Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment

Francesco Albarelli1,2, Matteo A. C. Rossi1,3, Dario Tamascelli1, and Marco G. Genoni1

1Quantum Technology Lab, Dipartimento di Fisica ``Aldo Pontremoli'', Università degli Studi di Milano, IT-20133, Milan, Italy
2Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
3QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland

We study quantum frequency estimation for $N$ qubits subjected to independent Markovian noise, via strategies based on time-continuous monitoring of the environment. Both physical intuition and an extended convexity property of the quantum Fisher information (QFI) suggest that these strategies are more effective than the standard ones based on the measurement of the unconditional state after the noisy evolution. Here we focus on initial GHZ states and on parallel or transverse noise. For parallel noise, i.e. dephasing, we show that perfectly efficient time-continuous photo-detection allows to recover the unitary (noiseless) QFI, and thus to obtain a Heisenberg scaling for every value of the monitoring time. For finite detection efficiency, one falls back to the noisy standard quantum limit scaling, but with a constant enhancement due to an effective reduced dephasing. Also in the transverse noise case we obtain that the Heisenberg scaling is recovered for perfectly efficient detectors, and we find that both homodyne and photo-detection based strategies are optimal. For finite detectors efficiency, our numerical simulations show that, as expected, an enhancement can be observed, but we cannot give any conclusive statement regarding the scaling. We finally describe in detail the stable and compact numerical algorithm that we have developed in order to evaluate the precision of such time-continuous estimation strategies, and that may find application in other quantum metrology schemes.

Parameter estimation and metrology are fundamental parts of science and technology.
Everyday life applications in telecommunication, transport and medicine are copious, as well as in fundamental research, with the paradigmatic example of gravitational waves detection.
Thanks to quantum mechanics, we are now able to design a new generation of ultra-precise measurement devices with the potential to greatly outperform the most advanced classical strategies.
However, the promised enhancement is easily lost whenever the quantum system interacts with an environment, questioning the possibility of implementing these metrological schemes in any practical relevant scenario.
Our paper presents a way to overcome this ostensible no-go theorem.

We consider frequency estimation for an ensemble of two-level atoms (the standard setup of Ramsey spectroscopy), initially prepared in an entangled (GHZ) state and each subject to independent noise, i.e. each interacting with its own separate environment.
We show that continuously measuring the modes of the environment allows to restore the promised quantum advantage, without requiring feedback or error correction schemes.
This holds true for two important classes of noise, parallel and transverse to the Hamiltonian imprinting the rotation.
Furthermore, even if inefficient detectors can in principle destroy the improvement in terms of the scaling in the number of atoms, we show that with our approach there is always a non-trivial gain in precision.

Our results provide a promising new path for noisy quantum metrology, paving the way for fruition of quantum-enhanced parameter estimation protocols in realistic setups.

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[1] J. F. Haase, A. Smirne, S. F. Huelga, J. Kołodynski, and R. Demkowicz-Dobrzanski, "Precision Limits in Quantum Metrology with Open Quantum Systems", Quantum Measurements and Quantum Metrology 5 1, 2 (2016).

[2] Emanuele Roccia, Valeria Cimini, Marco Sbroscia, Ilaria Gianani, Ludovica Ruggiero, Luca Mancino, Marco G. Genoni, Maria Antonietta Ricci, and Marco Barbieri, "Multiparameter quantum estimation of noisy phase shifts", arXiv:1805.02561 (2018).

[3] Lewis A. Clark, Adam Stokes, and Almut Beige, "Quantum jump metrology", Physical Review A 99 2, 022102 (2019).

[4] Shibdas Roy, "Fundamental noisy multiparameter quantum bounds", Scientific Reports 9 1, 1038 (2019).

[5] Yao Ma, Mi Pang, Libo Chen, and Wen Yang, "Improving quantum parameter estimation by monitoring quantum trajectories", Physical Review A 99 3, 032347 (2019).

[6] Louis Garbe, Simone Felicetti, Perola Milman, Thomas Coudreau, and Arne Keller, "Metrological advantage at finite temperature for Gaussian phase estimation", Physical Review A 99 4, 043815 (2019).

[7] Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, Simone De Liberato, and Franco Nori, "Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance", Physical Review A 98 6, 063815 (2018).

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