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arXiv:quant-ph/0508007 · Physical Review Letters

Rapid State Reduction of Quantum Systems Using Feedback Control

Joshua Combes, Kurt A. Jacobs

Published Jan 13, 2006• 86 citations
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Abstract

We consider using Hamiltonian feedback control to increase the speed at which a continuous measurement purifies (reduces) the state of a quantum system, and thus to increase the speed of the preparation of pure states. For a measurement of an observable with equispaced eigenvalues, we show that there exists a feedback algorithm which will speed up the rate of state reduction by at least a factor of 2(N + 1)/3.

DOI
10.1103/physrevlett.96.010504

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Reconsidering Rapid Qubit Purification by Feedback

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Abstract. This paper reconsiders the properties of a scheme for the rapid 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 maximise 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 mininises 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. Reconsidering Rapid Qubit Purification by Feedback 2 1.

•Mar 8, 2006•36 citations
We consider using Hamiltonian feedback control to increase the speed at which a continuous measurement purifies (reduces) the state of a quantum system, and thus to increase the speed of the preparation of pure states. For a measurement of an observable with equispaced eigenvalues, we show that there exists a feedback algorithm which will speed up the rate of state reduction by at least a factor of 2(N + 1)/3.
When one performs a continuous measurement, whether on a classical or quantum system, the measurement provides a certain average rate at which one becomes certain about the state of the system. For a quantum system this is an average rate at which the system is projected onto a pure state. We show that for a standard kind of continuous measurement, for a qubit this rate may be increased by applying unitary operations during the measurement (that is, by using Hamiltonian feedback), in contrast to the equivalent measurement on a classical bit, where reversible operations cannot be used to enhance the rate of entropy reduction. We determine the optimal feedback algorithm and discuss the Hamiltonian resources required.
We consider the use of feedback control during a measurement to increase the rate at which a single qubit is purified, and more generally the rate at which near-pure states may be prepared. We derive the optimal bang-bang algorithm for rapid state preparation from an initially completely mixed state when the measurement basis is unrestricted, and evaluate its performance numerically. We also consider briefly the case in which the measurement basis is fixed with respect to the state to be prepared, and describe the qualitative structure of the optimal bang-bang algorithm.
We consider the use of feedback control during a measurement to increase the rate at which a single qubit is purified, and more generally the rate at which near-pure states may be prepared. We derive the optimal bang-bang algorithm for rapid state preparation from an initially completely mixed state when the measurement basis is unrestricted, and evaluate its performance numerically. We also consider briefly the case in which the measurement basis is fixed with respect to the state to be prepared, and describe the qualitative structure of the optimal bang-bang algorithm.
Abstract. This paper reconsiders the properties of a scheme for the rapid 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 maximise 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 mininises 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. Reconsidering Rapid Qubit Purification by Feedback 2 1.