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

TL;DR

This paper investigates how feedback control using unbiased measurement bases can rapidly purify quantum systems, establishing bounds on the speed-up achievable and extending results to multi-qubit registers.

Contribution

It rigorously derives bounds on the speed-up factor for rapid purification using unbiased measurement bases and extends these results to multi-qubit systems.

Findings

Lower bound on speed-up factor: (2/3)(D+1)

Upper bound on speed-up factor: D^2/2

Speed-up scales linearly with number of qubits n

Abstract

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 $n$.