1. Outer product as a simple tensor
For vectors $u\in\mathbb R^3$ and $v\in\mathbb R^2$, the simple tensor $u\otimes v$ corresponds to the matrix $uv^T$.
$u$
$v$
2. Wedge product and signed area
For $a=(a_1,a_2)$ and $b=(b_1,b_2)$, $a\wedge b=(a_1b_2-a_2b_1)e_1\wedge e_2$.
3. Kronecker product
The Kronecker product is the block matrix $A\otimes B=[a_{ij}B]$.
$A$
$B$
4. Vectorization identity
Check $\operatorname{vec}(AXB)=(B^T\otimes A)\operatorname{vec}(X)$ using column stacking.
5. Two-qubit separability test
A two-qubit state $\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle$ is separable exactly when $\alpha\delta-\beta\gamma=0$.
6. Tensor rank-one construction
An order-3 rank-one tensor has entries $T_{ijk}=a_i b_j c_k$.
7. Independent-study practice tasks with answers
Task A. Compute $u\otimes v$ for $u=(1,2,3)^T$ and $v=(4,5)^T$.
Answer: $uv^T=\begin{bmatrix}4&5\\8&10\\12&15\end{bmatrix}$.
Answer: $uv^T=\begin{bmatrix}4&5\\8&10\\12&15\end{bmatrix}$.
Task B. Compute the signed area of $u=(2,1)$ and $v=(1,3)$.
Answer: $\det\begin{bmatrix}2&1\\1&3\end{bmatrix}=5$.
Answer: $\det\begin{bmatrix}2&1\\1&3\end{bmatrix}=5$.
Task C. Why is $I_2=e_1\otimes e_1+e_2\otimes e_2$ not simple?
Answer: It corresponds to the identity matrix, which has rank $2$, while a simple tensor corresponds to a rank-one outer product.
Answer: It corresponds to the identity matrix, which has rank $2$, while a simple tensor corresponds to a rank-one outer product.
Task D. Decide whether $\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$ is separable.
Answer: No. Its coefficient matrix has rank $2$.
Answer: No. Its coefficient matrix has rank $2$.