Fundamental limits on low-temperature quantum thermometry with finite resolution

Patrick P. Potts, Jonatan Bohr Brask, and Nicolas Brunner

Department of Applied Physics, University of Geneva, Switzerland

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While the ability to measure low temperatures accurately in quantum systems is important in a wide range of experiments, the possibilities and the fundamental limits of quantum thermometry are not yet fully understood theoretically. Here we develop a general approach to low-temperature quantum thermometry, taking into account restrictions arising not only from the sample but also from the measurement process. {We derive a fundamental bound on the minimal uncertainty for any temperature measurement that has a finite resolution. A similar bound can be obtained from the third law of thermodynamics. Moreover, we identify a mechanism enabling sub-exponential scaling, even in the regime of finite resolution. We illustrate this effect in the case of thermometry on a fermionic tight-binding chain with access to only two lattice sites, where we find a quadratic divergence of the uncertainty}. We also give illustrative examples of ideal quantum gases and a square-lattice Ising model, highlighting the role of phase transitions.

The precise measurement of low temperatures at small length scales is of great interest for modern technologies, promising benefits for applications in many fields ranging from quantum information to molecular biology. However, fundamental limitations on the precision of such measurements are not yet fully understood. Here we develop a general approach to low-temperature quantum thermometry which allows for taking into account the restrictions imposed by the measurement process. We find that the restriction of a finite resolution, a restriction which is present in most experiments, results in a fundamental bound on the precision of any temperature measurement.

Interestingly, this fundamental bound allows, at least in principle, for infinitely precise measurements as temperature approaches zero. This is in stark contrast to the exponential divergence of the measurement error, which dominates many low-temperature thermometry strategies. Our approach provides an intuitive understanding of the mechanism that allows overcoming this exponential scaling as well as a visual tool to easily identify such scenarios. Through these insights, we were able to demonstrate that measurements with finite resolution can indeed result in a polynomially diverging measurement error in an experimentally relevant physical setup. Our approach thus provides a promising route for identifying systems favorable for low-temperature quantum thermometry, paving the way for temperature measurements with a precision that is only limited by fundamental constraints.

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[1] A. Altland and B. D. Simons. Condensed Matter Field Theory, (Cambridge University Press 2010).

[2] S. Vinjanampathy and J. Anders. Quantum thermodynamics. Contemp. Phys. 57, 545 (2016).

[3] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk. The role of quantum information in thermodynamics – a topical review. J. Phys. A: Math. Theor. 49, 143001 (2016).

[4] L. D. Carlos and F. Palacio (Editors). Thermometry at the Nanoscale, (The Royal Society of Chemistry 2016).

[5] A. D. Pasquale and T. M. Stace. Quantum thermometry. In F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Editors), Thermodynamics in the quantum regime, volume 195 of Fundamental Theories of Physics, (Springer, Cham 2019).

[6] T. M. Stace. Quantum limits of thermometry. Phys. Rev. A 82, 011611 (2010).

[7] M. Brunelli, S. Olivares, and M. G. A. Paris. Qubit thermometry for micromechanical resonators. Phys. Rev. A 84, 032105 (2011).

[8] M. Brunelli, S. Olivares, M. Paternostro, and M. G. A. Paris. Qubit-assisted thermometry of a quantum harmonic oscillator. Phys. Rev. A 86, 012125 (2012).

[9] C. Sabín, A. White, L. Hackermuller, and I. Fuentes. Impurities as a quantum thermometer for a Bose-Einstein condensate. Sci. Rep. 4, 6436 (2014).

[10] M. Mehboudi, M. Moreno-Cardoner, G. D. Chiara, and A. Sanpera. Thermometry precision in strongly correlated ultracold lattice gases. New J. Phys. 17, 055020 (2015).

[11] L.-S. Guo, B.-M. Xu, J. Zou, and B. Shao. Improved thermometry of low-temperature quantum systems by a ring-structure probe. Phys. Rev. A 92, 052112 (2015).

[12] P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner. Quantum thermal machine as a thermometer. Phys. Rev. Lett. 119, 090603 (2017).

[13] A. De Pasquale, K. Yuasa, and V. Giovannetti. Estimating temperature via sequential measurements. Phys. Rev. A 96, 012316 (2017).

[14] S. Campbell, M. Mehboudi, G. D. Chiara, and M. Paternostro. Global and local thermometry schemes in coupled quantum systems. New J. Phys. 19, 103003 (2017).

[15] S. Campbell, M. G. Genoni, and S. Deffner. Precision thermometry and the quantum speed limit. Quantum Sci. Technol. 3, 025002 (2018).

[16] M. Płodzień, R. Demkowicz-Dobrzański, and T. Sowiński. Few-fermion thermometry. Phys. Rev. A 97, 063619 (2018).

[17] A. Sone, Q. Zhuang, and P. Cappellaro. Quantifying precision loss in local quantum thermometry via diagonal discord. Phys. Rev. A 98, 012115 (2018).

[18] M. G. A. Paris. Quantum estimation for quantum technology. Int. J. Quantum Inform. 07, 125 (2009).

[19] A. De Pasquale, D. Rossini, R. Fazio, and V. Giovannetti. Local quantum thermal susceptibility. Nat. Commun. 7, 12782 (2016).

[20] G. De Palma, A. De Pasquale, and V. Giovannetti. Universal locality of quantum thermal susceptibility. Phys. Rev. A 95, 052115 (2017).

[21] L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera. Individual quantum probes for optimal thermometry. Phys. Rev. Lett. 114, 220405 (2015).

[22] L. A. Correa, M. Perarnau-Llobet, K. V. Hovhannisyan, S. Hernández-Santana, M. Mehboudi, and A. Sanpera. Enhancement of low-temperature thermometry by strong coupling. Phys. Rev. A 96, 062103 (2017).

[23] F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, and J. P. Pekola. Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications. Rev. Mod. Phys. 78, 217 (2006).

[24] M. G. A. Paris. Achieving the Landau bound to precision of quantum thermometry in systems with vanishing gap. J. Phys. A: Math. Theor. 49, 03LT02 (2016).

[25] P. Zanardi, P. Giorda, and M. Cozzini. Information-theoretic differential geometry of quantum phase transitions. Phys. Rev. Lett. 99, 100603 (2007).

[26] L. D. Landau and E. M. Lifshitz. Course of Theoretical Physics, Vol. 5: Statistical Physics, Part 1, (Pergamon Press, Oxford 1959).

[27] H. B. Callen. Thermodynamics and an Introduction to Thermostatistics, (Wiley, New York 1985).

[28] N. W. Ashcroft and N. D. Mermin. Solid state physics, (Harcourt College Publishers 1976).

[29] U. Marzolino and D. Braun. Precision measurements of temperature and chemical potential of quantum gases. Phys. Rev. A 88, 063609 (2013).

[30] F. Fröwis, P. Sekatski, and W. Dür. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett. 116, 090801 (2016).

[31] K. V. Hovhannisyan and L. A. Correa. Measuring the temperature of cold many-body quantum systems. Phys. Rev. B 98, 045101 (2018).

[32] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information, (Cambridge University Press, New York 2010).

[33] A. Monras and F. Illuminati. Measurement of damping and temperature: Precision bounds in gaussian dissipative channels. Phys. Rev. A 83, 012315 (2011).

[34] E. Martín-Martínez, A. Dragan, R. B. Mann, and I. Fuentes. Berry phase quantum thermometer. New J. Phys. 15, 053036 (2013).

[35] S. Jevtic, D. Newman, T. Rudolph, and T. M. Stace. Single-qubit thermometry. Phys. Rev. A 91, 012331 (2015).

[36] T. H. Johnson, F. Cosco, M. T. Mitchison, D. Jaksch, and S. R. Clark. Thermometry of ultracold atoms via nonequilibrium work distributions. Phys. Rev. A 93, 053619 (2016).

[37] L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri. Quantum simulation of single-qubit thermometry using linear optics. Phys. Rev. Lett. 118, 130502 (2017).

[38] S. Razavian, C. Benedetti, M. Bina, Y. Akbari-Kourbolagh, and M. G. A. Paris. Quantum thermometry by single-qubit dephasing. ArXiv:1807.11810.

[39] V. Cavina, L. Mancino, A. De Pasquale, I. Gianani, M. Sbroscia, R. I. Booth, E. Roccia, R. Raimondi, V. Giovannetti, and M. Barbieri. Bridging thermodynamics and metrology in nonequilibrium quantum thermometry. Phys. Rev. A 98, 050101 (2018).

[40] H. Cramér. Mathematical Methods of Statistics. PMS-9 /​ Princeton Landmarks in Mathematics, (Princeton University Press 1946).

[41] E. T. Jaynes. Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957).

[42] P. Zanardi, M. G. A. Paris, and L. Campos Venuti. Quantum criticality as a resource for quantum estimation. Phys. Rev. A 78, 042105 (2008).

[43] L. Masanes and J. Oppenheim. A general derivation and quantification of the third law of thermodynamics. Nat. Commun. 8, 14538 (2017).

[44] H. Wilming and R. Gallego. Third law of thermodynamics as a single inequality. Phys. Rev. X 7, 041033 (2017).

[45] K. Weierstrass. Über die analytische Darstellbarkeit sogenannter willkürlicher Funktionen einer reellen Veränderlichen. In Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II), page 789, (Verl. d. Kgl. Akad. d. Wiss. 1885).

[46] I. Peschel. Calculation of reduced density matrices from correlation functions. J. Phys. A 36, L205 (2003).

[47] F. London. Superfluids, volume II, (Wiley & Sons 1954).

[48] L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117 (1944).

[49] V. Mukherjee, A. Zwick, A. Ghosh, X. Chen, and G. Kurizki. Enhanced precision of low-temperature quantum thermometry via dynamical control. ArXiv:1711.09660.

[50] L. Seveso and M. G. A. Paris. Trade-off between information and disturbance in qubit thermometry. Phys. Rev. A 97, 032129 (2018).

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[13] Karen V. Hovhannisyan and Luis A. Correa, "Measuring the temperature of cold many-body quantum systems", Physical Review B 98 4, 045101 (2018).

[14] Antonella De Pasquale and Thomas M. Stace, "Quantum Thermometry", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 503 (2018).

[15] Sholeh Razavian, Claudia Benedetti, Matteo Bina, Yahya Akbari-Kourbolagh, and Matteo G. A. Paris, "Quantum thermometry by single-qubit dephasing", arXiv:1807.11810.

[16] Alexander Holm Kiilerich, Antonella De Pasquale, and Vittorio Giovannetti, "Dynamical approach to ancilla-assisted quantum thermometry", arXiv:1807.11268, Physical Review A 98 4, 042124 (2018).

[17] Marcin Płodzień, Rafał Demkowicz-Dobrzański, and Tomasz Sowiński, "Few-fermion thermometry", Physical Review A 97 6, 063619 (2018).

[18] Luigi Seveso and Matteo G. A. Paris, "Trade-off between information and disturbance in qubit thermometry", Physical Review A 97 3, 032129 (2018).

[19] Michele M. Feyles, Luca Mancino, Marco Sbroscia, Ilaria Gianani, and Marco Barbieri, "Dynamical role of quantum signatures in quantum thermometry", Physical Review A 99 6, 062114 (2019).

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