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).
 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. 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 Yao Ma, Mi Pang, Libo Chen, and Wen Yang, "Improving quantum parameter estimation by monitoring quantum trajectories", Physical Review A 99 3, 032347 (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).
 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).
The above citations are from Crossref's cited-by service (last updated 2019-05-20 19:40:59) and SAO/NASA ADS (last updated 2019-05-20 19:41:01). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.