Quantum metrology with full and fast quantum control

Pavel Sekatski1, Michalis Skotiniotis1,2, Janek Kołodyński3, and Wolfgang Dür1

1Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
2Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona), Spain
3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We establish general limits on how precise a parameter, e.g. frequency or the strength of a magnetic field, can be estimated with the aid of full and fast quantum control. We consider uncorrelated noisy evolutions of N qubits and show that fast control allows to fully restore the Heisenberg scaling (~1/N^2) for all rank-one Pauli noise except dephasing. For all other types of noise the asymptotic quantum enhancement is unavoidably limited to a constant-factor improvement over the standard quantum limit (~1/N) even when allowing for the full power of fast control. The latter holds both in the single-shot and infinitely-many repetitions scenarios. However, even in this case allowing for fast quantum control helps to increase the improvement factor. Furthermore, for frequency estimation with finite resource we show how a parallel scheme utilizing any fixed number of entangled qubits but no fast quantum control can be outperformed by a simple, easily implementable, sequential scheme which only requires entanglement between one sensing and one auxiliary qubit.

► BibTeX data

► References

[1] Jonathan P. Dowling and Kaushik P. Seshadreesan. Quantum Optical Technologies for Metrology, Sensing, and Imaging. J. Lightwave Technol., 33 (12): 2359–2370, June 2015. ISSN 0733-8724. 10.1109/​JLT.2014.2386795.

[2] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature Photon., 5: 222–229, 2011. 10.1038/​nphoton.2011.35.

[3] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum-enhanced measurements: Beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004. 10.1126/​science.1104149.

[4] V. Bužek, R. Derka, and S. Massar. Optimal quantum clocks. Phys. Rev. Lett., 82: 2207–2210, Mar 1999. 10.1103/​PhysRevLett.82.2207.

[5] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Phys. Rev. Lett., 96: 010401, 2006. 10.1103/​PhysRevLett.96.010401.

[6] B. C. Sanders and G. J. Milburn. Optimal quantum measurements for phase estimation. Phys. Rev. Lett., 75: 2944–2947, Oct 1995. 10.1103/​PhysRevLett.75.2944.

[7] D. W. Berry and H. M. Wiseman. Optimal states and almost optimal adaptive measurements for quantum interferometry. Phys. Rev. Lett., 85: 5098–5101, Dec 2000. 10.1103/​PhysRevLett.85.5098.

[8] Asher Peres and Petra F. Scudo. Entangled quantum states as direction indicators. Phys. Rev. Lett., 86: 4160–4162, Apr 2001. 10.1103/​PhysRevLett.86.4160.

[9] E. Bagan, M. Baig, and R. Muñoz Tapia. Quantum reverse engineering and reference-frame alignment without nonlocal correlations. Phys. Rev. A, 70: 030301, Sep 2004. 10.1103/​PhysRevA.70.030301.

[10] G. Chiribella, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi. Efficient use of quantum resources for the transmission of a reference frame. Phys. Rev. Lett., 93: 180503, Oct 2004a. 10.1103/​PhysRevLett.93.180503.

[11] Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Massimiliano F. Sacchi. Covariant quantum measurements that maximize the likelihood. Phys. Rev. A, 70: 062105, Dec 2004b. 10.1103/​PhysRevA.70.062105.

[12] G. Chiribella, G. M. D'Ariano, and M. F. Sacchi. Optimal estimation of group transformations using entanglement. Phys. Rev. A, 72: 042338, Oct 2005. 10.1103/​PhysRevA.72.042338.

[13] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde. Entanglement-free heisenberg-limited phase estimation. Nature, 450: 393, 2007. 10.1038/​nature06257.

[14] B. Yurke. Input states for enhancement of fermion interferometer sensitivity. Phys. Rev. Lett., 56: 1515–1517, Apr 1986. 10.1103/​PhysRevLett.56.1515.

[15] 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: 3865–3868, Nov 1997. 10.1103/​PhysRevLett.79.3865.

[16] Konrad Banaszek, Rafał Demkowicz-Dobrzański, and Ian A. Walmsley. Quantum states made to measure. Nature Photon., 3: 673–676, 2009. 10.1038/​nphoton.2009.223.

[17] Lorenzo Maccone and Vittorio Giovannetti. Quantum metrology: Beauty and the noisy beast. Nature Phys., 7: 376–377, 2011. doi:10.1038/​nphys1976.

[18] Akio Fujiwara and Hiroshi Imai. A fibre bundle over manifolds of quantum channels and its application to quantum statistics. Journal of Physics A: Mathematical and Theoretical, 41 (25): 255304, 2008. 10.1088/​1751-8113/​41/​25/​255304.

[19] 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, March 2011. 10.1038/​nphys1958.

[20] B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho. Quantum metrological limits via a variational approach. Phys. Rev. Lett., 109: 190404, Nov 2012. 10.1103/​PhysRevLett.109.190404.

[21] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă. The elusive heisenberg limit in quantum-enhanced metrology. Nat. Commun., 3: 1063, 2012. 10.1038/​ncomms2067.

[22] J. Kołodyński and R. Demkowicz-Dobrzański. Efficient tools for quantum metrology with uncorrelated noise. New J. Phys., 15 (7): 073043, 2013. 10.1088/​1367-2630/​15/​7/​073043.

[23] S. Alipour, M. Mehboudi, and A. T. Rezakhani. Quantum metrology in open systems: Dissipative cramér-rao bound. Phys. Rev. Lett., 112: 120405, Mar 2014. 10.1103/​PhysRevLett.112.120405.

[24] Sergey Knysh, Vadim N. Smelyanskiy, and Gabriel A. Durkin. Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state. Phys. Rev. A, 83: 021804, Feb 2011. 10.1103/​PhysRevA.83.021804.

[25] Sergey I Knysh, Edward H Chen, and Gabriel A Durkin. True limits to precision via unique quantum probe. preprint, arXiv: 1402.0495[quant–ph], 2014. URL https:/​/​arxiv.org/​abs/​1402.0495.

[26] John Preskill. Quantum clock synchronization and quantum error correction. preprint, arXiv: 0010098[quant–ph], 2000. URL http:/​/​arxiv.org/​abs/​quant-ph/​0010098.

[27] W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus. Improved quantum metrology using quantum error correction. Phys. Rev. Lett., 112: 080801, Feb 2014. 10.1103/​PhysRevLett.112.080801.

[28] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin. Quantum error correction for metrology. Phys. Rev. Lett., 112: 150802, Apr 2014. 10.1103/​PhysRevLett.112.150802.

[29] G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker. Increasing sensing resolution with error correction. Phys. Rev. Lett., 112: 150801, Apr 2014. 10.1103/​PhysRevLett.112.150801.

[30] Roee Ozeri. Heisenberg limited metrology using quantum error-correction codes. preprint, arxiv: 1310.3432[quant–ph], 2013. URL https:/​/​arxiv.org/​abs/​1310.3432.

[31] Xiao-Ming Lu, Sixia Yu, and CH Oh. Robust quantum metrological schemes based on protection of quantum fisher information. Nat. Commun., 6: 7282, 2015. 10.1038/​ncomms8282.

[32] David A. Herrera-Martí, Tuvia Gefen, Dorit Aharonov, Nadav Katz, and Alex Retzker. Quantum error-correction-enhanced magnetometer overcoming the limit imposed by relaxation. Phys. Rev. Lett., 115: 200501, Nov 2015. 10.1103/​PhysRevLett.115.200501.

[33] Tuvia Gefen, David A. Herrera-Martí, and Alex Retzker. Parameter estimation with efficient photodetectors. Phys. Rev. A, 93: 032133, Mar 2016. 10.1103/​PhysRevA.93.032133.

[34] Martin B. Plenio and Susana F. Huelga. Sensing in the presence of an observed environment. Phys. Rev. A, 93: 032123, Mar 2016. 10.1103/​PhysRevA.93.032123.

[35] P Sekatski, M Skotiniotis, and W Dür. Dynamical decoupling leads to improved scaling in noisy quantum metrology. New J. Phys., 18 (7): 073034, 2016. 10.1088/​1367-2630/​18/​7/​073034.

[36] Duger Ulam-Orgikh and Masahiro Kitagawa. Spin squeezing and decoherence limit in ramsey spectroscopy. Phys. Rev. A, 64: 052106, Oct 2001. 10.1103/​PhysRevA.64.052106.

[37] Rafal Demkowicz-Dobrzański and Lorenzo Maccone. Using entanglement against noise in quantum metrology. Phys. Rev. Lett., 113: 250801, Dec 2014. 10.1103/​PhysRevLett.113.250801.

[38] R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín. Noisy metrology beyond the standard quantum limit. Phys. Rev. Lett., 111: 120401, Sep 2013. 10.1103/​PhysRevLett.111.120401.

[39] Lorenza Viola and Seth Lloyd. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A, 58: 2733–2744, Oct 1998. 10.1103/​PhysRevA.58.2733.

[40] Lorenza Viola, Emanuel Knill, and Seth Lloyd. Dynamical decoupling of open quantum systems. Phys. Rev. Lett., 82: 2417–2421, Mar 1999. 10.1103/​PhysRevLett.82.2417.

[41] Lorenza Viola and Emanuel Knill. Robust dynamical decoupling of quantum systems with bounded controls. Phys. Rev. Lett., 90: 037901, Jan 2003. 10.1103/​PhysRevLett.90.037901.

[42] Kaveh Khodjasteh and Lorenza Viola. Dynamically error-corrected gates for universal quantum computation. Phys. Rev. Lett., 102: 080501, Feb 2009. 10.1103/​PhysRevLett.102.080501.

[43] Kaveh Khodjasteh, Daniel A. Lidar, and Lorenza Viola. Arbitrarily accurate dynamical control in open quantum systems. Phys. Rev. Lett., 104: 090501, Mar 2010. 10.1103/​PhysRevLett.104.090501.

[44] Jacob R. West, Daniel A. Lidar, Bryan H. Fong, and Mark F. Gyure. High fidelity quantum gates via dynamical decoupling. Phys. Rev. Lett., 105: 230503, Dec 2010. 10.1103/​PhysRevLett.105.230503.

[45] Howard M Wiseman and Gerard J Milburn. Quantum Measurement and Control. Cambridge University Press, 2009. ISBN 0521804426. 10.1017/​CBO9780511813948.

[46] Giulio Chiribella. Optimal networks for quantum metrology: semidefinite programs and product rules. New Journal of Physics, 14 (12): 125008, 2012. 10.1088/​1367-2630/​14/​12/​125008.

[47] Alexandr Sergeevich, Anushya Chandran, Joshua Combes, Stephen D. Bartlett, and Howard M. Wiseman. Characterization of a qubit hamiltonian using adaptive measurements in a fixed basis. Phys. Rev. A, 84: 052315, Nov 2011. 10.1103/​PhysRevA.84.052315.

[48] Mankei Tsang. Ziv-zakai error bounds for quantum parameter estimation. Phys. Rev. Lett., 108: 230401, Jun 2012. 10.1103/​PhysRevLett.108.230401.

[49] Richard D Gill and Boris Y Levit. Applications of the van Trees inequality: a Bayesian Cramér-Rao bound. Bernoulli, 1(1/​2): 59–79, 1995. 10.2307/​3318681.

[50] C. W. Helstrom. Quantum Detection and Estimation Theory. Academic Press, 1976. ISBN 0123400503.

[51] A. S. Holevo. Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Series in Statistics and Probability, 1980. 10.1007/​978-88-7642-378-9.

[52] Samuel L. Braunstein and Carlton M. Caves. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72: 3439–3443, May 1994. 10.1103/​PhysRevLett.72.3439.

[53] Luca Pezzé and Augusto Smerzi. Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett., 102: 100401, Mar 2009. 10.1103/​PhysRevLett.102.100401.

[54] Bernd Lücke, Jan Peise, Giuseppe Vitagliano, Jan Arlt, Luis Santos, Géza Tóth, and Carsten Klempt. Detecting multiparticle entanglement of dicke states. Phys. Rev. Lett., 112: 155304, Apr 2014. 10.1103/​PhysRevLett.112.155304.

[55] Helmut Strobel, Wolfgang Muessel, Daniel Linnemann, Tilman Zibold, David B. Hume, Luca Pezzè, Augusto Smerzi, and Markus K. Oberthaler. Fisher information and entanglement of non-gaussian spin states. Science, 345 (6195): 424–427, 2014. 10.1126/​science.1250147.

[56] Diego Paiva Pires, Marco Cianciaruso, Lucas C. Céleri, Gerardo Adesso, and Diogo O. Soares-Pinto. Generalized geometric quantum speed limits. Phys. Rev. X, 6: 021031, Jun 2016. 10.1103/​PhysRevX.6.021031.

[57] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho. Quantum speed limit for physical processes. Phys. Rev. Lett., 110: 050402, Jan 2013. 10.1103/​PhysRevLett.110.050402.

[58] Florian Fröwis and Wolfgang Dür. Measures of macroscopicity for quantum spin systems. New J. Phys., 14 (9): 093039, 2012. 10.1088/​1367-2630/​14/​9/​093039.

[59] M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge university press, 2010. 10.1017/​CBO9780511976667.

[60] Robert Alicki and Karl Lendi. Quantum Dynamical Semigroups and Applications. Springer, 1987. 10.1007/​3-540-18276-4.

[61] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. 10.1093/​acprof:oso/​9780199213900.001.0001.

[62] Yuichiro Matsuzaki, Simon C. Benjamin, and Joseph Fitzsimons. Magnetic field sensing beyond the standard quantum limit under the effect of decoherence. Phys. Rev. A, 84: 012103, Jul 2011. 10.1103/​PhysRevA.84.012103.

[63] Alex W. Chin, Susana F. Huelga, and Martin B. Plenio. Quantum metrology in non-markovian environments. Phys. Rev. Lett., 109: 233601, Dec 2012. 10.1103/​PhysRevLett.109.233601.

[64] Katarzyna Macieszczak. Zeno limit in frequency estimation with non-markovian environments. Phys. Rev. A, 92: 010102, Jul 2015. 10.1103/​PhysRevA.92.010102.

[65] Andrea Smirne, Jan Kołodyński, Susana F. Huelga, and Rafał Demkowicz-Dobrzański. Ultimate precision limits for noisy frequency estimation. Phys. Rev. Lett., 116: 120801, Mar 2016. 10.1103/​PhysRevLett.116.120801.

[66] Carole Addis, Elsi-Mari Laine, Clemens Gneiting, and Sabrina Maniscalco. Problem of coherent control in non-Markovian open quantum systems. Phys. Rev. A, 94: 052117, Nov 2016. 10.1103/​PhysRevA.94.052117.

[67] 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 (12): 1695–1716, 2007. 10.1080/​09500340701352581.

[68] J. B. Brask, R. Chaves, and J. Kołodyński. Improved quantum magnetometry beyond the standard quantum limit. Phys. Rev. X, 5: 031010, Jul 2015. 10.1103/​PhysRevX.5.031010.

[69] T. H. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, and R. Hanson. Universal control and error correction in multi-qubit spin registers in diamond. Nat. Nanotechnol., 9 (3): 171–176, March 2014. 10.1038/​nnano.2014.2.

[70] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbruggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann, and J. Wrachtrup. Quantum error correction in a solid-state hybrid spin register. Nature, 506 (7487): 204–207, February 2014. ISSN 0028-0836. 10.1038/​nature12919.

[71] Thomas Unden, Priya Balasubramanian, Daniel Louzon, Yuval Vinkler, Martin B. Plenio, Matthew Markham, Daniel Twitchen, Alastair Stacey, Igor Lovchinsky, Alexander O. Sushkov, Mikhail D. Lukin, Alex Retzker, Boris Naydenov, Liam P. McGuinness, and Fedor Jelezko. Quantum metrology enhanced by repetitive quantum error correction. Phys. Rev. Lett., 116: 230502, Jun 2016. 10.1103/​PhysRevLett.116.230502.

[72] F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik. Autonomous Quantum Error Correction and Application to Quantum Sensing with Trapped Ions. preprint, arXiv: 1702.08673[quant–ph], 2017. URL http:/​/​arxiv.org/​abs/​1702.08673.

[73] R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski. Adaptive quantum metrology under general Markovian noise. preprint, arXiv: 1704.06280[quant–ph], 2017. URL http:/​/​arxiv.org/​abs/​1704.06280.

[74] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. Achieving the Heisenberg limit in quantum metrology using quantum error correction. preprint, arXiv: 1706.02445[quant–ph], 2017. URL http:/​/​arxiv.org/​abs/​1706.02445.

[75] Ingemar Bengtsson and Karol Życzkowski. Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2006. 10.1017/​CBO9780511535048.

Cited by

[1] Jeremy T. Young, Alexey V. Gorshkov, and I. B. Spielman, "Feedback-stabilized dynamical steady states in the Bose-Hubbard model", Physical Review Research 3 4, 043075 (2021).

[2] N. H. Abdel-Wahab, T. A. S. Ibrahim, Magdy E. Amin, and Ahmed Salah, "Influence of intrinsic decoherence on quantum metrology of two atomic systems in the presence of dipole–dipole interaction", Optical and Quantum Electronics 56 1, 105 (2024).

[3] Emanuele Roccia, Valeria Cimini, Marco Sbroscia, Ilaria Gianani, Ludovica Ruggiero, Luca Mancino, Marco G. Genoni, Maria Antonietta Ricci, and Marco Barbieri, "Multiparameter approach to quantum phase estimation with limited visibility", Optica 5 10, 1171 (2018).

[4] Simon Morelli, Ayaka Usui, Elizabeth Agudelo, and Nicolai Friis, "Bayesian parameter estimation using Gaussian states and measurements", Quantum Science and Technology 6 2, 025018 (2021).

[5] David Layden, Sisi Zhou, Paola Cappellaro, and Liang Jiang, "Ancilla-Free Quantum Error Correction Codes for Quantum Metrology", Physical Review Letters 122 4, 040502 (2019).

[6] Chungwei Lin, Yanting Ma, and Dries Sels, "Optimal control for quantum metrology via Pontryagin's principle", Physical Review A 103 5, 052607 (2021).

[7] Hai-Yuan Hong, Xiu-Juan Lu, and Sen Kuang, "Feedback control and quantum error correction assisted quantum multi-parameter estimation", Chinese Physics B 32 4, 040603 (2023).

[8] Francisco Revson F. Pereira, Stefano Mancini, and Giuliano G. La Guardia, "Stabilizer codes for open quantum systems", Scientific Reports 13 1, 10540 (2023).

[9] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino, "Photonic quantum metrology", AVS Quantum Science 2 2, 024703 (2020).

[10] María García Díaz, Benjamin Desef, Matteo Rosati, Dario Egloff, John Calsamiglia, Andrea Smirne, Michaelis Skotiniotis, and Susana F. Huelga, "Accessible coherence in open quantum system dynamics", Quantum 4, 249 (2020).

[11] Stephen D Bartlett, Gavin K Brennen, and Akimasa Miyake, "Robust symmetry-protected metrology with the Haldane phase", Quantum Science and Technology 3 1, 014010 (2018).

[12] M. R. Perelshtein, N. S. Kirsanov, V. V. Zemlyanov, A. V. Lebedev, G. Blatter, V. M. Vinokur, and G. B. Lesovik, "Linear Ascending Metrological Algorithm", Physical Review Research 3 1, 013257 (2021).

[13] Pavel Sekatski and Martí Perarnau-Llobet, "Optimal nonequilibrium thermometry in Markovian environments", Quantum 6, 869 (2022).

[14] Kianna Wan and Robert Lasenby, "Bounds on adaptive quantum metrology under Markovian noise", Physical Review Research 4 3, 033092 (2022).

[15] Shingo Kukita, Yuichiro Matsuzaki, and Yasushi Kondo, "Heisenberg-Limited Quantum Metrology Using Collective Dephasing", Physical Review Applied 16 6, 064026 (2021).

[16] Francesco Albarelli, Matteo A. C. Rossi, Matteo G. A. Paris, and Marco G. Genoni, Toward a Science Campus in Milan 127 (2018) ISBN:978-3-030-01628-9.

[17] Yuxiang Yang, "Memory Effects in Quantum Metrology", Physical Review Letters 123 11, 110501 (2019).

[18] Diego P. Pires, Isabela A. Silva, Eduardo R. deAzevedo, Diogo O. Soares-Pinto, and Jefferson G. Filgueiras, "Coherence orders, decoherence, and quantum metrology", Physical Review A 98 3, 032101 (2018).

[19] P. Sekatski, S. Wölk, and W. Dür, "Optimal distributed sensing in noisy environments", Physical Review Research 2 2, 023052 (2020).

[20] Victor Montenegro, Marco G. Genoni, Abolfazl Bayat, and Matteo G. A. Paris, "Quantum metrology with boundary time crystals", Communications Physics 6 1, 304 (2023).

[21] Han Xu, Junning Li, Liqiang Liu, Yu Wang, Haidong Yuan, and Xin Wang, "Generalizable control for quantum parameter estimation through reinforcement learning", npj Quantum Information 5 1, 82 (2019).

[22] S Wölk, P Sekatski, and W Dür, "Noisy distributed sensing in the Bayesian regime", Quantum Science and Technology 5 4, 045003 (2020).

[23] Chungwei Lin, Yanting Ma, and Dries Sels, "Application of Pontryagin's maximum principle to quantum metrology in dissipative systems", Physical Review A 105 4, 042621 (2022).

[24] Manish Chaudhary, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, and Tim Byrnes, "Macroscopic maximally-entangled-state preparation between two atomic ensembles", Physical Review A 108 3, 032420 (2023).

[25] Philippe Faist, Mischa P. Woods, Victor V. Albert, Joseph M. Renes, Jens Eisert, and John Preskill, "Time-Energy Uncertainty Relation for Noisy Quantum Metrology", PRX Quantum 4 4, 040336 (2023).

[26] F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, "Dissipative quantum error correction and application to quantum sensing with trapped ions", Nature Communications 8 1, 1822 (2017).

[27] David Layden and Paola Cappellaro, "Spatial noise filtering through error correction for quantum sensing", npj Quantum Information 4 1, 30 (2018).

[28] Francesco Albarelli, Matteo A. C. Rossi, Dario Tamascelli, and Marco G. Genoni, "Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment", Quantum 2, 110 (2018).

[29] Jan Czajkowski, Krzysztof Pawłowski, and Rafał Demkowicz-Dobrzański, "Many-body effects in quantum metrology", New Journal of Physics 21 5, 053031 (2019).

[30] Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă, "Multi-parameter estimation beyond quantum Fisher information", Journal of Physics A: Mathematical and Theoretical 53 36, 363001 (2020).

[31] Mateo Casariego, Yasser Omar, and Mikel Sanz, "Bi‐Frequency Illumination: A Quantum‐Enhanced Protocol", Advanced Quantum Technologies 5 11, 2100051 (2022).

[32] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang, "Achieving the Heisenberg limit in quantum metrology using quantum error correction", Nature Communications 9 1, 78 (2018).

[33] Javid Naikoo, Ravindra W. Chhajlany, and Jan Kołodyński, "Multiparameter Estimation Perspective on Non-Hermitian Singularity-Enhanced Sensing", Physical Review Letters 131 22, 220801 (2023).

[34] Jinrong Wang, Shuange Wu, Liying Hou, Chengdong Mi, Xurong Shi, and Xuzhen Gao, "A low-noise, high-SNR and large-dynamic-range balanced homodyne detector for broadband squeezed light measurement", Results in Physics 57, 107356 (2024).

[35] Benedikt Tratzmiller, Qiong Chen, Ilai Schwartz, Susana F. Huelga, and Martin B. Plenio, "Limited-control metrology approaching the Heisenberg limit without entanglement preparation", Physical Review A 101 3, 032347 (2020).

[36] Ivan Rojkov, David Layden, Paola Cappellaro, Jonathan Home, and Florentin Reiter, "Bias in Error-Corrected Quantum Sensing", Physical Review Letters 128 14, 140503 (2022).

[37] Yink Loong Len, Tuvia Gefen, Alex Retzker, and Jan Kołodyński, "Quantum metrology with imperfect measurements", Nature Communications 13 1, 6971 (2022).

[38] Arne Hamann, Pavel Sekatski, and Wolfgang Dür, "Approximate decoherence free subspaces for distributed sensing", Quantum Science and Technology 7 2, 025003 (2022).

[39] Rui-Jie Cai, Wei Zhong, Lan Zhou, and Yu-Bo Sheng, "Ancilla-assisted frequency estimation under phase covariant noises with Greenberger–Horne–Zeilinger states", Quantum Information Processing 19 10, 359 (2020).

[40] Manish Chaudhary, Yuping Mao, Manikandan Kondappan, Amiel S. P. Paz, Valentin Ivannikov, and Tim Byrnes, "Stroboscopic quantum nondemolition measurements for enhanced entanglement generation between atomic ensembles", Physical Review A 105 2, 022443 (2022).

[41] Francesco Albarelli and Rafał Demkowicz-Dobrzański, "Probe Incompatibility in Multiparameter Noisy Quantum Metrology", Physical Review X 12 1, 011039 (2022).

[42] Valeria Cimini, Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino, "Benchmarking Bayesian quantum estimation", Quantum Science and Technology 9 3, 035035 (2024).

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

[44] Vishal Katariya and Mark M. Wilde, "Geometric distinguishability measures limit quantum channel estimation and discrimination", Quantum Information Processing 20 2, 78 (2021).

[45] Vishal Katariya and Mark M Wilde, "RLD Fisher information bound for multiparameter estimation of quantum channels", New Journal of Physics 23 7, 073040 (2021).

[46] Priya Batra, M. Harshanth Ram, and T.S. Mahesh, "Recommender system expedited quantum control optimization", Physics Open 14, 100127 (2023).

[47] Michael H. Goerz, Sebastián C. Carrasco, and Vladimir S. Malinovsky, "Quantum Optimal Control via Semi-Automatic Differentiation", Quantum 6, 871 (2022).

[48] Sisi Zhou, Zi-Wen Liu, and Liang Jiang, "New perspectives on covariant quantum error correction", Quantum 5, 521 (2021).

[49] Stanisław Kurdziałek, Wojciech Górecki, Francesco Albarelli, and Rafał Demkowicz-Dobrzański, "Using Adaptiveness and Causal Superpositions Against Noise in Quantum Metrology", Physical Review Letters 131 9, 090801 (2023).

[50] Pietro Liuzzo-Scorpo, Luis A Correa, Felix A Pollock, Agnieszka Górecka, Kavan Modi, and Gerardo Adesso, "Energy-efficient quantum frequency estimation", New Journal of Physics 20 6, 063009 (2018).

[51] S. Danilin, A. V. Lebedev, A. Vepsäläinen, G. B. Lesovik, G. Blatter, and G. S. Paraoanu, "Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom", npj Quantum Information 4 1, 29 (2018).

[52] Zibo Miao, Yu Chen, and Haidong Yuan, 2020 Chinese Automation Congress (CAC) 5351 (2020) ISBN:978-1-7281-7687-1.

[53] Victor Montenegro, Utkarsh Mishra, and Abolfazl Bayat, "Global Sensing and Its Impact for Quantum Many-Body Probes with Criticality", Physical Review Letters 126 20, 200501 (2021).

[54] Andrea López-Incera, Pavel Sekatski, and Wolfgang Dür, "All macroscopic quantum states are fragile and hard to prepare", Quantum 3, 118 (2019).

[55] Agnieszka Górecka, Felix A Pollock, Pietro Liuzzo-Scorpo, Rosanna Nichols, Gerardo Adesso, and Kavan Modi, "Noisy frequency estimation with noisy probes", New Journal of Physics 20 8, 083008 (2018).

[56] Krzysztof Chabuda, Jacek Dziarmaga, Tobias J. Osborne, and Rafał Demkowicz-Dobrzański, "Tensor-network approach for quantum metrology in many-body quantum systems", Nature Communications 11 1, 250 (2020).

[57] Nelson Filipe Costa, Yasser Omar, Aidar Sultanov, and Gheorghe Sorin Paraoanu, "Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors", EPJ Quantum Technology 8 1, 16 (2021).

[58] Jing Liu, Mao Zhang, Hongzhen Chen, Lingna Wang, and Haidong Yuan, "Optimal Scheme for Quantum Metrology", Advanced Quantum Technologies 5 1, 2100080 (2022).

[59] Nathan Shettell and Damian Markham, "Graph States as a Resource for Quantum Metrology", Physical Review Letters 124 11, 110502 (2020).

[60] Wojciech Górecki, Sisi Zhou, Liang Jiang, and Rafał Demkowicz-Dobrzański, "Optimal probes and error-correction schemes in multi-parameter quantum metrology", Quantum 4, 288 (2020).

[61] W. Wang, Z.-J. Chen, X. Liu, W. Cai, Y. Ma, X. Mu, X. Pan, Z. Hua, L. Hu, Y. Xu, H. Wang, Y. P. Song, X.-B. Zou, C.-L. Zou, and L. Sun, "Quantum-enhanced radiometry via approximate quantum error correction", Nature Communications 13 1, 3214 (2022).

[62] Francisco Riberi, Leigh M Norris, Félix Beaudoin, and Lorenza Viola, "Frequency estimation under non-Markovian spatially correlated quantum noise", New Journal of Physics 24 10, 103011 (2022).

[63] Kok Chuan Tan, S. Omkar, and Hyunseok Jeong, "Quantum-error-correction-assisted quantum metrology without entanglement", Physical Review A 100 2, 022312 (2019).

[64] Jie Tang, HuiCun Yu, Ying Liu, ZhiFeng Deng, JiaHao Li, YueXiang Cao, JiaHua Wei, and Lei Shi, "Bayesian quantum parameter estimation with Gaussian states and homodyne measurements in a dissipative environment", Results in Physics 47, 106383 (2023).

[65] 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).

[66] Nathan Shettell, William J Munro, Damian Markham, and Kae Nemoto, "Practical limits of error correction for quantum metrology", New Journal of Physics 23 4, 043038 (2021).

[67] Liying Bao and Bo Qi, 2020 39th Chinese Control Conference (CCC) 5818 (2020) ISBN:978-9-8815-6390-3.

[68] A. R. Shlyakhov, V. V. Zemlyanov, M. V. Suslov, A. V. Lebedev, G. S. Paraoanu, G. B. Lesovik, and G. Blatter, "Quantum metrology with a transmon qutrit", Physical Review A 97 2, 022115 (2018).

[69] Jasminder S. Sidhu and Pieter Kok, "Geometric perspective on quantum parameter estimation", AVS Quantum Science 2 1, 014701 (2020).

[70] Sisi Zhou and Liang Jiang, "Optimal approximate quantum error correction for quantum metrology", Physical Review Research 2 1, 013235 (2020).

[71] Daniel Braun, Gerardo Adesso, Fabio Benatti, Roberto Floreanini, Ugo Marzolino, Morgan W. Mitchell, and Stefano Pirandola, "Quantum-enhanced measurements without entanglement", Reviews of Modern Physics 90 3, 035006 (2018).

[72] Stefano Gherardini, Matthias M. Müller, Simone Montangero, Tommaso Calarco, and Filippo Caruso, "Information flow and error scaling for fully quantum control", Physical Review Research 4 2, 023027 (2022).

[73] Wojciech Górecki, Rafał Demkowicz-Dobrzański, Howard M. Wiseman, and Dominic W. Berry, "π -Corrected Heisenberg Limit", Physical Review Letters 124 3, 030501 (2020).

[74] Sisi Zhou and Liang Jiang, "Asymptotic Theory of Quantum Channel Estimation", PRX Quantum 2 1, 010343 (2021).

[75] K. El Anouz, A. El Allati, and N. Metwally, "Different indicators for Markovian and non-Markovian dynamics", Physics Letters A 384 5, 126122 (2020).

[76] J F Haase, A Smirne, J Kołodyński, R Demkowicz-Dobrzański, and S F Huelga, "Fundamental limits to frequency estimation: a comprehensive microscopic perspective", New Journal of Physics 20 5, 053009 (2018).

[77] A Smirne, A Lemmer, M B Plenio, and S F Huelga, "Improving the precision of frequency estimation via long-time coherences", Quantum Science and Technology 4 2, 025004 (2019).

[78] 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).

[79] Theodoros Kapourniotis and Animesh Datta, "Fault-tolerant quantum metrology", Physical Review A 100 2, 022335 (2019).

[80] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski, "Adaptive Quantum Metrology under General Markovian Noise", Physical Review X 7 4, 041009 (2017).

[81] Florian Fröwis, Pavel Sekatski, Wolfgang Dür, Nicolas Gisin, and Nicolas Sangouard, "Macroscopic quantum states: Measures, fragility, and implementations", Reviews of Modern Physics 90 2, 025004 (2018).

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

[83] Sisi Zhou, Argyris Giannisis Manes, and Liang Jiang, "Achieving metrological limits using ancilla-free quantum error-correcting codes", Physical Review A 109 4, 042406 (2024).

[84] Aleksander Kubica and Rafał Demkowicz-Dobrzański, "Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem", Physical Review Letters 126 15, 150503 (2021).

[85] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio, "Colloquium: Quantum coherence as a resource", Reviews of Modern Physics 89 4, 041003 (2017).

[86] Jing Liu and Haidong Yuan, "Quantum parameter estimation with optimal control", Physical Review A 96 1, 012117 (2017).

[87] Nicolai Friis, Davide Orsucci, Michalis Skotiniotis, Pavel Sekatski, Vedran Dunjko, Hans J. Briegel, and Wolfgang Dür, "Flexible resources for quantum metrology", New Journal of Physics 19 6, 063044 (2017).

[88] Pavel Sekatski, Michalis Skotiniotis, and Wolfgang Dür, "Dynamical decoupling leads to improved scaling in noisy quantum metrology", New Journal of Physics 18 7, 073034 (2016).

[89] Rosanna Nichols, Thomas R. Bromley, Luis A. Correa, and Gerardo Adesso, "Practical quantum metrology in noisy environments", Physical Review A 94 4, 042101 (2016).

[90] Jessica Bavaresco, Patryk Lipka-Bartosik, Pavel Sekatski, and Mohammad Mehboudi, "Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations", arXiv:2311.01513, (2023).

[91] Rozhin Yousefjani, Rosanna Nichols, Shahriar Salimi, and Gerardo Adesso, "Estimating phase with a random generator: Strategies and resources in multiparameter quantum metrology", Physical Review A 95 6, 062307 (2017).

[92] Marcin Jarzyna and Marcin Zwierz, "Parameter estimation in the presence of the most general Gaussian dissipative reservoir", Physical Review A 95 1, 012109 (2017).

[93] W. L. Boyajian, M. Skotiniotis, W. Dür, and B. Kraus, "Compressed quantum metrology for the Ising Hamiltonian", Physical Review A 94 6, 062326 (2016).

[94] Shane Dooley, William J. Munro, and Kae Nemoto, "Quantum metrology including state preparation and readout times", Physical Review A 94 5, 052320 (2016).

[95] Jukka Kiukas, Kazuya Yuasa, and Daniel Burgarth, "Remote parameter estimation in a quantum spin chain enhanced by local control", Physical Review A 95 5, 052132 (2017).

[96] P. Sekatski, M. Skotiniotis, and W. Dür, "Improved Sensing with a Single Qubit", Physical Review Letters 118 17, 170801 (2017).

[97] Fabricio Toscano, Wellison P. Bastos, and Ruynet L. de Matos Filho, "Attainability of the quantum information bound in pure-state models", Physical Review A 95 4, 042125 (2017).

[98] Le Yang, Xi Chen, Ming Zhang, and Hong-Yi Dai, "Quantum Parameter Estimation: From Experimental Design to Constructive Algorithm", Communications in Theoretical Physics 68 5, 641 (2017).

[99] Vishal Katariya, "Limits on Parameter Estimation of Quantum Channels", arXiv:2201.01738, (2022).

[100] Simon Morelli, Ayaka Usui, Elizabeth Agudelo, and Nicolai Friis, "Bayesian parameter estimation using Gaussian states and measurements", arXiv:2009.03709, (2020).

[101] Stephen D. Bartlett, Gavin K. Brennen, and Akimasa Miyake, "Robust symmetry-protected metrology with the Haldane phase", arXiv:1608.08221, (2016).

[102] Yu Liu, Zijun Shu, Martin B. Plenio, and Jianming Cai, "Adiabatic quantum parameter amplification for generic robust quantum sensing", arXiv:1612.01653, (2016).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-18 02:41:49) and SAO/NASA ADS (last updated successfully 2024-06-18 02:41:50). The list may be incomplete as not all publishers provide suitable and complete citation data.