How do you write the Hadamard operator on two qubits in braket notation?

Question

I understand how to write the Hadamard operator on one qubit in braket notation using $$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} H=\sum_{i,j} \bra{w_{j}}H\ket{v_{i}} \ket{w_{j}} \bra{v_{i}}$$ where $$H_{ij}= \sum_{i,j} \bra{w_{j}}H\ket{v_{i}} $$

From my understanding, $\ket{w_{i}}=\ket{v_{i}}=\ket{0}$ and $\ket{w_{j}}=\ket{v_{j}}=\ket{1}$

But I don't understand how to translate the matrix into braket notation when we range over $i$ and $j$ in ${0,1,2,3}$. Does the new basis become $\ket{00}, \ket{01}, \ket{10}, \ket{11}$?

Answer 1

Hadamard in Dirac notation is $$ H=\frac{1}{\sqrt{2}}(|0\rangle\langle 0|+|0\rangle\langle 1|+|1\rangle\langle 0|-|1\rangle\langle 1|). $$ For two qubits, you take the tensor product $H\otimes H$. You get $$ \frac12\left(|00\rangle\langle 00|+|00\rangle\langle 01|+|00\rangle\langle 10|+|00\rangle\langle 11| +|01\rangle\langle 00|-|01\rangle\langle 01|+|01\rangle\langle 10|-|01\rangle\langle 11| +|10\rangle\langle 00|+|10\rangle\langle 01|-|10\rangle\langle 10|-|10\rangle\langle 11| +|11\rangle\langle 00|-|11\rangle\langle 01|-|11\rangle\langle 10|+|11\rangle\langle 11|\right) $$ (essentially, you're looking for combinations $|1\rangle\langle 1|$ to put a negative sign on). If I were to write this as a formula, I'd have $$ H=\frac{1}{\sqrt{2}}\sum_{x,y\in\{0,1\}}(-1)^{x\cdot y}|x\rangle\langle y| $$ and $$ H\otimes H=\frac{1}{2}\sum_{x_1,x_2,y_1,y_2\in\{0,1\}}(-1)^{x_1\cdot y_1}(-1)^{x_2\cdot y_2}|x_1x_2\rangle\langle y_1y_2|. $$