Adiabatic critical quantum metrology cannot reach the Heisenberg limit even when shortcuts to adiabaticity are applied

Karol Gietka, Friederike Metz, Tim Keller, and Jing Li

Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

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We show that the quantum Fisher information attained in an adiabatic approach to critical quantum metrology cannot lead to the Heisenberg limit of precision and therefore $regular$ quantum metrology under optimal settings is always superior. Furthermore, we argue that even though shortcuts to adiabaticity can arbitrarily decrease the time of preparing critical ground states, they cannot be used to achieve or overcome the Heisenberg limit for quantum parameter estimation in adiabatic critical quantum metrology. As case studies, we explore the application of counter-diabatic driving to the Landau-Zener model and the quantum Rabi model.

Critical metrology, which relies on the extreme sensitivity of critical ground states to small changes of Hamiltonian parameters, is believed to give rise to an enhanced precision compared to regular quantum metrology. However, the required time resources for preparing the critical ground state are usually not considered and will result in critical quantum metrology protocols which will inevitably operate on time scales beyond the decoherence times of an experiment. In order to decrease the time of an adiabatic protocol one could harness shortcuts-to-adiabaticity and, in principle, overcome the Heisenberg limit of precision. However, in our article we show that not only the adiabatic approach to critical quantum metrology cannot lead to the Heisenberg limit of precision but also argue that even though shortcuts-to-adiabaticity can arbitrarily decrease the time of preparing critical ground states, they cannot be used to achieve or overcome the Heisenberg limit for quantum parameter estimation in adiabatic critical quantum metrology.

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

[2] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum-enhanced measurements: beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004a. 10.1126/​science.1104149.

[3] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum-enhanced measurements: beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004b. 10.1126/​science.1104149.

[4] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă. The elusive heisenberg limit in quantum-enhanced metrology. Nat. Commun, 3 (1): 1–8, 2012. 10.1038/​ncomms2067.

[5] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński. Chapter four - quantum limits in optical interferometry. In E. Wolf, editor, Progress in Optics, volume 60, pages 345 – 435. Elsevier, 2015. 10.1016/​bs.po.2015.02.003.

[6] E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino. Photonic quantum metrology. AVS Quantum Science, 2 (2): 024703, 2020. 10.1116/​5.0007577.

[7] 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, 2018a. 10.1103/​RevModPhys.90.035005.

[8] BM Escher, RL de Matos Filho, and L. Davidovich. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys., 7 (5): 406–411, 2011. 10.1038/​nphys1958.

[9] 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. 10.1103/​PhysRevLett.111.120401.

[10] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola. Quantum-enhanced measurements without entanglement. Rev. Mod. Phys., 90: 035006, 2018. 10.1103/​RevModPhys.90.035006.

[11] C. Invernizzi, M. Korbman, L. Campos Venuti, and M. G. A. Paris. Optimal quantum estimation in spin systems at criticality. Phys. Rev. A, 78: 042106, 2008. 10.1103/​PhysRevA.78.042106.

[12] P. Zanardi, M. G. A. Paris, and L. Campos Venuti. Quantum criticality as a resource for quantum estimation. Phys. Rev. A, 78: 042105, 2008a. 10.1103/​PhysRevA.78.042105.

[13] P. A. Ivanov and D. Porras. Adiabatic quantum metrology with strongly correlated quantum optical systems. Phys. Rev. A, 88: 023803, 2013. 10.1103/​PhysRevA.88.023803.

[14] M. Tsang. Quantum transition-edge detectors. Phys. Rev. A, 88: 021801, 2013. 10.1103/​PhysRevA.88.021801.

[15] G. Salvatori, A. Mandarino, and M. G. A. Paris. Quantum metrology in lipkin-meshkov-glick critical systems. Phys. Rev. A, 90: 022111, 2014. 10.1103/​PhysRevA.90.022111.

[16] K. Macieszczak, Guţă, M., I. Lesanovsky, and J. P. Garrahan. Dynamical phase transitions as a resource for quantum enhanced metrology. Phys. Rev. A, 93: 022103, 2016. 10.1103/​PhysRevA.93.022103.

[17] M. Bina, I. Amelio, and M. G. A. Paris. Dicke coupling by feasible local measurements at the superradiant quantum phase transition. Phys. Rev. E, 93: 052118, 2016. 10.1103/​PhysRevE.93.052118.

[18] S. Fernández-Lorenzo and D. Porras. Quantum sensing close to a dissipative phase transition: Symmetry breaking and criticality as metrological resources. Phys. Rev. A, 96: 013817, 2017. 10.1103/​PhysRevA.96.013817.

[19] T. L. Heugel, M. Biondi, O. Zilberberg, and R. Chitra. Quantum transducer using a parametric driven-dissipative phase transition. Phys. Rev. Lett., 123: 173601, 2019. 10.1103/​PhysRevLett.123.173601.

[20] L. Garbe, M. Bina, A. Keller, M. G. A. Paris, and S. Felicetti. Critical quantum metrology with a finite-component quantum phase transition. Phys. Rev. Lett., 124: 120504, 2020. 10.1103/​PhysRevLett.124.120504.

[21] Y. Chu, S. Zhang, B. Yu, and J. Cai. Dynamic framework for criticality-enhanced quantum sensing. Phys. Rev. Lett., 126: 010502, 2021. 10.1103/​PhysRevLett.126.010502.

[22] P. Zanardi, M. G. A. Paris, and L. Campos Venuti. Quantum criticality as a resource for quantum estimation. Phys. Rev. A, 78: 042105, 2008b. 10.1103/​PhysRevA.78.042105.

[23] M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and J. Zakrzewski. At the limits of criticality-based quantum metrology: Apparent super-heisenberg scaling revisited. Phys. Rev. X, 8: 021022, 2018. 10.1103/​PhysRevX.8.021022.

[24] O. Abah, R. Puebla, A. Kiely, G. De Chiara, M. Paternostro, and S. Campbell. Energetic cost of quantum control protocols. New J. Phys., 21 (10): 103048, 2019. 10.1088/​1367-2630/​ab4c8c.

[25] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga. Shortcuts to adiabaticity: Concepts, methods, and applications. Rev. Mod. Phys., 91: 045001, 2019. 10.1103/​RevModPhys.91.045001.

[26] Busch P. Time in Quantum Mechanics. Lecture Notes in Physics, volume 72, chapter The Time-Energy Uncertainty Relation. Springer, Berlin, Heidelberg, 2002.​10.1007/​3-540-45846-8_3.

[27] H. R. Lewis and PGL Leach. A direct approach to finding exact invariants for one-dimensional time-dependent classical hamiltonians. J. Math. Phys., 23 (12): 2371–2374, 1982. 10.1063/​1.525329.

[28] M. V. Berry. Transitionless quantum driving. J. Phys. A Math. Theor., 42 (36): 365303, 2009. 10.1088/​1751-8113/​42/​36/​365303.

[29] M. Demirplak and S. A. Rice. Adiabatic population transfer with control fields. J. Phys. Chem. A, 107 (46): 9937–9945, 2003. 10.1021/​jp030708a.

[30] S. Masuda and K. Nakamura. Fast-forward of adiabatic dynamics in quantum mechanics. Proc. R. Soc. A, 466 (2116): 1135–1154, 2010. 10.1098/​rspa.2009.0446.

[31] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga. Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity. Phys. Rev. Lett., 104: 063002, 2010a. 10.1103/​PhysRevLett.104.063002.

[32] E. Torrontegui, S. Ibáñez, Xi Chen, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga. Fast atomic transport without vibrational heating. Phys. Rev. A, 83: 013415, 2011. 10.1103/​PhysRevA.83.013415.

[33] Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga. Shortcut to adiabatic passage in two- and three-level atoms. Phys. Rev. Lett., 105: 123003, 2010b. 10.1103/​PhysRevLett.105.123003.

[34] E. Torrontegui, S. Ibánez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga. Shortcuts to adiabaticity. In Advances in atomic, molecular, and optical physics, volume 62, pages 117–169. Elsevier, 2013. 10.1016/​B978-0-12-408090-4.00002-5.

[35] C. Jarzynski. Generating shortcuts to adiabaticity in quantum and classical dynamics. Phys. Rev. A, 88: 040101, 2013. 10.1103/​PhysRevA.88.040101.

[36] A. del Campo. Shortcuts to adiabaticity by counterdiabatic driving. Phys. Rev. Lett., 111: 100502, 2013. 10.1103/​PhysRevLett.111.100502.

[37] S. Pang and A. N Jordan. Optimal adaptive control for quantum metrology with time-dependent hamiltonians. Nat. Commun, 8 (1): 1–9, 2017. 10.1038/​ncomms14695.

[38] M. Cabedo-Olaya, J. G. Muga, and S. Martínez-Garaot. Shortcut-to-adiabaticity-like techniques for parameter estimation in quantum metrology. Entropy, 22 (11): 1251, 2020.​10.3390/​e22111251.

[39] Z. Hou, Y. Jin, H. Chen, J.-F. Tang, C.-J. Huang, H. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo. ``super-heisenberg'' and heisenberg scalings achieved simultaneously in the estimation of a rotating field. Phys. Rev. Lett., 126: 070503, 2021. 10.1103/​PhysRevLett.126.070503.

[40] J. Yang, S. Pang, and A. N. Jordan. Quantum parameter estimation with the landau-zener transition. Phys. Rev. A, 96: 020301, 2017. 10.1103/​PhysRevA.96.020301.

[41] S. Campbell and S. Deffner. Trade-off between speed and cost in shortcuts to adiabaticity. Phys. Rev. Lett., 118: 100601, 2017. 10.1103/​PhysRevLett.118.100601.

[42] Y.-H. Chen, W. Qin, X. Wang, A. Miranowicz, and F. Nori. Shortcuts to adiabaticity for the quantum rabi model: Efficient generation of giant entangled cat states via parametric amplification. Phys. Rev. Lett., 126: 023602, 2021. 10.1103/​PhysRevLett.126.023602.

[43] C. W. Helstrom. Quantum detection and estimation theory. J. Stat. Phys., 1: 231–252, 1969. 10.1007/​BF01007479.

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

[45] S. Boixo, S. T. Flammia, C. M. Caves, and JM Geremia. Generalized limits for single-parameter quantum estimation. Phys. Rev. Lett., 98: 090401, 2007. 10.1103/​PhysRevLett.98.090401.

[46] 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, 2018b. 10.1103/​RevModPhys.90.035005.

[47] W. Hyllus, P.and Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi. Fisher information and multiparticle entanglement. Phys. Rev. A, 85: 022321, 2012. 10.1103/​PhysRevA.85.022321.

[48] G. Tóth. Multipartite entanglement and high-precision metrology. Phys. Rev. A, 85: 022322, 2012. 10.1103/​PhysRevA.85.022322.

[49] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler. Fisher information and entanglement of non-gaussian spin states. Science, 345 (6195): 424–427, 2014. 10.1126/​science.1250147.

[50] S. Sachdev. Quantum phase transitions. Handbook of Magnetism and Advanced Magnetic Materials, 2007. 10.1002/​9780470022184.

[51] W.-L. You, Y.-W. Li, and S.-J. Gu. Fidelity, dynamic structure factor, and susceptibility in critical phenomena. Phys. Rev. E, 76: 022101, 2007. 10.1103/​PhysRevE.76.022101.

[52] L. Campos Venuti and P. Zanardi. Quantum critical scaling of the geometric tensors. Phys. Rev. Lett., 99: 095701, 2007. 10.1103/​PhysRevLett.99.095701.

[53] A. C. Santos and M. S. Sarandy. Generalized shortcuts to adiabaticity and enhanced robustness against decoherence. J. Phys. A, 51 (2): 025301, 2017. 10.1088/​1751-8121/​aa96f1.

[54] J. G. Muga, X. Chen, S. Ibáñez, I. Lizuain, and A. Ruschhaupt. Transitionless quantum drivings for the harmonic oscillator. J. Phys. B, 43 (8): 085509, 2010. 10.1088/​0953-4075/​43/​8/​085509.

[55] A. del Campo, M. M. Rams, and W. H. Zurek. Assisted finite-rate adiabatic passage across a quantum critical point: Exact solution for the quantum ising model. Phys. Rev. Lett., 109: 115703, 2012. 10.1103/​PhysRevLett.109.115703.

[56] S. Deffner and S. Campbell. Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control. J. Phys. A, 50 (45): 453001, 2017. 10.1088/​1751-8121/​aa86c6.

[57] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum limits to dynamical evolution. Phys. Rev. A, 67: 052109, 2003. 10.1103/​PhysRevA.67.052109.

[58] C. Zener. Non-adiabatic crossing of energy levels. Proc. Math. Phys. Eng., 137 (833): 696–702, 1932. 10.1098/​rspa.1932.0165.

[59] L. Innocenti, G. De Chiara, M. Paternostro, and R. Puebla. Ultrafast critical ground state preparation via bang–bang protocols. New J. Phys., 22 (9): 093050, 2020. 10.1088/​1367-2630/​abb1df.

[60] J. Zhang, F. M. Cucchietti, C. M. Chandrashekar, M. Laforest, C. A. Ryan, M. Ditty, A. Hubbard, J. K. Gamble, and R. Laflamme. Direct observation of quantum criticality in ising spin chains. Phys. Rev. A, 79: 012305, 2009. 10.1103/​PhysRevA.79.012305.

[61] M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch. High-fidelity quantum driving. Nat. Phys, 8 (2): 147–152, 2012. 10.1038/​nphys2170.

[62] G. C. Hegerfeldt. Driving at the quantum speed limit: Optimal control of a two-level system. Phys. Rev. Lett., 111: 260501, 2013. 10.1103/​PhysRevLett.111.260501.

[63] M.-J. Hwang, R. Puebla, and M. B. Plenio. Quantum phase transition and universal dynamics in the rabi model. Phys. Rev. Lett., 115: 180404, 2015. 10.1103/​PhysRevLett.115.180404.

[64] H. Zhong, Q. Xie, M. T. Batchelor, and C. Lee. Analytical eigenstates for the quantum rabi model. J. Phys. A, 46 (41): 415302, 2013. 10.1088/​1751-8113/​46/​41/​415302.

[65] Z.-J. Ying, M. Liu, H.-G. Luo, H.-Q. Lin, and J. Q. You. Ground-state phase diagram of the quantum rabi model. Phys. Rev. A, 92: 053823, 2015. 10.1103/​PhysRevA.92.053823.

[66] A. Monras. Optimal phase measurements with pure gaussian states. Phys. Rev. A, 73: 033821, 2006. 10.1103/​PhysRevA.73.033821.

[67] I. Frérot and T. Roscilde. Quantum critical metrology. Phys. Rev. Lett., 121: 020402, 2018. 10.1103/​PhysRevLett.121.020402.

[68] T. Hatomura. Shortcuts to adiabatic cat-state generation in bosonic josephson junctions. New J. Phys., 20 (1): 015010, 2018. 10.1088/​1367-2630/​aaa117.

[69] S. S. Mirkhalaf, E. Witkowska, and L. Lepori. Supersensitive quantum sensor based on criticality in an antiferromagnetic spinor condensate. Phys. Rev. A, 101: 043609, 2020. 10.1103/​PhysRevA.101.043609.

[70] S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch. QuantumOptics.jl: A julia framework for simulating open quantum systems. Comput. Phys. Commun, 227: 109–116, 2018. 10.1016/​j.cpc.2018.02.004.

[71] K. Husimi. Some formal properties of the density matrix. Soc. Jpn, 22 (4): 264–314, 1940. 10.11429/​ppmsj1919.22.4_264.

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