Consider the following function $$f_s: k \rightarrow \lvert \psi_k \rangle$$ where $s,k$ are bit strings, and $\lvert \psi_k \rangle$ is a $n$-qubit state. Assume the function is a one-to-one mapping.
Given only $\big(f, k, \lvert \psi_k \rangle \big) $ is there a way to produce a (zero knowledge) proof $P$ that only verifies that $\lvert \psi_k \rangle$ has been generated from $k$, but not learn any more information about $\lvert \psi_k \rangle$ or $s$?
Digression (to the classical world where factoring is difficult)
Consider a classical function $f_s: k \rightarrow p$ so a function $f$ does the following $f_p(q) = p*q*r*s$, where $p,q,r,s$ are all large primes, now given only $(f,q, n=p*q*r*s)$, one can easily check $q|n$, but cannot learn $p$ since that requires factoring $p*r*s$
I am looking for something similar for the quantum regime.