Rapid purification of quantum systems by measuring in a feedback-controlled unbiased basis
Joshua Combes, Howard M. Wiseman, Kurt A. Jacobs, A. J. O'Connor · Physical Review A · 2010
Rapid purification by feedback---specifically, reducing the mean impurity faster than by measurement alone---can be achieved by choosing 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. 96, 010504 (2006)] involving continuous measurement of the observable ${J}_{z}$ for a $D$-dimensional system. We rigorously rederive the lower bound $(2/3)(D+1)$ on the achievable speedup 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 speedup factor that scales linearly with $n$.