    \text{Tr}[A\otimes B] = \text{Tr}[A] \, \text{Tr}[B].
\end{align}
These relations allow to write
\begin{align}
    \text{Tr}[[A\otimes B, C\otimes D]] &= \text{Tr}[(A\otimes B)(C \otimes D)] - \text{Tr}[(C\otimes D)(A \otimes B)] = \nonumber \\
    &= \text{Tr}[AC \otimes BD] - \text{Tr}[CA \otimes DB] = \nonumber \\
    &= \text{Tr}[AC] \, \text{Tr}[BD] - \text{Tr}[CA] \, \text{Tr}[DB]. 
\end{align}
In a similar way, one can prove that 
\begin{align}
    \text{Tr}[(A\otimes B)(C\otimes D)(E \otimes F)] &= \text{Tr}[(AC \otimes BD)(E \otimes F)] - \text{Tr}[ACE \otimes BDF)] = \nonumber \\
    &= \text{Tr}[ACE] \, \text{Tr}[BDF]
\end{align}