So the question came up in a book I am working through. Given a circuit with $n$ qubits, construct a state with only $n$ possible measurement results, each of which has only $1$ of $n$ qubits as $1$, such as $|0001\rangle$, $|0010\rangle$, $|0100\rangle$, $|1000\rangle$, obviously normalized.
The only way I can think to do this is to take the all $|0\rangle$ input state, apply $\operatorname{H}$ to each qubit and then used multiple-controlled $\operatorname{CNOT}$ gates to affect the change on each qubit, but I feel like this won't lead to the desired end state.
To be clear, I am enquiring how to create a $W_n$ state can be arbitrarily prepared, given $n$ qubits.
