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Chinese Control Conference (CCC)

Improving metrological limit via weak measurement and quantum measurement reversal

Lijun Liu, Bo Qi, Shuming Cheng, Zairong Xi

Published Jul 1, 2015• 2 citations
Publisher

Abstract

Quantum parameter estimation is of fundamental importance in physics. Quantum resources, involving for instance entangled probes, offer a significant improvement in estimation precision over classical strategies for closed quantum systems. However, when noises such as decoherence are taken into account, the quantum gain may be jeopardized. In this paper, we show how the amplitude-damping decoherence degrades the estimation precision from the Heisenberg limit to the standard limit with the initial Greenberger-Horne-Zeilinger (GHZ) state being employed and demonstrate that the method of weak measurement and quantum measurement reversal can recover this estimation precision to the Heisenberg limit, even though at the price of a very low success probability.

DOI
10.1109/chicc.2015.7260956

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Quantum parameter estimation is of fundamental importance in physics. Quantum resources, involving for instance entangled probes, offer a significant improvement in estimation precision over classical strategies for closed quantum systems. However, when noises such as decoherence are taken into account, the quantum gain may be jeopardized. In this paper, we show how the amplitude-damping decoherence degrades the estimation precision from the Heisenberg limit to the standard limit with the initial Greenberger-Horne-Zeilinger (GHZ) state being employed and demonstrate that the method of weak measurement and quantum measurement reversal can recover this estimation precision to the Heisenberg limit, even though at the price of a very low success probability.
For single-parameter sensing, Greenberger-Horne-Zeilinger (GHZ) probes achieve optimal quantum-enhanced precision across the unknown parameter range, solely relying on parameter-independent separable measurement strategies for all values of the unknown parameter. However, in the multiparameter setting, a single GHZ probe not only fails to achieve quantum advantage but also the corresponding optimal measurement becomes complex and dependent on the unknown parameters. Here, we provide a recipe for multiparameter sensing with GHZ probes using quantum error correction techniques by treating all but one unknown parameters as noise, whose effects can be corrected. This strategy restores the core advantage of single parameter GHZ-based quantum sensing, namely reaching optimally quantum-enhanced precision for all unknown parameter values while keeping the measurements separable and fixed. Specifically, given one shielded ancilla qubit per GHZ probe, our protocol extracts optimal possible precision for any probe size. While this optimal precision is shot-noise limited for a single GHZ probe, we recover the Heisenberg scaling through use of multiple complementary GHZ probes. We demonstrate the effectiveness of the protocol with Bayesian estimation.
Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss. Quantum metrology employs the properties of quantum states to further enhance the accuracy of some of the most precise measurement schemes to date. Here, a method for estimating the upper bounds to achievable precision in quantum-enhanced metrology protocols in the presence of decoherence is presented.
Abstract We investigate the effect of the weak measurement (WM) and quantum measurement reversal (QMR) on the improvement of the precision of parameter estimation for a qubit interacting with various non-Markovian decoherence sources: amplitude damping, dephasing damping and random unitary damping, respectively. We derive the exact expressions of the optimal quantum Fisher information, show explicitly that the WM and QMR operation indeed helps for enhancing the precision of parameter estimation from amplitude damping and phase damping, but not for random unitary damping. In particular, we found that the enhancement effect is independent of the specific form of the spectral density of the reservoir, which extends the applicable scope of the WM and QMR approach.
Abstract Maximal quantum entanglement provided by N00N states is a unique resource in the quest for the ultimate precision in physical measurements. Such states, however, are fragile and prone to decoherence. Even in weak-measurement schemes, as we demonstrate in this work, the phase super-resolution provided by N00N states is achieved at a cost of an N -fold enhancement of amplitude damping. Still, as the analysis presented here shows, a partial collapse of N00N states induced by weak measurements can be reversed, despite the dramatic, N -fold enhancement of amplitude damping, through appropriate reversal operations on the post-measurement state, enabling a full restoration of the Heisenberg-limit phase super-resolution of N00N states.
Quantum sensing leverages non-classical resources to enhance precision. In particular, Greenberger-Horne-Zeilinger (GHZ) states can, in principle, attain the Heisenberg limit that surpasses the standard quantum limit. While many studies have examined how open-system noise-typically modeled with Lindblad master equations-degrades GHZ-based metrology, coherent control imperfections during state preparation and readout have received less attention. Here, we analyze the effect of detuning between actual and nominal spin frequencies in a GHZ-state preparation scheme employing a frequency selective pulse. We show that detuning induces coherent, systematic error that prevents GHZ sensing from reaching the Heisenberg limit. To mitigate this effect, we design a composite-pulse protocol that compensates for detuning-induced errors and improves the sensitivity under the effect of coherent error.
The enhancement of the precision of phase estimation in quantum metrology is investigated by employing weak measurement (WM) and quantum measurement reversal (QMR). We derive the exact expressions of the optimal quantum Fisher information (QFI) and success probability of phase estimation for an exactly solving model consisting of a qubit interacting with a structured reservoir. We show that the QFI can be obviously enhanced by means of the WM and QMR in different regimes. In addition, we also show that the magnitude of the decoherence involved in the WM and QMR can be a general complex number, which extends the applicable scope of the WM and QMR approach.