Suppose we are given two sequences of qudits, in some states unknown to us: $(|\psi_1\rangle, ... |\psi_n\rangle)$ and $(|\phi_1\rangle, ... |\phi_n\rangle)$. The qudits are not entangled to each other. What procedure can we use to determine whether these are the same sets? That is, how can we determine whether there exists a permutation $\pi$ s.t. for all $i$, $|\psi_i\rangle = |\phi_{\pi(i)}\rangle$?
As is standard we may assume multiple copies of these sets are given, and we wish to minimise the number of copies that we use.
Can we do any better than using the SWAP test on pairs of qubits $O(n^2)$ times?
