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arXiv:1005.4230 · Physical Review A

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

Published Aug 6, 2010• 26 citations
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Abstract

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 $n$.

DOI
10.1103/physreva.82.022307

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This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs (2003 Phys. Rev. A 67 030301(R)). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximize the average purity of the qubit for a given time , with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it minimizes the average time taken for a qubit to reach a given purity . In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result and the usefulness of the different approaches.
This paper discusses the use of continuous weak measurement and quantum feedback for the rapid purification of the quantum state of a model solid state qubit: a superconducting Cooper pair box. The feedback algorithm uses Jacobs' rapid purification protocol, which starts with a completely mixed state and applies controls to rotate the qubit Bloch vector onto the plane orthogonal to the measurement axis. This rotation maximises the rate of increase of the average purity of the state but can require large changes in the control fields to produce the required rotation. Since solid state qubits have finite controls and feedback channels have limited bandwidth, such rotations may not be practical. This paper applies Jacobs' protocol to the Cooper pair box with realistic control fields.
Feedback control protocols can stabilize and enhance the operation of quantum devices, however, unavoidable delays in the feedback loop adversely affect their performance. We introduce a quantum control methodology, combining open-loop control with quantum filtering, which is not constrained by feedback delays. For the problems studied (rapid purification and rapid measurement) we analytically derive lower bounds on the control performance that are comparable with the best corresponding bounds for feedback protocols.
We propose techniques for implementing two different rapid-state purification schemes, within the constraints present in a superconducting charge qubit system. Both schemes use a continuous measurement of charge (z)(z)(z) measurements and seek to minimize the time required to purify the conditional state. Our methods are designed to make the purification process relatively insensitive to rotations about the xxx-axis, due to the Josephson tunneling Hamiltonian. The first proposed method, based on the scheme of Jacobs [Phys. Rev. A 67, 030301(R) (2003)] uses the measurement results to control bias (z)(z)(z) pulses so as to rotate the Bloch vector onto the xxx-axis of the Bloch sphere. The second proposed method, based on the scheme of Wiseman and Ralph [New J. Phys. 8, 90 (2006)] uses a simple feedback protocol which tightly rotates the Bloch vector about an axis almost parallel with the measurement axis. We compare the performance of these and other techniques by a number of different measures.