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

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.


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[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. Bużek, 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.