Hadamard Followed by CNOT Creates Bell Entanglement from |00>

The first column of the Hadamard matrix in [L3] gives $H\mathrm{\mid }0⟩=(\mathrm{\mid }0⟩+\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}=\mathrm{\mid }+⟩$. Thus before CNOT the state is $(\mathrm{\mid }00⟩+\mathrm{\mid }10⟩)\mathrm{/}\sqrt{2}$ in the product order of [L4] and [L5].

By [L1], CNOT fixes $\mathrm{\mid }00⟩$ and sends $\mathrm{\mid }10⟩$ to $\mathrm{\mid }11⟩$. Linearity therefore gives the output $(\mathrm{\mid }00⟩+\mathrm{\mid }11⟩)\mathrm{/}\sqrt{2}$ in the Statement.

In the ordered product basis, the coefficient matrix of the output is $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, whose determinant is $1\mathrm{/}2\mathrm{\ne }0$ and whose rank is two. If a nonzero product vector is $a\otimes b$, its coefficient in row $i$, column $j$ is $a_i{b}_j$, so its coefficient matrix is $a{b}^{\mathsf{T}}$. Since $a,b\mathrm{\ne }0$, at least one column is nonzero and every column is a scalar multiple of $a$; its rank is exactly one. A nonzero scalar multiple of a product vector has the same rank-one property, so the output ray is not a product ray. By [L2] it is entangled.