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arXiv:0709.3867 · Physical Review A

Rapid-purification protocols for optical homodyning

Aravind Chiruvelli, Kurt A. Jacobs

Published Jan 8, 2008• 6 citations
arXivHTMLPDF

Abstract

We present a number of rapid-purification feedback protocols for optical homodyne detection of a single optical qubit. We derive first a protocol that speeds up the rate of increase of the average purity of the system, and find that as in the equivalent protocol for a nondissipative measurement, this generates a deterministic evolution for the purity in the limit of strong feedback. We also consider two analogs of the Wiseman-Ralph rapid-purification protocol in this setting, and show that as in that protocol they speed up the average time taken to reach a fixed level of purity. We also examine how the performance of these algorithms changes with detection efficiency, being an important practical consideration.

DOI
10.1103/physreva.77.012102

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We present a number of rapid-purification feedback protocols for optical homodyne detection of a single optical qubit. We derive first a protocol that speeds up the rate of increase of the average purity of the system, and find that as in the equivalent protocol for a nondissipative measurement, this generates a deterministic evolution for the purity in the limit of strong feedback. We also consider two analogs of the Wiseman-Ralph rapid-purification protocol in this setting, and show that as in that protocol they speed up the average time taken to reach a fixed level of purity. We also examine how the performance of these algorithms changes with detection efficiency, being an important practical consideration.
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.
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.
We consider qubit purification under simultaneous continuous measurement of the three non-commuting qubit operators σ(x), σ(y), σ(z). The purification dynamics is quantified by (i) the average purification rate and (ii) the mean time of reaching a given level of purity, 1-ε. Under ideal measurements (detector efficiency η=1), we show in the first case an asymptotic mean purification speed-up of 4 when compared with a standard (classical) single-detector measurement. However, by the second measure-the mean time of first passage T(ε) of the purity-the corresponding speed-up is only 2. We explain these speed-ups using the isotropy of the qubit evolution that provides an equivalence between the original measurement directions and three simultaneous measurements, one with an axis aligned along the Bloch vector and the other with axes in the two complementary directions. For inefficient detectors, η=1 - δ < 1, the mean time of first passage T(δ,ε)increases because qubit purification competes with an isotropic qubit dephasing. In the asymptotic high-purity limit (ε,δ≪1), we show that the increase possesses a scaling behaviour: ΔT(δ,ε) is a function only of the ratio δ/ε. The increase ΔT(δ,ε) is linear for small arguments, but becomes exponential ~exp(δ,2ε) for δ/ε large.
In this paper, we consider feedback control algorithms for the rapid purification of a bipartite state consisting of two qubits, when the observer has access to only one of the qubits. We show (1) the algorithm maximizing the average purification rate is different from that for a projective measurement and (2) that the weak measurement protocol for bipartite systems contrasts with known single qubit protocols. We also reveal that the rate of increase in purity may be made deterministic for bipartite states.
Quantum feedback control protocols can improve the operation of quantum devices. Here we examine the performance of a purification protocol when there are imperfections in the controls. The ideal feedback protocol produces an x -eigenstate from a mixed state in the minimum time, and is known as rapid state preparation. The imperfections we examine include time delays in the feedback loop, finite strength feedback, calibration errors and inefficient detection. We analyse these imperfections using the Wiseman–Milburn feedback master equation and related formalism. We find that the protocol is most sensitive to time delays in the feedback loop. For systems with slow dynamics, however, our analysis suggests that inefficient detection would be the bigger problem. We also show how system imperfections, such as dephasing and damping, can be included in a model via the feedback master equation.
The preparation stage of optical qubits is an essential task in all the experimental setups employed for the test and demonstration of quantum optics principles. We consider a deterministic protocol for the preparation of qubits as a superposition of vacuum and one photon number states, which has the advantage to reduce the amount of resources required via phase-sensitive measurements using a local oscillator ('dyne detection'). We investigate the performances of the protocol using different phase measurement schemes: homodyne, heterodyne, and adaptive dyne detection (involving a feedback loop). First, we define a suitable figure of merit for the prepared state and we obtain an analytical expression for that in terms of the phase measurement considered. Further, we study limitations that the phase measurement can exhibit, such as delay or limited resources in the feedback strategy. Finally, we evaluate the figure of merit of the protocol for different mode-shapes handily available in an experimental setup. We show that even in the presence of such limitations simple feedback algorithms can perform surprisingly well, outperforming the protocols when simple homodyne or heterodyne schemes are employed.