Non-informative Bayesian Quantum Thermometry

We study the Bayesian approach to thermometry with no prior knowledge about the expected temperature scale, through the example of energy measurements on fully or partially thermalized qubit probes. We show that the most common Bayesian estimators, namely the mean and the median, lead to high-temperature divergences when used for uninformed thermometry. To circumvent this and achieve better overall accuracy, we propose two new estimators based on an optimization of relative deviations. Their global temperature-averaged behavior matches a modified van Trees bound, which complements the Cram\'er-Rao bound for smaller probe numbers and unrestricted temperature ranges. Furthermore, we show that, using partially thermalized probes, one can increase the range of temperatures to which the thermometer is sensitive at the cost of the local accuracy.We study the Bayesian approach to thermometry with no prior knowledge about the expected temperature scale, through the example of energy measurements on fully or partially thermalized qubit probes. We show that the most common Bayesian estimators, namely the mean and the median, lead to high-temperature divergences when used for uninformed thermometry. To circumvent this and achieve better overall accuracy, we propose two new estimators based on an optimization of relative deviations. Their global temperature-averaged behavior matches a modified van Trees bound, which complements the Cram\'er-Rao bound for smaller probe numbers and unrestricted temperature ranges. Furthermore, we show that, using partially thermalized probes, one can increase the range of temperatures to which the thermometer is sensitive at the cost of the local accuracy.

arXiv:2108.07025Physical Review ANov 18, 202125 citations

Direct temperature readout in nonequilibrium quantum thermometry

Yan Xie, Junjie Liu

Quantum thermometry aims to measure temperature in nanoscale quantum systems, paralleling classical thermometry. However, temperature is not a quantum observable, and most theoretical studies have therefore concentrated on analyzing fundamental precision limits set by the quantum Fisher information through the quantum Cramer-Rao bound. In contrast, whether a direct temperature readout can be achieved in quantum thermometry remains largely unexplored, particularly under the nonequilibrium conditions prevalent in real-world applications. To address this, we develop a direct temperature readout scheme based on a thermodynamic inference strategy. The scheme integrates two conceptual developments: (i) By applying the maximum entropy principle with the thermometer's mean energy as a constraint, we assign a reference temperature to the nonequilibrium thermometer. We demonstrate that this reference temperature outperforms a commonly used effective temperature defined through equilibrium analogy. (ii) We obtain positive semi-definite error functions that lower-bound the deviation of the reference temperature from the true temperature-in analogy to the quantum Cramer-Rao bound for the mean squared error-and vanish upon thermalization with the sample. Combining the reference temperature with these error functions, we introduce a notion of corrected dynamical temperature which furnishes a postprocessed temperature readout under nonequilibrium conditions. This corrected dynamical temperature can be evaluated adaptively without prior knowledge of the actual temperature. We validate the corrected dynamical temperature in a qubit-based thermometer under a range of nonequilibrium initial states, confirming its capability to estimate the true temperature. Importantly, we find that increasing quantum coherence can enhance the precision of this readout.Quantum thermometry aims to measure temperature in nanoscale quantum systems, paralleling classical thermometry. However, temperature is not a quantum observable, and most theoretical studies have therefore concentrated on analyzing fundamental precision limits set by the quantum Fisher information through the quantum Cramer-Rao bound. In contrast, whether a direct temperature readout can be achieved in quantum thermometry remains largely unexplored, particularly under the nonequilibrium conditions prevalent in real-world applications. To address this, we develop a direct temperature readout scheme based on a thermodynamic inference strategy. The scheme integrates two conceptual developments: (i) By applying the maximum entropy principle with the thermometer's mean energy as a constraint, we assign a reference temperature to the nonequilibrium thermometer. We demonstrate that this reference temperature outperforms a commonly used effective temperature defined through equilibrium analogy. (ii) We obtain positive semi-definite error functions that lower-bound the deviation of the reference temperature from the true temperature-in analogy to the quantum Cramer-Rao bound for the mean squared error-and vanish upon thermalization with the sample. Combining the reference temperature with these error functions, we introduce a notion of corrected dynamical temperature which furnishes a postprocessed temperature readout under nonequilibrium conditions. This corrected dynamical temperature can be evaluated adaptively without prior knowledge of the actual temperature. We validate the corrected dynamical temperature in a qubit-based thermometer under a range of nonequilibrium initial states, confirming its capability to estimate the true temperature. Importantly, we find that increasing quantum coherence can enhance the precision of this readout.

quant-pharXiv:2601.07198Physical Review AJan 13, 20261 citation

Bayesian estimation for collisional thermometry

Gabriel O. Alves, Gabriel Teixeira Landi

Quantum thermometry exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. This highlights the need for constructing concrete estimation strategies. Of particular importance is collisional thermometry, where a series of ancillae are sent sequentially to probe the system's temperature. In this paper we put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cram\'er-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cram\'er-Rao bound associated with Van Trees and Sch\"utzenberger.Quantum thermometry exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. This highlights the need for constructing concrete estimation strategies. Of particular importance is collisional thermometry, where a series of ancillae are sent sequentially to probe the system's temperature. In this paper we put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cram\'er-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cram\'er-Rao bound associated with Van Trees and Sch\"utzenberger.

arXiv:2106.12072Physical Review AJan 19, 20226 citations

Quantum Thermometry

Antonella De Pasquale, Thomas M. Stace

We discuss the application of techniques of quantum estimation theory and quantum metrology to thermometry. The ultimate limit to the precision at which the temperature of a system at thermal equilibrium can be determined is related to the heat capacity when global measurements are performed on the system. We prove that if technical or practical limitations restrict our capabilities to local probing, the highest achievable accuracy to temperature estimation reduces to a sort of mesoscopic version of the heat capacity. Adopting a more practical perspective, we also discuss the relevance of qubit systems as optimal quantum thermometers, in order to retrieve the temperature, or to discriminate between two temperatures, characterizing a thermal reservoir. We show that quantum coherence and entanglement in a probe system can facilitate faster, or more accurate measurements of temperature. While not surprising given this has been demonstrated in phase estimation, temperature is not a conventional quantum observable, therefore these results extend the theory of parameter estimation to measurement of non-Hamiltonian quantities. Finally we point out the advantages brought by a less standard estimation technique based on sequential measurements, when applied to quantum thermometry.We discuss the application of techniques of quantum estimation theory and quantum metrology to thermometry. The ultimate limit to the precision at which the temperature of a system at thermal equilibrium can be determined is related to the heat capacity when global measurements are performed on the system. We prove that if technical or practical limitations restrict our capabilities to local probing, the highest achievable accuracy to temperature estimation reduces to a sort of mesoscopic version of the heat capacity. Adopting a more practical perspective, we also discuss the relevance of qubit systems as optimal quantum thermometers, in order to retrieve the temperature, or to discriminate between two temperatures, characterizing a thermal reservoir. We show that quantum coherence and entanglement in a probe system can facilitate faster, or more accurate measurements of temperature. While not surprising given this has been demonstrated in phase estimation, temperature is not a conventional quantum observable, therefore these results extend the theory of parameter estimation to measurement of non-Hamiltonian quantities. Finally we point out the advantages brought by a less standard estimation technique based on sequential measurements, when applied to quantum thermometry.

arXiv:1807.05762Fundamental theories of physicsJan 1, 201826 citations

Global quantum thermometry based on the optimal biased bound

Shoukang Chang, Yunbin Yan, Lu Wang +8 more

Thermometry is a fundamental parameter estimation problem that is crucial for the advancement of natural sciences. One widely adopted approach to address this issue is the local thermometry theory, which employs classical and quantum Cramér-Rao bounds as benchmarks for thermometric precision. However, this theory is constrained to decrease temperature fluctuations around a known temperature value, hardly tackling the precision thermometry problem over a wide temperature range. To overcome this limitation, we derive two basic bounds on thermometry precision within a global framework, i.e., classical and quantum optimal biased bounds. By implementing energy measurements on a thermal equilibrium system, the quantum optimal biased bound can be saturated. Furthermore, we demonstrate their thermometry performance through two specific applications: a noninteracting spin- 1 / 2 gas and a thermalized quantum harmonic oscillator. Our results indicate that, compared to local thermometry, global thermometry provides superior temperature estimation performance. Notably, global error bound approaches its local approximation under asymptotic cases. Published by the American Physical Society 2024Thermometry is a fundamental parameter estimation problem that is crucial for the advancement of natural sciences. One widely adopted approach to address this issue is the local thermometry theory, which employs classical and quantum Cramér-Rao bounds as benchmarks for thermometric precision. However, this theory is constrained to decrease temperature fluctuations around a known temperature value, hardly tackling the precision thermometry problem over a wide temperature range. To overcome this limitation, we derive two basic bounds on thermometry precision within a global framework, i.e., classical and quantum optimal biased bounds. By implementing energy measurements on a thermal equilibrium system, the quantum optimal biased bound can be saturated. Furthermore, we demonstrate their thermometry performance through two specific applications: a noninteracting spin- 1 / 2 gas and a thermalized quantum harmonic oscillator. Our results indicate that, compared to local thermometry, global thermometry provides superior temperature estimation performance. Notably, global error bound approaches its local approximation under asymptotic cases. Published by the American Physical Society 2024

arXiv:2305.08397Physical Review ResearchNov 19, 20245 citations

Bayesian quantum thermometry based on thermodynamic length

Mathias R. Jørgensen, Jan Kołodyński, Mohammad Mehboudi +2 more

In this work we propose a theory of temperature estimation of quantum systems, which is applicable in the regime of non-negligible prior temperature uncertainty and limited measurement data. In this regime the problem of establishing a well-defined measure of estimation precision becomes nontrivial. Furthermore, the construction of a suitable criterion for optimal measurement design must be reexamined to account for the prior uncertainty. We propose a fully Bayesian approach to temperature estimation based on the concept of thermodynamic length, which solves both these problems. As an illustration of this framework, we consider thermal spin- $1/2$ particles and investigate the fundamental difference between two cases: on the one hand, when the spins are probing the temperature of a heat reservoir and, on the other, when the spins themselves constitute the sample.In this work we propose a theory of temperature estimation of quantum systems, which is applicable in the regime of non-negligible prior temperature uncertainty and limited measurement data. In this regime the problem of establishing a well-defined measure of estimation precision becomes nontrivial. Furthermore, the construction of a suitable criterion for optimal measurement design must be reexamined to account for the prior uncertainty. We propose a fully Bayesian approach to temperature estimation based on the concept of thermodynamic length, which solves both these problems. As an illustration of this framework, we consider thermal spin-$1/2$ particles and investigate the fundamental difference between two cases: on the one hand, when the spins are probing the temperature of a heat reservoir and, on the other, when the spins themselves constitute the sample.

arXiv:2108.05901Physical Review AApr 1, 202226 citations

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particles and investigate the fundamental difference between two cases: on the one hand, when the spins are probing the temperature of a heat reservoir and, on the other, when the spins themselves constitute the sample.