Let's say I have a ket which is a momentum eigenket $| p \rangle$ and then I measure the position and obtain $|x' \rangle$.
$$ | p \rangle = \int | x \rangle \langle x | p \rangle dx \to | x' \rangle $$
My question is what is the minimum number of ancilla qubits required to simulate this transformation unitarily?
Note: Since the cardinality of kets involved here is $\aleph_1$ I am unaware how to implement this
While talking about knowing the position exactly is a nice theoretical ideal, in practice, you cannot do that. You'll really be asking: "In which 'bin' of width $\delta x$ where $x$ spans from $x_{\min}$ to $x_{\max}$ is the particle confined to?". This means that there's $(x_{\max}-x_{\min})/\delta x$ bins, and so you basically need $$ \log_2\left((x_{\max}-x_{\min})/\delta x\right) $$ qubits to represent that information. Hence, this is the number of ancillas you would need.
