Suppose I have a three-qubit entangled state of the following form:
$$ |00\rangle|\psi_1\rangle + |01\rangle|\psi_2\rangle + |10\rangle|\psi_3\rangle + |11\rangle|\psi_4\rangle $$
I refer to the first two qubits as address qubits. The third qubit is the data qubit.
I want to be able to extract some arbitrary $|\psi_i\rangle$, but collapsing the first two qubits and post-selecting has a success probability of $1/4$. And this probability gets even worse with more address qubits: $1/2^n$.
Trying amplitude amplification on the address qubits does not work as the amplitudes of the data qubit also get modified.
Is there any procedure that allows me to increase the success probability of $1/2^n$? Or maybe some proof that it is not possible to do such thing?