Rapid-purification by feedback --- specifically, reducing the mean impurityfaster than by measurement alone --- can be achieved by making the eigenbasisof the density matrix to be unbiased relative to the measurement basis. Here wefurther 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 thatscales linearly with $n$.

Rapid purification of quantum systems by measuring in a feedback-controlled unbiased basis