Optimal probes and error-correction schemes in multi-parameter quantum metrology

Wojciech Górecki1, Sisi Zhou2,3,4, Liang Jiang2,3,4, and Rafał Demkowicz-Dobrzański1

1Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
2Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
3Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA
4Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

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Abstract

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

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[1] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.010401

[2] M. G. A. Paris, Quantum estimation for quantum technologies, Int. J. Quantum Inf. 07, 125 (2009).
https:/​/​doi.org/​10.1142/​S0219749909004839

[3] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[4] G. Toth and I. Apellaniz, Quantum metrology from a quantum information science perspective, J. Phys. A: Math. Theor. 47, 424006 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424006

[5] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, in Prog. Optics, Vol. 60, edited by E. Wolf (Elsevier, 2015) pp. 345–435.
https:/​/​doi.org/​10.1016/​bs.po.2015.02.003

[6] R. Schnabel, Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684, 1 (2017).
https:/​/​doi.org/​10.1016/​j.physrep.2017.04.001

[7] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.035002

[8] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
https:/​/​doi.org/​10.1103/​RevModPhys.90.035005

[9] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12, 724 (2018).
https:/​/​doi.org/​10.1038/​s41566-018-0301-6

[10] C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
https:/​/​doi.org/​10.1103/​PhysRevD.23.1693

[11] M. Holland and K. Burnett, Interferometric detection of optical phase shifts at the heisenberg limit, Phys. Rev. Lett. 71, 1355 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.1355

[12] H. Lee, P. Kok, and J. P. Dowling, A quantum rosetta stone for interferometry, J. Mod. Optic. 49, 2325 (2002).
https:/​/​doi.org/​10.1080/​0950034021000011536

[13] D. Wineland, J. Bollinger, W. Itano, F. Moore, and D. Heinzen, Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A 46, R6797 (1992).
https:/​/​doi.org/​10.1103/​PhysRevA.46.R6797

[14] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.231102

[15] J. Bollinger, W. M. Itano, D. Wineland, and D. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54, R4649 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.R4649

[16] D. Leibfried, M. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. Itano, J. Jost, C. Langer, and D. Wineland, Toward heisenberg-limited spectroscopy with multiparticle entangled states, Science 304, 1476 (2004).
https:/​/​doi.org/​10.1126/​science.1097576

[17] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit, Science 306, 1330 (2004).
https:/​/​doi.org/​10.1126/​science.1104149

[18] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.3865

[19] D. W. Berry and H. M. Wiseman, Optimal states and almost optimal adaptive measurements for quantum interferometry, Phys. Rev. Lett. 85, 5098 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.85.5098

[20] M. de Burgh and S. D. Bartlett, Quantum methods for clock synchronization: Beating the standard quantum limit without entanglement, Phys. Rev. A 72, 042301 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.042301

[21] A. Fujiwara and H. Imai, A fibre bundle over manifolds of quantum channels and its application to quantum statistics, J. Phys. A: Math. Theor. 41, 255304 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​25/​255304

[22] R. Demkowicz-Dobrzański, U. Dorner, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, and I. Walmsley, Quantum phase estimation with lossy interferometers, Phys. Rev. A 80, 013825 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.013825

[23] B. Escher, R. de Matos Filho, and L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nat. Phys. 7, 406 (2011).
https:/​/​doi.org/​10.1038/​nphys1958

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

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

[26] S. I. Knysh, E. H. Chen, and G. A. Durkin, True limits to precision via unique quantum probe, arXiv:1402.0495 (2014).
arXiv:1402.0495

[27] R. Demkowicz-Dobrzański and L. Maccone, Using entanglement against noise in quantum metrology, Phys. Rev. Lett. 113, 250801 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.250801

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

[29] W. Dür, M. Skotiniotis, F. Froewis, and B. Kraus, Improved quantum metrology using quantum error correction, Phys. Rev. Lett. 112, 080801 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.080801

[30] R. Ozeri, Heisenberg limited metrology using quantum error-correction codes. arXiv:1310.3432 (2013).
arXiv:1310.3432

[31] G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Increasing sensing resolution with error correction, Phys. Rev. Lett. 112, 150801 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.150801

[32] T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchinsky, A. O. Sushkov, et al., Quantum metrology enhanced by repetitive quantum error correction, Phys. Rev. Lett. 116, 230502 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.230502

[33] F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, Dissipative quantum error correction and application to quantum sensing with trapped ions, Nat. Commun. 8, 1822 (2017).
https:/​/​doi.org/​10.1038/​s41467-017-01895-5

[34] P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, Quantum metrology with full and fast quantum control, Quantum 1, 27 (2017).
https:/​/​doi.org/​10.22331/​q-2017-09-06-27

[35] R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general markovian noise, Phys. Rev. X 7, 041009 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.041009

[36] 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).
https:/​/​doi.org/​10.1038/​s41467-017-02510-3

[37] D. Layden and P. Cappellaro, Spatial noise filtering through error correction for quantum sensing, npj Quantum Inf. 4, 30 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0082-2

[38] D. Layden, S. Zhou, P. Cappellaro, and L. Jiang, Ancilla-free quantum error correction codes for quantum metrology, Phys. Rev. Lett. 122, 040502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.040502

[39] T. Kapourniotis and A. Datta, Fault-tolerant quantum metrology, Phys. Rev. A 100, 022335 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022335

[40] K. C. Tan, S. Omkar, and H. Jeong, Quantum-error-correction-assisted quantum metrology without entanglement, Phys. Rev. A 100, 022312 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022312

[41] S. Zhou and L. Jiang, The theory of entanglement-assisted metrology for quantum channels, arXiv:2003.10559 (2020a).
arXiv:2003.10559

[42] Y. Chen, H. Chen, J. Liu, Z. Miao, and H. Yuan, Fluctuation-enhanced quantum metrology, arXiv:2003.13010 (2020).
arXiv:2003.13010

[43] T. Baumgratz and A. Datta, Quantum enhanced estimation of a multidimensional field, Phys. Rev. Lett. 116, 030801 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.030801

[44] M. Tsang, R. Nair, and X.-M. Lu, Quantum theory of superresolution for two incoherent optical point sources, Phys. Rev. X 6, 031033 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.031033

[45] P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett. 111, 070403 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.070403

[46] M. Gessner, L. Pezzè, and A. Smerzi, Sensitivity bounds for multiparameter quantum metrology, Phys. Rev. Lett. 121, 130503 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.130503

[47] M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental quantum limit to waveform estimation, Phys. Rev. Lett. 106, 090401 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.090401

[48] D. W. Berry, M. J. W. Hall, and H. M. Wiseman, Stochastic heisenberg limit: Optimal estimation of a fluctuating phase, Phys. Rev. Lett. 111, 113601 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.113601

[49] K. Matsumoto, A new approach to the cramér-rao-type bound of the pure-state model, J. Phys. A.: Math. Theor. 35, 3111 (2002).
https:/​/​doi.org/​10.1088/​0305-4470/​35/​13/​307

[50] M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, Optimal estimation of joint parameters in phase space, Phys. Rev. A 87, 012107 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.012107

[51] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052108

[52] H. Yuan, Sequential feedback scheme outperforms the parallel scheme for hamiltonian parameter estimation, Phys. Rev. Lett. 117, 160801 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.160801

[53] N. Kura and M. Ueda, Finite-error metrological bounds on multiparameter hamiltonian estimation, Phys. Rev. A 97, 012101 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.012101

[54] J. Liu and H. Yuan, Control-enhanced multiparameter quantum estimation, Phys. Rev. A 96, 042114 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.042114

[55] R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Multiparameter gaussian quantum metrology, Phys. Rev. A 98, 012114 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.012114

[56] W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss-Feig, Distributed quantum metrology with linear networks and separable inputs, Phys. Rev. Lett. 121, 043604 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.043604

[57] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[58] C. W. Helstrom, Quantum detection and estimation theory (Academic press, 1976).

[59] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North Holland, Amsterdam, 1982).

[60] R. Demkowicz-Dobrzanski, W. Gorecki, and M. Guta, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical (2020).
https:/​/​doi.org/​10.1088/​1751-8121/​ab8ef3

[61] H. Nagaoka and M. Hayashi, Asymptotic Theory of Quantum Statistical Inference (World Scientific Singapore, 2005) Chap. 8.

[62] J. Suzuki, Explicit formula for the holevo bound for two-parameter qubit-state estimation problem, J. Math. Phys. 57, 042201 (2016).
https:/​/​doi.org/​10.1063/​1.4945086

[63] M. Guţă and A. Jenčová, Local asymptotic normality in quantum statistics, Comm. Math. Phys. 276, 341 (2007).
https:/​/​doi.org/​10.1007/​s00220-007-0340-1

[64] K. Yamagata, A. Fujiwara, R. D. Gill, et al., Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Statist. 41, 2197 (2013).
https:/​/​doi.org/​10.1214/​13-AOS1147

[65] A. Fujiwara, Multi-parameter pure state estimation based on the right logarithmic derivative, Math. Eng. Tech. Rep 94, 94 (1994).

[66] F. Albarelli, J. F. Friel, and A. Datta, Evaluating the holevo cramér-rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.200503

[67] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119 (1976).
https:/​/​doi.org/​10.1007/​BF01608499

[68] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level systems, J. Math. Phys. 17, 821 (1976).
https:/​/​doi.org/​10.1063/​1.522979

[69] H.-P. Breuer, F. Petruccione, et al., The theory of open quantum systems (Oxford University Press on Demand, 2002).

[70] S. M. Kay, Fundamentals of statistical signal processing: estimation theory (Prentice Hall, 1993).

[71] R. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.61.042312

[72] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.900

[73] M. Grant and S. Boyd, Cvx: Matlab software for disciplined convex programming,.
http:/​/​cvxr.com/​cvx/​

[74] S. Zhou and L. Jiang, Optimal approximate quantum error correction for quantum metrology, Phys. Rev. Research 2, 013235 (2020b).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013235

[75] W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, ${\pi}$-corrected heisenberg limit, Phys. Rev. Lett. 124, 030501 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.030501

[76] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett. 81, 2594 (1998).
https:/​/​doi.org/​10.1103/​PhysRevLett.81.2594

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[2] Francesco Albarelli, Jamie F. Friel, and Animesh Datta, "Evaluating the Holevo Cramér-Rao Bound for Multiparameter Quantum Metrology", Physical Review Letters 123 20, 200503 (2019).

[3] Francesco Albarelli, Mankei Tsang, and Animesh Datta, "Upper bounds on the Holevo Cramér-Rao bound for multiparameter quantum parametric and semiparametric estimation", arXiv:1911.11036.

[4] 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, 126311 (2020).

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

[6] Yingkai Ouyang, Nathan Shettell, and Damian Markham, "Robust quantum metrology with explicit symmetric states", arXiv:1908.02378.

[7] Rafal Demkowicz-Dobrzanski, Wojciech Gorecki, and Madalin Guta, "Multi-parameter estimation beyond Quantum Fisher Information", arXiv:2001.11742.

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

[9] Sisi Zhou and Liang Jiang, "The theory of entanglement-assisted metrology for quantum channels", arXiv:2003.10559.

[10] Aleksander Kubica and Rafal Demkowicz-Dobrzanski, "Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem", arXiv:2004.11893.

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[12] Le Bin Ho, Hideaki Hakoshima, Yuichiro Matsuzaki, Masayuki Matsuzaki, and Yasushi Kondo, "Multiparameter quantum estimation under dephasing noise", arXiv:2004.00720.

[13] Yingkai Ouyang and Narayanan Rengaswamy, "Weight Distribution of Classical Codes Influences Robust Quantum Metrology", arXiv:2007.02859.

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