The Hadamard Operator on one qubit is:
\begin{align*}
H = \tfrac{1}{\sqrt{2}}\left[\,\left(\color{darkgreen}{|0\rangle + |1\rangle}\right)\color{darkblue}{\langle 0|}+\left(\color{darkgreen}{|0\rangle – |1\rangle}\right)\color{darkblue}{\langle 1|}\,\right]
\end{align*}
Show that:
\begin{align*}
H^{\otimes n} = \frac{1}{\sqrt{2^n}}\sum_{x,y}(-1)^{x \cdot y}\,\left|x\rangle \langle y\right|
\end{align*}
I can evaluate things like $H \otimes H$ in practice, but I don't know how to get a general formula for $H^{\otimes n}$. Are there any tricks I could use?
Best Answer
The Keyword here is mathematical induction: Suppose that the formula holds for some $n$ and show that therefore it holds for $n+1$. If you additionally show that it holds for $n=2$, you have shown the general formula for arbitrary $n$.