Quantum entanglement

Remarks

Example: let $H_A = H_B = \C^2$ with basis $|0\rangle, |1\rangle$, and $\psi = (|00\rangle + |11\rangle)/\sqrt{2}$. Suppose $\psi = (a_0|0\rangle + a_1|1\rangle) \otimes (b_0|0\rangle + b_1|1\rangle)$. Comparing coefficients gives $a_0 b_0 = a_1 b_1 = 1/\sqrt{2}$ and $a_0 b_1 = a_1 b_0 = 0$. The last two equations force one of $a_0, b_1$ and one of $a_1, b_0$ to vanish, which contradicts the first two, so $\psi$ is entangled.

Non-example: $(|00\rangle + |01\rangle)/\sqrt{2} = |0\rangle \otimes (|0\rangle + |1\rangle)/\sqrt{2}$ is a product state, so it is not entangled.

For pure states the two definitions agree: the projection $|\psi\rangle\langle\psi|$ is separable exactly when $\psi$ is a product vector. In finite dimension, a vector $\psi = \sum_{i,j} c_{ij} e_i \otimes f_j$ is a product state exactly when the coefficient matrix $(c_{ij})$ has rank $1$.