  Rapid-purification by feedback --- specifically, reducing the mean impurity
faster than by measurement alone --- can be achieved by making the eigenbasis
of the density matrix to be unbiased relative to the measurement basis. Here we
further examine the protocol introduced by Combes and Jacobs [Phys.Rev.Lett.
{\bf 96}, 010504 (2006)] involving continuous measurement of the observable
$J_z$ for a $D$-dimensional system. We rigorously re-derive the lower bound
$(2/3)(D+1)$ on the achievable speed-up factor, and also an upper bound, namely
$D^2/2$, for all feedback protocols that use measurements in unbiased bases.
Finally we extend our results to $n$ independent measurements on a register of
$n$ qubits, and derive an upper bound on the achievable speed-up factor that
scales linearly with $n$.
