Let ${\lvert e_1 \rangle, \ldots, \lvert e_m \rangle}$ be a basis for the first arbitary space $V$, and write $A = \sum_{i,j=1}^{m} a_{ij} \lvert e_i \rangle\langle e_j \rvert$ (the $(i,j)$-th element of $A$ is $a_{ij}$).
Let ${\lvert f_1 \rangle, \ldots, \lvert f_n \rangle}$ be a basis for the second space $W$, and write $B = \sum_{k,l=1}^{n} b_{kl} \lvert f_k \rangle\langle f_l \rvert$.
Then a basis for the combined system is $\lvert e_i \rangle \otimes \lvert f_j \rangle$, for $i = 1, \ldots, m$ and $j = 1, \ldots, n$. The operator $A \otimes B$ on $V \otimes W$ is given by:
\begin{align}A \otimes B &= \left(\sum_{i,j} a_{ij} \lvert e_i \rangle \langle e_j \rvert \right) \otimes \left(\sum_{k,l} b_{kl} \lvert f_k \rangle \langle f_l \rvert \right)\\ &= \sum_{i,j,k,l} a_{ij} b_{kl} \lvert e_i \rangle \langle e_j \rvert \otimes \lvert f_k \rangle \langle f_l \rvert \\&= \sum_{i,j,k,l} a_{ij} b_{kl} (\lvert e_i \rangle \otimes \langle f_k \rvert) ( \lvert e_j \rangle \otimes\langle f_l \rvert) \end{align}
How is the last step mathematically justified? Intuitively you group column vectors which are bras and linear operators which are kets together. Still I do not get what rule is behind all this?
Best Answer
The last step should be
$$\sum_{i,j,k,l} a_{ij}b_{kl}(\lvert e_i \rangle \otimes \color{red}{\lvert} f_k \color{red}{\rangle}) (\color{red}{\langle} e_j \color{red}{\rvert} \otimes\langle f_l \rvert)$$
For if $\lvert v\rangle \otimes \lvert w\rangle$ is a simple tensor in $V\otimes W$, $$(\lvert e_i\rangle \langle e_j\rvert \otimes \lvert f_k\rangle \langle f_l\rvert)(\lvert v\rangle \otimes \lvert w\rangle) = \lvert e_i\rangle\langle e_j\rvert v\rangle \otimes \lvert f_k\rangle \langle f_l\rvert w\rangle = (\lvert e_i\rangle \otimes \lvert f_k\rangle)\langle e_j\rvert v\rangle \langle f_l\rvert w\rangle$$ that is, $(\lvert e_i\rangle \otimes \lvert f_k\rangle)(\langle e_j\rvert \otimes \langle f_l\rvert)(\lvert v\rangle \otimes \lvert w\rangle)$. Thus $$\lvert e_i\rangle\langle e_j\rvert \otimes \lvert f_k\rangle \langle f_l\rvert = (\lvert e_i \rangle \otimes \lvert f_k\rangle)(\langle e_j \rvert \otimes \langle f_l\rvert)$$