Interplay Between Time and Energy in Bosonic Noisy Quantum Metrology

Górecki, Wojciech; Albarelli, Francesco; Felicetti, Simone; Di Candia, Roberto; Maccone, Lorenzo

doi:10.1103/PRXQuantum.6.020351

Interplay Between Time and Energy in Bosonic Noisy Quantum Metrology

Journal Article · Fri Jun 13 00:00:00 EDT 2025 · PRX Quantum

DOI:https://doi.org/10.1103/PRXQuantum.6.020351· OSTI ID:2497828

^[1]; ^[2]; ^[3]; Di Candia, Roberto ^[4]; ^[5]

Istituto Nazionale di Fisica Nucleare (INFN), Pavia (Italy)
Scuola Normale Superiore (SNS), Pisa (Italy)
National Research Council (CNR), Rome (Italy); Sapienza Univ., Rome (Italy)
Aalto Univ. (Finland); Univ. of Pavia (Italy)
Istituto Nazionale di Fisica Nucleare (INFN), Pavia (Italy); Univ. of Pavia (Italy)

Quantum entanglement and coherence often allow for protocols that outperform classical ones in estimating a system’s parameter. When using infinite-dimensional probes (such as a bosonic mode), one could, in principle, obtain infinite precision in a finite time for both classical and quantum protocols, which makes it hard to quantify potential quantum advantage. However, such a situation is unphysical, as it would require infinite resources, so one needs to impose some additional constraint: typically the average energy employed by the probe is finite. Here we treat both energy and time as a resource, showing that, in the presence of noise, there is a nontrivial interplay between the average energy and the time devoted to the estimation. Our results are valid for the most general metrological schemes (e.g., adaptive schemes, which may involve entanglement with external ancillae or any kind of continuous measurement). We apply recently derived precision bounds for all parameters characterizing the paradigmatic case of a bosonic mode, subject to Lindbladian noise. We show how the time employed in the estimation should be partitioned in order to achieve the best possible precision. In most cases, the optimal performance may be obtained without the necessity of adaptivity or entanglement with ancilla. We compare results with classical strategies. Interestingly, for temperature estimation, applying a fast-prepare-and-measure protocol with Fock states provides better scaling with the number of photons than any classical strategy.

View Accepted Manuscript (DOE)

Research Organization:: Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP)

Grant/Contract Number:: 89243024CSC000002; AC02-07CH11359

OSTI ID:: 2497828

Report Number(s):: FERMILAB-PUB--24-0894-SQMS-V; oai:inspirehep.net:2835144; arXiv:2409.18791

Journal Information:: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 2 Vol. 6; ISSN 2691-3399

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (75)

Quantum Optics Walls, D. F.; Milburn, Gerard J. https://doi.org/10.1007/978-3-540-28574-8	book	January 2008
Quantum detection and estimation theory Helstrom, Carl W. Journal of Statistical Physics, Vol. 1, Issue 2 https://doi.org/10.1007/BF01007479	journal	January 1969
Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation Hayashi, Masahito Communications in Mathematical Physics, Vol. 304, Issue 3 https://doi.org/10.1007/s00220-011-1239-4	journal	April 2011
Attaining the Ultimate Precision Limit in Quantum State Estimation Yang, Yuxiang; Chiribella, Giulio; Hayashi, Masahito Communications in Mathematical Physics, Vol. 368, Issue 1 https://doi.org/10.1007/s00220-019-03433-4	journal	April 2019
Quantum Limits in Optical Interferometry Demkowicz-Dobrzański, Rafal; Jarzyna, Marcin; Kołodyński, Jan Progress in Optics https://doi.org/10.1016/bs.po.2015.02.003	book	January 2015
Parameter estimation and system identification for continuously-observed quantum systems Nurdin, Hendra I.; Guţǎ, Mădălin Annual Reviews in Control, Vol. 54 https://doi.org/10.1016/j.arcontrol.2022.04.012	journal	January 2022
Classes of Gaussian states for squeezing estimation Souza, Leonardo A. M. Physics Letters A, Vol. 504 https://doi.org/10.1016/j.physleta.2024.129431	journal	April 2024
Squeezed states of light and their applications in laser interferometers Schnabel, Roman Physics Reports, Vol. 684 https://doi.org/10.1016/j.physrep.2017.04.001	journal	April 2017
The elusive Heisenberg limit in quantum-enhanced metrology Demkowicz-Dobrzański, Rafał; Kołodyński, Jan; Guţă, Mădălin Nature Communications, Vol. 3, Issue 1 https://doi.org/10.1038/ncomms2067	journal	January 2012
Advances in quantum metrology Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo Nature Photonics, Vol. 5, Issue 4 https://doi.org/10.1038/nphoton.2011.35	journal	March 2011
General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology Escher, B. M.; de Matos Filho, R. L.; Davidovich, L. Nature Physics, Vol. 7, Issue 5 https://doi.org/10.1038/nphys1958	journal	March 2011
Achieving the Heisenberg limit in quantum metrology using quantum error correction Zhou, Sisi; Zhang, Mengzhen; Preskill, John Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-017-02510-3	journal	January 2018
Observation of a quantum phase transition in the quantum Rabi model with a single trapped ion Cai, M. -L.; Liu, Z. -D.; Zhao, W. -D. Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-21425-8	journal	February 2021
Ultimate precision limit of noise sensing and dark matter search Shi, Haowei; Zhuang, Quntao npj Quantum Information, Vol. 9, Issue 1 https://doi.org/10.1038/s41534-023-00693-w	journal	March 2023
Quantum-enhanced dark matter detection with in-cavity control: mitigating the Rayleigh curse Shi, Haowei; Brady, Anthony J.; Górecki, Wojciech npj Quantum Information, Vol. 11, Issue 1 https://doi.org/10.1038/s41534-025-01004-1	journal	March 2025
Advances in photonic quantum sensing Pirandola, S.; Bardhan, B. R.; Gehring, T. Nature Photonics, Vol. 12, Issue 12 https://doi.org/10.1038/s41566-018-0301-6	journal	November 2018
Collective quantum enhancement in critical quantum sensing Alushi, Uesli; Coppo, Alessandro; Brosco, Valentina Communications Physics, Vol. 8, Issue 1 https://doi.org/10.1038/s42005-025-01975-9	journal	February 2025
Fundamental limits to frequency estimation: a comprehensive microscopic perspective Haase, J. F.; Smirne, A.; Kołodyński, J. New Journal of Physics, Vol. 20, Issue 5 https://doi.org/10.1088/1367-2630/aab67f	journal	May 2018
Noisy atomic magnetometry in real time Amorós-Binefa, Júlia; Kołodyński, Jan New Journal of Physics, Vol. 23, Issue 12 https://doi.org/10.1088/1367-2630/ac3b71	journal	December 2021
Probe thermometry with continuous measurements Boeyens, Julia; Annby-Andersson, Björn; Bakhshinezhad, Pharnam New Journal of Physics, Vol. 25, Issue 12 https://doi.org/10.1088/1367-2630/ad0e8a	journal	December 2023
Operational significance of nonclassicality in nonequilibrium Gaussian quantum thermometry Mirkhalaf, Safoura; Mehboudi, Mohammad; Nafari Qaleh, Zohre New Journal of Physics, Vol. 26, Issue 2 https://doi.org/10.1088/1367-2630/ad23a1	journal	February 2024
A fibre bundle over manifolds of quantum channels and its application to quantum statistics Fujiwara, Akio; Imai, Hiroshi Journal of Physics A: Mathematical and Theoretical, Vol. 41, Issue 25 https://doi.org/10.1088/1751-8113/41/25/255304	journal	May 2008
Quantum metrology from a quantum information science perspective Tóth, Géza; Apellaniz, Iagoba Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 42 https://doi.org/10.1088/1751-8113/47/42/424006	journal	October 2014
Gaussian quantum estimation of the loss parameter in a thermal environment Jonsson, Robert; Di Candia, Roberto Journal of Physics A: Mathematical and Theoretical, Vol. 55, Issue 38 https://doi.org/10.1088/1751-8121/ac83fa	journal	August 2022
Optimal estimation of pure states with displaced-null measurements Girotti, Federico; Godley, Alfred; Guţă, Mădălin Journal of Physics A: Mathematical and Theoretical, Vol. 57, Issue 24 https://doi.org/10.1088/1751-8121/ad4c2b	journal	June 2024
Critical quantum metrology with fully-connected models: from Heisenberg to Kibble–Zurek scaling Garbe, Louis; Abah, Obinna; Felicetti, Simone Quantum Science and Technology, Vol. 7, Issue 3 https://doi.org/10.1088/2058-9565/ac6ca5	journal	May 2022
Optimal quantum metrology of two-photon absorption Karsa, Athena; Nair, Ranjith; Chia, Andy Quantum Science and Technology, Vol. 9, Issue 3 https://doi.org/10.1088/2058-9565/ad466b	journal	May 2024
Asymptotic Theory of Quantum Channel Estimation Zhou, Sisi; Jiang, Liang PRX Quantum, Vol. 2, Issue 1 https://doi.org/10.1103/PRXQuantum.2.010343	journal	March 2021
Optimal Quantum Thermometry with Coarse-Grained Measurements Hovhannisyan, Karen V.; Jørgensen, Mathias R.; Landi, Gabriel T. PRX Quantum, Vol. 2, Issue 2 https://doi.org/10.1103/PRXQuantum.2.020322	journal	May 2021
Criticality-Enhanced Quantum Sensing via Continuous Measurement Ilias, Theodoros; Yang, Dayou; Huelga, Susana F. PRX Quantum, Vol. 3, Issue 1 https://doi.org/10.1103/PRXQuantum.3.010354	journal	March 2022
Two-photon-absorption measurements in the presence of single-photon losses Panahiyan, Shahram; Muñoz, Carlos Sánchez; Chekhova, Maria V. Physical Review A, Vol. 106, Issue 4 https://doi.org/10.1103/PhysRevA.106.043706	journal	October 2022
Invasiveness of nonequilibrium pure-dephasing quantum thermometry Albarelli, Francesco; Paris, Matteo G. A.; Vacchini, Bassano Physical Review A, Vol. 108, Issue 6 https://doi.org/10.1103/PhysRevA.108.062421	journal	December 2023
Temperature-enhanced critical quantum metrology Ostermann, Laurin; Gietka, Karol Physical Review A, Vol. 109, Issue 5 https://doi.org/10.1103/PhysRevA.109.L050601	journal	May 2024
Universal time scalings of sensitivity in Markovian quantum metrology Das, Arpan; Górecki, Wojciech; Demkowicz-Dobrzański, Rafał Physical Review A, Vol. 111, Issue 2 https://doi.org/10.1103/PhysRevA.111.L020403	journal	February 2025
Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping Cochrane, P. T.; Milburn, G. J.; Munro, W. J. Physical Review A, Vol. 59, Issue 4 https://doi.org/10.1103/PhysRevA.59.2631	journal	April 1999
State estimation for large ensembles Gill, Richard D.; Massar, Serge Physical Review A, Vol. 61, Issue 4 https://doi.org/10.1103/PhysRevA.61.042312	journal	March 2000
Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement Kim, M. S.; Son, W.; Bužek, V. Physical Review A, Vol. 65, Issue 3 https://doi.org/10.1103/PhysRevA.65.032323	journal	February 2002
Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states Adesso, G.; Dell’Anno, F.; De Siena, S. Physical Review A, Vol. 79, Issue 4 https://doi.org/10.1103/PhysRevA.79.040305	journal	April 2009
Universality of the Heisenberg limit for estimates of random phase shifts Hall, Michael J. W.; Berry, Dominic W.; Zwierz, Marcin Physical Review A, Vol. 85, Issue 4 https://doi.org/10.1103/PhysRevA.85.041802	journal	April 2012
Enhanced estimation of loss in the presence of Kerr nonlinearity Rossi, Matteo A. C.; Albarelli, Francesco; Paris, Matteo G. A. Physical Review A, Vol. 93, Issue 5 https://doi.org/10.1103/PhysRevA.93.053805	journal	May 2016
Bounding the quantum limits of precision for phase estimation with loss and thermal noise Gagatsos, Christos N.; Bash, Boulat A.; Guha, Saikat Physical Review A, Vol. 96, Issue 6 https://doi.org/10.1103/PhysRevA.96.062306	journal	December 2017
Sub-Heisenberg Estimation Strategies Are Ineffective Giovannetti, Vittorio; Maccone, Lorenzo Physical Review Letters, Vol. 108, Issue 21 https://doi.org/10.1103/PhysRevLett.108.210404	journal	May 2012
Quantum Measurement Bounds beyond the Uncertainty Relations Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo Physical Review Letters, Vol. 108, Issue 26 https://doi.org/10.1103/PhysRevLett.108.260405	journal	June 2012
Quantum Metrology in Non-Markovian Environments Chin, Alex W.; Huelga, Susana F.; Plenio, Martin B. Physical Review Letters, Vol. 109, Issue 23 https://doi.org/10.1103/PhysRevLett.109.233601	journal	December 2012
Quantum Speed Limits in Open System Dynamics del Campo, A.; Egusquiza, I. L.; Plenio, M. B. Physical Review Letters, Vol. 110, Issue 5 https://doi.org/10.1103/PhysRevLett.110.050403	journal	January 2013
Hardware-Efficient Autonomous Quantum Memory Protection Leghtas, Zaki; Kirchmair, Gerhard; Vlastakis, Brian Physical Review Letters, Vol. 111, Issue 12 https://doi.org/10.1103/PhysRevLett.111.120501	journal	September 2013
Using Entanglement Against Noise in Quantum Metrology Demkowicz-Dobrzański, Rafal; Maccone, Lorenzo Physical Review Letters, Vol. 113, Issue 25 https://doi.org/10.1103/PhysRevLett.113.250801	journal	December 2014
Individual Quantum Probes for Optimal Thermometry Correa, Luis A.; Mehboudi, Mohammad; Adesso, Gerardo Physical Review Letters, Vol. 114, Issue 22 https://doi.org/10.1103/PhysRevLett.114.220405	journal	June 2015
Ultimate Precision of Adaptive Noise Estimation Pirandola, Stefano; Lupo, Cosmo Physical Review Letters, Vol. 118, Issue 10 https://doi.org/10.1103/PhysRevLett.118.100502	journal	March 2017
π -Corrected Heisenberg Limit Górecki, Wojciech; Demkowicz-Dobrzański, Rafał; Wiseman, Howard M. Physical Review Letters, Vol. 124, Issue 3 https://doi.org/10.1103/PhysRevLett.124.030501	journal	January 2020
Critical Quantum Metrology with a Finite-Component Quantum Phase Transition Garbe, Louis; Bina, Matteo; Keller, Arne Physical Review Letters, Vol. 124, Issue 12 https://doi.org/10.1103/PhysRevLett.124.120504	journal	March 2020
Quantum Optical Metrology of Correlated Phase and Loss Birchall, Patrick M.; Allen, Euan J.; Stace, Thomas M. Physical Review Letters, Vol. 124, Issue 14 https://doi.org/10.1103/PhysRevLett.124.140501	journal	April 2020
Noisy Quantum Metrology Enhanced by Continuous Nondemolition Measurement Rossi, Matteo A. C.; Albarelli, Francesco; Tamascelli, Dario Physical Review Letters, Vol. 125, Issue 20 https://doi.org/10.1103/PhysRevLett.125.200505	journal	November 2020
Searching for Dark Matter with a Superconducting Qubit Dixit, Akash V.; Chakram, Srivatsan; He, Kevin Physical Review Letters, Vol. 126, Issue 14 https://doi.org/10.1103/PhysRevLett.126.141302	journal	April 2021
Quantum Metrology of Noisy Spreading Channels Górecki, Wojciech; Riccardi, Alberto; Maccone, Lorenzo Physical Review Letters, Vol. 129, Issue 24 https://doi.org/10.1103/PhysRevLett.129.240503	journal	December 2022
Measurement Noise Susceptibility in Quantum Estimation Kurdziałek, Stanisław; Demkowicz-Dobrzański, Rafał Physical Review Letters, Vol. 130, Issue 16 https://doi.org/10.1103/PhysRevLett.130.160802	journal	April 2023
Using Adaptiveness and Causal Superpositions Against Noise in Quantum Metrology Kurdziałek, Stanisław; Górecki, Wojciech; Albarelli, Francesco Physical Review Letters, Vol. 131, Issue 9 https://doi.org/10.1103/PhysRevLett.131.090801	journal	August 2023
Stimulated Emission of Signal Photons from Dark Matter Waves Agrawal, Ankur; Dixit, Akash V.; Roy, Tanay Physical Review Letters, Vol. 132, Issue 14 https://doi.org/10.1103/PhysRevLett.132.140801	journal	April 2024
Optimality and Noise Resilience of Critical Quantum Sensing Alushi, U.; Górecki, W.; Felicetti, S. Physical Review Letters, Vol. 133, Issue 4 https://doi.org/10.1103/PhysRevLett.133.040801	journal	July 2024
Wave-function approach to dissipative processes in quantum optics Dalibard, Jean; Castin, Yvan; Mølmer, Klaus Physical Review Letters, Vol. 68, Issue 5 https://doi.org/10.1103/PhysRevLett.68.580	journal	February 1992
Improvement of Frequency Standards with Quantum Entanglement Huelga, S. F.; Macchiavello, C.; Pellizzari, T. Physical Review Letters, Vol. 79, Issue 20 https://doi.org/10.1103/PhysRevLett.79.3865	journal	November 1997
Quantum Metrology Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo Physical Review Letters, Vol. 96, Issue 1 https://doi.org/10.1103/PhysRevLett.96.010401	journal	January 2006
Optimal Quantum Estimation of Loss in Bosonic Channels Monras, Alex; Paris, Matteo G. A. Physical Review Letters, Vol. 98, Issue 16 https://doi.org/10.1103/PhysRevLett.98.160401	journal	April 2007
Quantum sensing of open systems: Estimation of damping constants and temperature Wang, J.; Davidovich, L.; Agarwal, G. S. Physical Review Research, Vol. 2, Issue 3 https://doi.org/10.1103/PhysRevResearch.2.033389	journal	September 2020
Quantum metrology of two-photon absorption Sánchez Muñoz, Carlos; Frascella, Gaetano; Schlawin, Frank Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033250	journal	September 2021
Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations Bavaresco, Jessica; Lipka-Bartosik, Patryk; Sekatski, Pavel Physical Review Research, Vol. 6, Issue 2 https://doi.org/10.1103/PhysRevResearch.6.023305	journal	June 2024
Adaptive Quantum Metrology under General Markovian Noise Demkowicz-Dobrzański, Rafał; Czajkowski, Jan; Sekatski, Pavel Physical Review X, Vol. 7, Issue 4 https://doi.org/10.1103/PhysRevX.7.041009	journal	October 2017
Quantum sensing Degen, C. L.; Reinhard, F.; Cappellaro, P. Reviews of Modern Physics, Vol. 89, Issue 3 https://doi.org/10.1103/RevModPhys.89.035002	journal	July 2017
Quantum metrology with nonclassical states of atomic ensembles Pezzè, Luca; Smerzi, Augusto; Oberthaler, Markus K. Reviews of Modern Physics, Vol. 90, Issue 3 https://doi.org/10.1103/RevModPhys.90.035005	journal	September 2018
Sample-optimal tomography of quantum states Haah, Jeongwan; Harrow, Aram W.; Ji, Zhengfeng IEEE Transactions on Information Theory https://doi.org/10.1109/TIT.2017.2719044	journal	January 2017
Quantum-Enhanced Measurements: Beating the Standard Quantum Limit Giovannetti, V. Science, Vol. 306, Issue 5700 https://doi.org/10.1126/science.1104149	journal	November 2004
Local Asymptotic Normality and Optimal Estimation for D-Dimensional Quantum Systems Kahn, Jonas; GutŢĂ, MĂDĂLin Quantum Stochastics and Information https://doi.org/10.1142/9789812832962_0014	conference	August 2008
Quantum Estimation for Quantum Technology Paris, Matteo G. A. International Journal of Quantum Information, Vol. 07, Issue supp01 https://doi.org/10.1142/S0219749909004839	journal	January 2009
Kramers–Kronig relations and precision limits in quantum phase estimation Gianani, Ilaria; Albarelli, Francesco; Verna, Adriano Optica, Vol. 8, Issue 12 https://doi.org/10.1364/OPTICA.440438	journal	December 2021
Optimal nonequilibrium thermometry in Markovian environments Sekatski, Pavel; Perarnau-Llobet, Martí Quantum, Vol. 6 https://doi.org/10.22331/q-2022-12-07-869	journal	December 2022

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