Let $V$ be a finite dimensional vector space
For $A \in End(V \otimes V)$, i've noticed that often times people index the coefficients with 4 varibles, something like $A^{a,b}_{c,d}$. That being said, $A$ is a linear endomorphism, so we can represent it as a square matrix, so why use four indicies? Can $A$ always be expressed in block diagonal form? I realize $A = A_1 \otimes A_2$ for $A_1,A_2 \in End(V)$... Is it common to write the matrix $A$ as either block diagonal form or as $A_1 \otimes A_2$ where $dim(A_i) = \frac{dim(A)}{2}$? Then, if we take the second partial trace or second transpose, it only effects the second block in the matrix, or the second summand, respectively??
Am I thinking about this all wrong? Insights and references appreciated!
You could use two indices to represent an element of $\mathrm{End}(V\otimes V)$ as a matrix, but given a basis of $V$, the corresponding basis of $V\otimes V$ is doubly-indexed, which means a matrix representing an endomorphism of $V\otimes V$ might benefit from using two indices for input and two indices for output.
For $v\in V$ and $A\in\mathrm{End}(V)$ we may rewrite $Av$ as $\sum A_i^j v_j$ or simply $A_i^j v_j$ (Einstein notation), where $v_j$ are the coordinates of $v$. Well, an element $\xi\in V\otimes V$ has doubly-indexed coordinates, so we may write
$$ A\xi=A_{cd}^{ab}\xi_{ab} $$
when $A\in\mathrm{End}(V\otimes V)$.