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.
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.
 J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, ``Optimal frequency measurements with maximally correlated states,'' Phys. Rev. A 54, R4649 (1996).
 K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, ``Experimental demonstration of a squeezing-enhanced power-recycled michelson interferometer for gravitational wave detection,'' Phys. Rev. Lett. 88, 231102 (2002).
 S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, ``Improvement of Frequency Standards with Quantum Entanglement,'' Phys. Rev. Lett. 79, 3865 (1997).
 B. M. Escher, R. L. de Matos Filho, and L. Davidovich, ``General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,'' Nat. Phys. 7, 406 (2011).
 J. Kołodyński and R. Demkowicz-Dobrzański, ``Efficient tools for quantum metrology with uncorrelated noise,'' New J. Phys. 15, 073043 (2013).
 Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, ``Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,'' Phys. Rev. A 84, 012103 (2011).
 A. W. Chin, S. F. Huelga, and M. B. Plenio, ``Quantum Metrology in Non-Markovian Environments,'' Phys. Rev. Lett. 109, 233601 (2012).
 A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, ``Ultimate Precision Limits for Noisy Frequency Estimation,'' Phys. Rev. Lett. 116, 120801 (2016).
 J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, ``Fundamental limits to frequency estimation: a comprehensive microscopic perspective,'' New J. Phys. 20, 053009 (2018a).
 A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, ``Noisy frequency estimation with noisy probes,'' New J. Phys. 20, 083008 (2018).
 R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, ``Noisy Metrology beyond the Standard Quantum Limit,'' Phys. Rev. Lett. 111, 120401 (2013).
 P. Sekatski, M. Skotiniotis, and W. Dür, ``Dynamical decoupling leads to improved scaling in noisy quantum metrology,'' New J. Phys. 18, 073034 (2016).
 E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, ``Quantum Error Correction for Metrology,'' Phys. Rev. Lett. 112, 150802 (2014).
 G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, ``Increasing Sensing Resolution with Error Correction,'' Phys. Rev. Lett. 112, 150801 (2014).
 W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, ``Improved Quantum Metrology Using Quantum Error Correction,'' Phys. Rev. Lett. 112, 080801 (2014).
 Y. Matsuzaki and S. Benjamin, ``Magnetic-field sensing with quantum error detection under the effect of energy relaxation,'' Phys. Rev. A 95, 032303 (2017).
 S. Zhou, M. Zhang, J. Preskill, and L. Jiang, ``Achieving the Heisenberg limit in quantum metrology using quantum error correction,'' Nat. Commun. 9, 78 (2018).
 H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, New York, 2010).
 J. F. Ralph, K. Jacobs, and C. D. Hill, ``Frequency tracking and parameter estimation for robust quantum state estimation,'' Phys. Rev. A 84, 052119 (2011).
 P. Six, P. Campagne-Ibarcq, L. Bretheau, B. Huard, and P. Rouchon, ``Parameter estimation from measurements along quantum trajectories,'' in 2015 54th IEEE Conf. Decis. Control, Cdc (IEEE, 2015) p. 7742.
 L. Cortez, A. Chantasri, L. P. García-Pintos, J. Dressel, and A. N. Jordan, ``Rapid estimation of drifting parameters in continuously measured quantum systems,'' Phys. Rev. A 95, 012314 (2017).
 J. F. Ralph, S. Maskell, and K. Jacobs, ``Multiparameter estimation along quantum trajectories with sequential Monte Carlo methods,'' Phys. Rev. A 96, 052306 (2017).
 M. Tsang, H. M. Wiseman, and C. M. Caves, ``Fundamental quantum limit to waveform estimation,'' Phys. Rev. Lett. 106, 090401 (2011).
 S. Ng, S. Z. Ang, T. A. Wheatley, H. Yonezawa, A. Furusawa, E. H. Huntington, and M. Tsang, ``Spectrum analysis with quantum dynamical systems,'' Phys. Rev. A 93, 042121 (2016).
 S. Gammelmark and K. Mølmer, ``Fisher Information and the Quantum Cramér-Rao Sensitivity Limit of Continuous Measurements,'' Phys. Rev. Lett. 112, 170401 (2014).
 K. Macieszczak, M. Guţă, I. Lesanovsky, and J. P. Garrahan, ``Dynamical phase transitions as a resource for quantum enhanced metrology,'' Phys. Rev. A 93, 022103 (2016).
 F. Albarelli, M. A. C. Rossi, M. G. A. Paris, and M. G. Genoni, ``Ultimate limits for quantum magnetometry via time-continuous measurements,'' New J. Phys. 19, 123011 (2017).
 J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, ``Quantum Kalman Filtering and the Heisenberg Limit in Atomic Magnetometry,'' Phys. Rev. Lett. 91, 250801 (2003).
 K. Mølmer and L. B. Madsen, ``Estimation of a classical parameter with Gaussian probes: Magnetometry with collective atomic spins,'' Phys. Rev. A 70, 052102 (2004).
 C. Catana and M. Guţă, ``Heisenberg versus standard scaling in quantum metrology with markov generated states and monitored environment,'' Phys. Rev. A 90, 012330 (2014).
 J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, ``Quantum limits on postselected, probabilistic quantum metrology,'' Phys. Rev. A 89, 052117 (2014).
 L. Zhang, A. Datta, and I. A. Walmsley, ``Precision Metrology Using Weak Measurements,'' Phys. Rev. Lett. 114, 210801 (2015).
 G. B. Alves, B. M. Escher, R. L. de Matos Filho, N. Zagury, and L. Davidovich, ``Weak-value amplification as an optimal metrological protocol,'' Phys. Rev. A 91, 062107 (2015).
 S. Alipour, M. Mehboudi, and A. T. Rezakhani, ``Quantum Metrology in Open Systems: Dissipative Cramér-Rao Bound,'' Phys. Rev. Lett. 112, 120405 (2014).
 M. Beau and A. del Campo, ``Nonlinear Quantum Metrology of Many-Body Open Systems,'' Phys. Rev. Lett. 119, 010403 (2017).
 F. Albarelli, Continuous measurements and nonclassicality as resources for quantum technologies, PhD thesis, Università degli Studi di Milano (2018).
 L. Seveso, F. Albarelli, M. G. Genoni, and M. G. A. Paris, in preparation (2018).
 E. Andersson, J. D. Cresser, and M. J. W. Hall, ``Finding the Kraus decomposition from a master equation and vice versa,'' J. Mod. Opt. 54, 1695 (2007).
 F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, ``ContinuousMeasurementFI,'' (2018), https://github.com/matteoacrossi/ContinuousMeasurementFI.
 J. F. Haase, A. Smirne, S. F. Huelga, J. Kołodyński, and R. Demkowicz-Dobrzański, ``Precision Limits in Quantum Metrology with Open Quantum Systems,'' Quantum Meas. Quantum Metrol. 5, 13 (2018b).
 C. Ahn, H. Wiseman, and K. Jacobs, ``Quantum error correction for continuously detected errors with any number of error channels per qubit,'' Phys. Rev. A 70, 024302 (2004).
 N. Akerman, S. Kotler, Y. Glickman, and R. Ozeri, ``Reversal of photon-scattering errors in atomic qubits,'' Phys. Rev. Lett. 109, 103601 (2012).
 S. S. Szigeti, A. R. R. Carvalho, J. G. Morley, and M. R. Hush, ``Ignorance is bliss: General and robust cancellation of decoherence via no-knowledge quantum feedback,'' Phys. Rev. Lett. 113, 020407 (2014).
 M. G. Genoni, S. Mancini, and A. Serafini, ``Optimal feedback control of linear quantum systems in the presence of thermal noise,'' Phys. Rev. A 87, 042333 (2013).
 M. G. Genoni, J. Zhang, J. Millen, P. F. Barker, and A. Serafini, ``Quantum cooling and squeezing of a levitating nanosphere via time-continuous measurements,'' New J. Phys. 17, 073019 (2015).
 D. K. Burgarth, V. Giovannetti, A. N. Kato, and K. Yuasa, ``Quantum estimation via sequential measurements,'' New J. Phys. 17, 113055 (2015).
 C. Catana, L. Bouten, and M. Guţă, ``Fisher informations and local asymptotic normality for continuous-time quantum Markov processes,'' J. Phys. A 48, 365301 (2015).
 M. Guţă and J. Kiukas, ``Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics,'' J. Mat. Phys. 58, 052201 (2017).
 F. Fröwis, M. Skotiniotis, B. Kraus, and W. Dür, ``Optimal quantum states for frequency estimation,'' New J. Phys. 16, 083010 (2014).
 R. Demkowicz-Dobrzański and L. Maccone, ``Using Entanglement Against Noise in Quantum Metrology,'' Phys. Rev. Lett. 113, 250801 (2014).
 M. Sbroscia, I. Gianani, L. Mancino, E. Roccia, Z. Huang, L. Maccone, C. Macchiavello, and M. Barbieri, ``Experimental ancilla-assisted phase estimation in a noisy channel,'' Phys. Rev. A 97, 032305 (2018).
 L. Diósi and L. Ferialdi, ``General Non-Markovian Structure of Gaussian Master and Stochastic Schrödinger Equations,'' Phys. Rev. Lett. 113, 200403 (2014).
 Yao Ma, Mi Pang, Libo Chen, and Wen Yang, "Improving quantum parameter estimation by monitoring quantum trajectories", Physical Review A 99 3, 032347 (2019).
 Alexander Predko, Francesco Albarelli, and Alessio Serafini, "Time-local optimal control for parameter estimation in the Gaussian regime", Physics Letters A 384 13, 126268 (2020).
 Matteo A. C. Rossi, Francesco Albarelli, Dario Tamascelli, and Marco G. Genoni, "Noisy Quantum Metrology Enhanced by Continuous Nondemolition Measurement", Physical Review Letters 125 20, 200505 (2020).
 Luigi Seveso, Francesco Albarelli, Marco G Genoni, and Matteo G A Paris, "On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank", Journal of Physics A: Mathematical and Theoretical 53 2, 02LT01 (2020).
 Liqiang Liu and Haidong Yuan, "Achieving higher precision in quantum parameter estimation with feedback controls", Physical Review A 102 1, 012208 (2020).
 Kai Bai, Zhen Peng, Hong-Gang Luo, and Jun-Hong An, "Retrieving Ideal Precision in Noisy Quantum Optical Metrology", Physical Review Letters 123 4, 040402 (2019).
 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).
 F. Albarelli, M. Barbieri, M.G. Genoni, and I. Gianani, "A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging", Physics Letters A 384 12, 126311 (2020).
 Sisi Zhou and Liang Jiang, "Optimal approximate quantum error correction for quantum metrology", Physical Review Research 2 1, 013235 (2020).
 Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino, "Photonic quantum metrology", AVS Quantum Science 2 2, 024703 (2020).
 Lahcen Bakmou, Mohammed Daoud, and Rachid ahl laamara, "Multiparameter quantum estimation theory in quantum Gaussian states", Journal of Physics A: Mathematical and Theoretical 53 38, 385301 (2020).
 Lewis A. Clark, Adam Stokes, and Almut Beige, "Quantum jump metrology", Physical Review A 99 2, 022102 (2019).
 Shibdas Roy, "Fundamental noisy multiparameter quantum bounds", Scientific Reports 9 1, 1038 (2019).
 Marco G Genoni and Tommaso Tufarelli, "Non-orthogonal bases for quantum metrology", Journal of Physics A: Mathematical and Theoretical 52 43, 434002 (2019).
 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).
 Ivonne Guevara and Howard M. Wiseman, "Completely positive quantum trajectories with applications to quantum state smoothing", Physical Review A 102 5, 052217 (2020).
 Francesco Albarelli, Matteo A. C. Rossi, and Marco G. Genoni, "Quantum frequency estimation with conditional states of continuously monitored independent dephasing channels", International Journal of Quantum Information 18 01, 1941013 (2020).
 Zibo Miao, Yu Chen, and Haidong Yuan, 2019 IEEE 58th Conference on Decision and Control (CDC) 407 (2019) ISBN:978-1-7281-1398-2.
 Yi Peng and Heng Fan, "Achieving the Heisenberg limit under general Markovian noise using quantum error correction without ancilla", Quantum Information Processing 19 8, 266 (2020).
 Valeria Cimini, Ilaria Gianani, Nicolò Spagnolo, Fabio Leccese, Fabio Sciarrino, and Marco Barbieri, "Calibration of Quantum Sensors by Neural Networks", Physical Review Letters 123 23, 230502 (2019).
 Nathan Shettell and Damian Markham, "Graph States as a Resource for Quantum Metrology", Physical Review Letters 124 11, 110502 (2020).
 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.
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