B2 - qubits

\begin{matrix} | 0 ⟩ = (\begin{matrix} 1 \\ 0 \end{matrix}) \\ | 1 ⟩ = (\begin{matrix} 0 \\ 1 \end{matrix}) \end{matrix}

⟨ 0 | 1 ⟩ = 0 = ⟨ 1 | 0 ⟩

\begin{array}{r} | 0 ⟩ ⟨ 0 | = (\begin{array}{c} 1 \\ 0 \end{array}) (\begin{array}{c} 1 & 0 \end{array}) + (\begin{array}{c} 0 \\ 1 \end{array}) (\begin{array}{c} 0 & 1 \end{array}) = (\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}) = | 1 ⟩ ⟨ 1 | \end{array}

| ψ_{1} ⟩ \otimes | ψ_{2} ⟩ = (\begin{matrix} a_{1} \\ b_{1} \end{matrix}) \otimes (\begin{matrix} a_{2} \\ b_{2} \end{matrix}) = (\begin{matrix} a_{1} (\begin{matrix} a_{2} \\ b_{2} \end{matrix}) \\ b_{2} (\begin{matrix} a_{2} \\ b_{2} \end{matrix}) \end{matrix}) = (\begin{matrix} a_{1} a_{2} \\ a_{1} b_{2} \\ b_{1} a_{2} \\ b_{1} b_{2} \end{matrix})