Bell state

Example

Let $|0\rangle, |1\rangle$ be the standard basis of $\C^2$, and write $|jk\rangle = |j\rangle \otimes |k\rangle$ in $\C^2 \otimes \C^2$. The four Bell states are $$|\Phi^{\pm}\rangle = \frac{|00\rangle \pm |11\rangle}{\sqrt{2}}, \qquad |\Psi^{\pm}\rangle = \frac{|01\rangle \pm |10\rangle}{\sqrt{2}}.$$ They are unit vectors and pairwise orthogonal, so they form an orthonormal basis of $\C^2 \otimes \C^2$, the Bell basis. Each of them is entangled. For $|\Phi^+\rangle$, the coefficient matrix in the standard basis is $\tfrac{1}{\sqrt 2} I$, which has rank $2$, so it is not a product vector. The same holds for the other three, whose coefficient matrices are also invertible.