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IEEE Conference on Decision and Control (CDC)

Quantum linear feedback control with entanglement assistance

Naoki Yamamoto, Tomoaki Mikami

Published Dec 1, 2015• 0 citations
Publisher

Abstract

The great advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the so-called standard quantum limit. In this paper, we study an analogous strategy utilizing entanglement in a feedback control problem and show that, in the problem of cooling a nano-mechanical oscillator, it indeed can lower the control cost function below the limit attainable by the standard control method.

DOI
10.1109/cdc.2015.7403127

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The great advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the so-called standard quantum limit. In this paper, we study an analogous strategy utilizing entanglement in a feedback control problem and show that, in the problem of cooling a nano-mechanical oscillator, it indeed can lower the control cost function below the limit attainable by the standard control method.
The main advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the standard quantum limit. In this paper, we propose an analogous method utilizing entanglement for the purpose of feedback control. The system considered is a general linear dynamical quantum system, where the control goal can be systematically formulated as a linear quadratic Gaussian control problem based on the quantum Kalman filtering method; in this setting, an entangled input probe field is effectively used to reduce the estimation error and accordingly the control cost function. In particular, we show that, in the problem of cooling an opto-mechanical oscillator, the entanglement-assisted feedback control can lower the stationary occupation number of the oscillator below the limit attainable by the controller with a coherent probe field and furthermore beats the controller with an optimized squeezed probe field.
Abstract Feedback control is an essential component of many modern technologies and provides a key capability for emergent quantum technologies. We extend existing approaches of direct feedback control in which the controller applies a function directly proportional to the output signal (P feedback), to strategies in which feedback determined by an integrated output signal (I feedback), and to strategies in which feedback consists of a combination of P and I terms. The latter quantum PI feedback constitutes the analog of the widely used proportional-integral feedback of classical control. All of these strategies are experimentally feasible and require no complex state estimation. We apply the resulting formalism to two canonical quantum feedback control problems, namely, generation of an entangled state of two remote qubits, and stabilization of a harmonic oscillator under thermal noise under conditions of arbitrary measurement efficiency. These two problems allow analysis of the relative benefits of P, I, and PI feedback control. We find that for the two-qubit remote entanglement generation the best strategy can be a combined PI strategy when the measurement efficiency is less than one. In contrast, for harmonic state stabilization we find that P feedback shows the best performance when actuation of both position and momentum feedback is possible, while when only actuation of position is available, I feedback consistently shows the best performance, although feedback delay is shown to improve the performance of a P strategy here.
Quantum metrology provides a fundamental limit on the precision of multi-parameter estimation, called the Heisenberg limit, which has been achieved in noiseless quantum systems. However, for systems subject to noises, it is hard to achieve this limit since noises are inclined to destroy quantum coherence and entanglement. In this paper, a combined control scheme with feedback and quantum error correction (QEC) is proposed to achieve the Heisenberg limit in the presence of spontaneous emission, where the feedback control is used to protect a stabilizer code space containing an optimal probe state and an additional control is applied to eliminate the measurement incompatibility among three parameters. Although an ancilla system is necessary for the preparation of the optimal probe state, our scheme does not require the ancilla system to be noiseless. In addition, the control scheme in this paper has a low-dimensional code space. For the three components of a magnetic field, it can achieve the highest estimation precision with only a 2-dimensional code space, while at least a 4-dimensional code space is required in the common optimal error correction protocols.
We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. The two approaches to feedback are to a certain extent complementary, and they may be combined to achieve the best results. The resulting combined feedback, performed on a linear system, is closely analogous to classical LQG control theory with residual feedback.
Nano-electro-mechanical devices are now rapidly approaching the point where it will be possible to observe quantum mechanical behavior. However, for such behavior to be visible it is necessary to reduce the thermal motion of these devices down to temperatures in the millikelvin range. Here we consider the use of feedback control for this purpose. We analyze an experimentally realizable situation in which the position of the resonator is continuously monitored by a Single-Electron Transistor. Because the resonator is harmonic, it is possible to use a classical description of the measurement process, and we discuss both the quantum and classical descriptions. Because of this the optimal feedback algorithm can be calculated using classical control theory. We examine the quantum state of the controlled oscillator, and the achievable effective temperature. Our estimates indicate that with current experimental technology, feedback cooling is likely to bring the required milliKelvin temperatures within reach.
We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. Direct feedback may be combined with feedback by estimation, and the resulting combination, performed on a linear system, is closely analogous to classical linear-quadratic-Gaussian control theory with residual feedback.