Tensor product of Hadamard Operators

Question

The Hadamard Operator on one qubit is:

$$\begin{align*} H = \frac{1}{\sqrt{2}}\left[(|0\rangle + |1\rangle)\langle 0|+(|0\rangle - |1\rangle)\langle 1|\right] \end{align*}$$

Show that:

$$\begin{align*} H^{\otimes n} = \frac{1}{\sqrt{2^n}}\sum_{x,y}(-1)^{x \cdot y}|x\rangle \langle y| \end{align*}$$

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?

This post imported from StackExchange Physics at 2014-03-09 09:18 (UCT), posted by SE-user user82235

Answer 1

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 $n$.

This post imported from StackExchange Physics at 2014-03-09 09:18 (UCT), posted by SE-user pressure

Comments

Thanks, it's been a long time since I used induction, I forgot it existed

This post imported from StackExchange Physics at 2014-03-09 09:18 (UCT), posted by SE-user user82235