Thermometry of Gaussian quantum systems using Gaussian measurements

Marina F.B. Cenni1, Ludovico Lami2, Antonio Acín1,3, and Mohammad Mehboudi4

1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
2Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
3ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010, Barcelona, Spain
4Département de Physique Appliquée, Université de Genève, 1205 Genève, Switzerland

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Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
22 pages. More details are introduced compared to version 1, namely an example with maximum likelihood estimator for thermometry is added


We study the problem of estimating the temperature of Gaussian systems with feasible measurements, namely Gaussian and photo-detection-like measurements. For Gaussian measurements, we develop a general method to identify the optimal measurement numerically, and derive the analytical solutions in some relevant cases. For a class of single-mode states that includes thermal ones, the optimal Gaussian measurement is either Heterodyne or Homodyne, depending on the temperature regime. This is in contrast to the general setting, in which a projective measurement in the eigenbasis of the Hamiltonian is optimal regardless of temperature. In the general multi-mode case, and unlike the general unrestricted scenario where joint measurements are not helpful for thermometry (nor for any parameter estimation task), it is open whether joint Gaussian measurements provide an advantage over local ones. We conjecture that they are not useful for thermal systems, supported by partial analytical and numerical evidence. We further show that Gaussian measurements become optimal in the limit of large temperatures, while {on/off} photo-detection-like measurements do it for when the temperature tends to zero. Our results therefore pave the way for effective thermometry of Gaussian quantum systems using $\textit{experimentally realizable measurements}$.

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[1] Mohammad Mehboudi, Anna Sanpera, and Luis A Correa. Thermometry in the quantum regime: recent theoretical progress. Journal of Physics A: Mathematical and Theoretical, 52 (30): 303001, jul 2019a. https:/​/​​10.1088/​1751-8121/​ab2828.

[2] Antonella De Pasquale and Thomas M. Stace. Quantum thermometry. In Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso, editors, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, pages 503–527. Springer International Publishing, Cham, 2018. https:/​/​​10.1007/​978-3-319-99046-0_21.

[3] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature photonics, 5 (4): 222–229, 2011. https:/​/​​10.1038/​nphoton.2011.35.

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

[5] Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information science perspective. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424006, oct 2014. https:/​/​​10.1088/​1751-8113/​47/​42/​424006.

[6] Luis A. Correa, Mohammad Mehboudi, Gerardo Adesso, and Anna Sanpera. Individual quantum probes for optimal thermometry. Phys. Rev. Lett., 114: 220405, Jun 2015. https:/​/​​10.1103/​PhysRevLett.114.220405.

[7] Mathias R. Jorgensen, Patrick P. Potts, Matteo G. A. Paris, and Jonatan B. Brask. Tight bound on finite-resolution quantum thermometry at low temperatures. Phys. Rev. Research, 2: 033394, Sep 2020. https:/​/​​10.1103/​PhysRevResearch.2.033394.

[8] Karen V. Hovhannisyan, Mathias R. J0rgensen, Gabriel T. Landi, Álvaro M. Alhambra, Jonatan B. Brask, and Martí Perarnau-Llobet. Optimal quantum thermometry with coarse-grained measurements. PRX Quantum, 2: 020322, May 2021. https:/​/​​10.1103/​PRXQuantum.2.020322.

[9] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84: 621–669, May 2012. https:/​/​​10.1103/​RevModPhys.84.621.

[10] Maciej Lewenstein, Anna Sanpera, Veronica Ahufinger, Bogdan Damski, Aditi Sen(De), and Ujjwal Sen. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Advances in Physics, 56 (2): 243–379, 2007. https:/​/​​10.1080/​00018730701223200.

[11] Maciej Lewenstein, Anna Sanpera, and Veronica Ahufinger. Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems. Oxford University Press, 2012. https:/​/​​10.1093/​acprof:oso/​9780199573127.001.0001.

[12] Alessio Serafini. Quantum continuous variables: a primer of theoretical methods. CRC press, 2017.

[13] Eleni Diamanti and Anthony Leverrier. Distributing secret keys with quantum continuous variables: Principle, security and implementations. Entropy, 17 (9): 6072–6092, 2015. ISSN 1099-4300. https:/​/​​10.3390/​e17096072.

[14] Miguel Navascués and Antonio Acín. Securitybounds for continuous variables quantum key distribution. Phys. Rev. Lett., 94: 020505, Jan 2005. https:/​/​​10.1103/​PhysRevLett.94.020505.

[15] Alex Monras. Phase space formalism for quantum estimation of gaussian states. arXiv preprint arXiv:1303.3682, 2013. https:/​/​​10.48550/​arXiv.1303.3682.

[16] Zhang Jiang. Quantum fisher information for states in exponential form. Phys. Rev. A, 89: 032128, Mar 2014. https:/​/​​10.1103/​PhysRevA.89.032128.

[17] Rosanna Nichols, Pietro Liuzzo-Scorpo, Paul A. Knott, and Gerardo Adesso. Multiparameter gaussian quantum metrology. Phys. Rev. A, 98: 012114, Jul 2018. https:/​/​​10.1103/​PhysRevA.98.012114.

[18] Mohammad Mehboudi, Aniello Lampo, Christos Charalambous, Luis A. Correa, Miguel Ángel García-March, and Maciej Lewenstein. Using polarons for sub-nk quantum nondemolition thermometry in a bose-einstein condensate. Phys. Rev. Lett., 122: 030403, Jan 2019b. https:/​/​​10.1103/​PhysRevLett.122.030403.

[19] Guim Planella, Marina F. B. Cenni, Antonio Acín, and Mohammad Mehboudi. Bath-induced correlations enhance thermometry precision at low temperatures. Phys. Rev. Lett., 128: 040502, Jan 2022. https:/​/​​10.1103/​PhysRevLett.128.040502.

[20] Luis A. Correa, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Senaida Hernández-Santana, Mohammad Mehboudi, and Anna Sanpera. Enhancement of low-temperature thermometry by strong coupling. Phys. Rev. A, 96: 062103, Dec 2017. https:/​/​​10.1103/​PhysRevA.96.062103.

[21] Karen V. Hovhannisyan and Luis A. Correa. Measuring the temperature of cold many-body quantum systems. Phys. Rev. B, 98: 045101, Jul 2018. https:/​/​​10.1103/​PhysRevB.98.045101.

[22] Victor Mukherjee, Analia Zwick, Arnab Ghosh, Xi Chen, and Gershon Kurizki. Enhanced precision bound of low-temperature quantum thermometry via dynamical control. Communications Physics, 2 (1): 1–8, 2019. https:/​/​​10.1038/​s42005-019-0265-y.

[23] Eduardo Martín-Martínez, Andrzej Dragan, Robert B Mann, and Ivette Fuentes. Berry phase quantum thermometer. New Journal of Physics, 15 (5): 053036, may 2013. https:/​/​​10.1088/​1367-2630/​15/​5/​053036.

[24] Alexander S. Holevo. Probabilistic and Statistical Aspects of Quantum Theory. Publications of the Scuola Normale Superiore. Scuola Normale Superiore, Pisa, Italy, 2011. https:/​/​​10.1007/​978-88-7642-378-9.

[25] Stephen Barnett and Paul M. Radmore. Methods in Theoretical Quantum Optics. Oxford Series in Optical and Imaging Sciences. Clarendon Press, 2002. ISBN 9780198563617. https:/​/​​10.1093/​acprof:oso/​9780198563617.001.0001.

[26] Whitney K. Newey and Daniel McFadden. Chapter 36 large sample estimation and hypothesis testing. 4: 2111–2245, 1994. ISSN 1573-4412. https:/​/​​10.1016/​S1573-4412(05)80005-4.

[27] O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun. Quantum parameter estimation using general single-mode gaussian states. Phys. Rev. A, 88: 040102, Oct 2013. https:/​/​​10.1103/​PhysRevA.88.040102.

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

[29] Changhun Oh, Changhyoup Lee, Carsten Rockstuhl, Hyunseok Jeong, Jaewan Kim, Hyunchul Nha, and Su-Yong Lee. Optimal gaussian measurements for phase estimation in single-mode gaussian metrology. npj Quantum Information, 5 (1): 1–9, 2019. https:/​/​​10.1038/​s41534-019-0124-4.

[30] Simon Morelli, Ayaka Usui, Elizabeth Agudelo, and Nicolai Friis. Bayesian parameter estimation using gaussian states and measurements. Quantum Science and Technology, 6 (2): 025018, mar 2021. https:/​/​​10.1088/​2058-9565/​abd83d.

[31] Neel Kanth Kundu, Matthew R. McKay, and Ranjan K. Mallik. Machine-learning-based parameter estimation of gaussian quantum states. IEEE Transactions on Quantum Engineering, 3: 1–13, 2022. https:/​/​​10.1109/​TQE.2021.3137559.

[32] Géza Giedke and J. Ignacio Cirac. Characterization of gaussian operations and distillation of gaussian states. Phys. Rev. A, 66: 032316, Sep 2002. https:/​/​​10.1103/​PhysRevA.66.032316.

[33] John Williamson. On the algebraic problem concerning the normal forms of linear dynamical systems. American Journal of Mathematics, 58 (1): 141–163, 1936. ISSN 00029327, 10806377. https:/​/​​10.2307/​2371062.

[34] Ludovico Lami, Christoph Hirche, Gerardo Adesso, and Andreas Winter. From log-determinant inequalities to gaussian entanglement via recoverability theory. IEEE Transactions on Information Theory, 63 (11): 7553–7568, 2017. https:/​/​​10.1109/​TIT.2017.2737546.

[35] Aniello Lampo, Soon Hoe Lim, Miguel Ángel García-March, and Maciej Lewenstein. Bose polaron as an instance of quantum Brownian motion. Quantum, 1: 30, September 2017. ISSN 2521-327X. https:/​/​​10.22331/​q-2017-09-27-30.

[36] Muhammad Miskeen Khan, Mohammad Mehboudi, Hugo Terças, Maciej Lewenstein, and Miguel Angel Garcia-March. Subnanokelvin thermometry of an interacting $d$-dimensional homogeneous bose gas. Phys. Rev. Research, 4: 023191, Jun 2022. https:/​/​​10.1103/​PhysRevResearch.4.023191.

[37] Luigi Malagò and Giovanni Pistone. Information geometry of the gaussian distribution in view of stochastic optimization. page 150–162, 2015. 10.1145/​2725494.2725510.

[38] Saleh Rahimi-Keshari, Timothy C. Ralph, and Carlton M. Caves. Sufficient conditions for efficient classical simulation of quantum optics. Phys. Rev. X, 6: 021039, Jun 2016. https:/​/​​10.1103/​PhysRevX.6.021039.

[39] Stephen M. Barnett, Lee S. Phillips, and David T. Pegg. Imperfect photodetection as projection onto mixed states. Optics Communications, 158 (1): 45–49, 1998. ISSN 0030-4018. https:/​/​​10.1016/​S0030-4018(98)00511-2.

[40] Jesús Rubio, Janet Anders, and Luis A. Correa. Global quantum thermometry. Phys. Rev. Lett., 127: 190402, Nov 2021. https:/​/​​10.1103/​PhysRevLett.127.190402.

[41] Mohammad Mehboudi, Mathias R Jorgensen, Stella Seah, Jonatan B Brask, Jan Kołodyński, and Martí Perarnau-Llobet. Fundamental limits in bayesian thermometry and attainability via adaptive strategies. Physical Review Letters, 128 (13): 130502, 2022. https:/​/​​10.1103/​PhysRevLett.128.130502.

[42] Mathias R. Jorgensen, Jan Kołodyński, Mohammad Mehboudi, Martí Perarnau-Llobet, and Jonatan B. Brask. Bayesian quantum thermometry based on thermodynamic length. Phys. Rev. A, 105: 042601, Apr 2022. https:/​/​​10.1103/​PhysRevA.105.042601.

[43] Gabriel O. Alves and Gabriel T. Landi. Bayesian estimation for collisional thermometry. Phys. Rev. A, 105: 012212, Jan 2022. https:/​/​​10.1103/​PhysRevA.105.012212.

[44] Alexander S. Holevo. Quantum systems, channels, information. de Gruyter, 2019. https:/​/​​10.1515/​9783110273403.

[45] Jukka Kiukas and Jussi Schultz. Informationally complete sets of gaussian measurements. Journal of Physics A: Mathematical and Theoretical, 46 (48): 485303, nov 2013. https:/​/​​10.1088/​1751-8113/​46/​48/​485303.

[46] R. Simon, N. Mukunda, and Biswadeb Dutta. Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms. Phys. Rev. A, 49: 1567–1583, Mar 1994. 10.1103/​PhysRevA.49.1567.

[47] Marco G. Genoni, Ludovico Lami, and Alessio Serafini. Conditional and unconditional gaussian quantum dynamics. Contemporary Physics, 57 (3): 331–349, 2016. https:/​/​​10.1080/​00107514.2015.1125624.

[48] Matteo G. A. Paris. Quantum estimation for quantum technology. International Journal of Quantum Information, 7 (supp01): 125–137, 2009. https:/​/​​10.1142/​S0219749909004839.

[49] Jan Kolodynski. Precision bounds in noisy quantum metrology. arXiv:1409.0535, 2014. https:/​/​​10.48550/​arXiv.1409.0535.

[50] Samuel L. Braunstein. Homodyne statistics. Phys. Rev. A, 42: 474–481, Jul 1990. https:/​/​​10.1103/​PhysRevA.42.474.

[51] Jérôme Lodewyck. Dispositif de distribution quantique de clé avec des états cohérents à longueur d'onde télécom. PhD thesis, Université Paris Sud-Paris XI, 2006. URL https:/​/​​tel-00130680.

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