Anti-symmetrization is cheap and useful

We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-$r$ qudit state $ρ=\sum_{i=1}^r λ_i |φ_i\rangle\!\langleφ_i|$, prepare a copy of the state proportional to $Π_{\mathrm{anti}}^r (|φ_1\rangle\otimes \ldots\otimes|φ_r\rangle)$, with