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arXiv:2507.15458 · Physical Review A

Optimal strategies for transient and equilibrium quantum thermometry using Gaussian and non-Gaussian probes

Asghar Ullah, M. Tahir Naseem, Özgür Esat Müstecaplıoğlu

Published Oct 23, 2025• 2 citations
arXivHTMLPDF

Abstract

We study temperature estimation using quantum probes, including single-mode initial states and two-mode states generated via stimulated parametric down-conversion in a nonlinear crystal at finite temperature. We explore both transient and equilibrium regimes and compare the performance of Gaussian and non-Gaussian probe states for temperature estimation. In the nonequilibrium regime, we show that single-mode non-Gaussian probe states---such as Fock, odd cat, and Gottesman-Kitaev-Preskill states---can significantly enhance the speed of estimation, particularly at short interaction times. In the two-mode setting, entangled states such as the two-mode squeezed vacuum, NOON state, and entangled cat state can enable access to temperature information at earlier times. In the equilibrium regime, we analyze temperature estimation using two-mode squeezed thermal states, which outperform single-mode strategies. We evaluate practical measurement strategies and find that energy-based observables yield optimal precision, population difference observables provide near-optimal precision, while quadrature-based measurements are suboptimal. The precision gain arises from squeezing, which suppresses fluctuations in the population difference.

DOI
10.1103/62ks-19fs

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We study temperature estimation using quantum probes, including single-mode initial states and two-mode states generated via stimulated parametric down-conversion in a nonlinear crystal at finite temperature. We explore both transient and equilibrium regimes and compare the performance of Gaussian and non-Gaussian probe states for temperature estimation. In the nonequilibrium regime, we show that single-mode non-Gaussian probe states---such as Fock, odd cat, and Gottesman-Kitaev-Preskill states---can significantly enhance the speed of estimation, particularly at short interaction times. In the two-mode setting, entangled states such as the two-mode squeezed vacuum, NOON state, and entangled cat state can enable access to temperature information at earlier times. In the equilibrium regime, we analyze temperature estimation using two-mode squeezed thermal states, which outperform single-mode strategies. We evaluate practical measurement strategies and find that energy-based observables yield optimal precision, population difference observables provide near-optimal precision, while quadrature-based measurements are suboptimal. The precision gain arises from squeezing, which suppresses fluctuations in the population difference.
We present a comprehensive analysis of the performance of different classes of Gaussian states in the estimation of Gaussian phase-insensitive dissipative channels. In particular, we investigate the optimal estimation of the damping constant and reservoir temperature. We show that, for two-mode squeezed vacuum probe states, the quantum-limited accuracy of both parameters can be achieved simultaneously. Moreover, we show that for both parameters two-mode squeezed vacuum states are more efficient than coherent, thermal, or single-mode squeezed states. This suggests that at high-energy regimes, two-mode squeezed vacuum states are optimal within the Gaussian setup. This optimality result indicates a stronger form of compatibility for the estimation of the two parameters. Indeed, not only the minimum variance can be achieved at fixed probe states, but also the optimal state is common to both parameters. Additionally, we explore numerically the performance of non-Gaussian states for particular parameter values to find that maximally entangled states within ddd-dimensional cutoff subspaces ($d\ensuremath{\leqslant}6$) perform better than any randomly sampled states with similar energy. However, we also find that states with very similar performance and energy exist with much less entanglement than the maximally entangled ones.
Abstract We provide new operational significance of nonclassicality in nonequilibrium temperature estimation of bosonic baths with Gaussian probe states and Gaussian dynamics. We find a bound on the thermometry performance using classical probe states. Then we show that by using nonclassical probe states, single-mode and two-mode squeezed vacuum states, one can profoundly improve the classical limit. Interestingly, we observe that this improvement can also be achieved by using Gaussian measurements. Hence, we propose a fully Gaussian protocol for enhanced thermometry, which can simply be realized and used in quantum optics platforms.
We study low-temperature non-equilibrium quantum thermometry with a bosonic probe: a quantum harmonic oscillator strongly coupled to a bosonic bath at temperature TTT through a Drude--Ohmic spectral density. We treat the probe--bath dynamics both exactly, using the quadratic solution of Boyanovsky and Jasnow, and within a renormalized Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) master equation. From the time-dependent covariance matrix we extract the quantum Fisher information (QFI) for general single-mode Gaussian probe states, including squeezed ones. In the strong-coupling, non-Markovian regime the QFI is non-monotonic in time, displaying bath-memory revivals that make a finite interrogation time t∗>0t^*>0t∗>0 strictly optimal. By contrast, we prove that the Markovian QFI rises monotonically to its stationary value and develops no interior optimum, so that its optimum is always pinned to the boundary t∗→∞t^*\to\inftyt∗→∞; this complements existing Markovian precision-rate bounds, which concern (F(t)/t)(\mathcal F(t)/t)(F(t)/t) rather than the single-shot QFI (F(t))(\mathcal F(t))(F(t)). Squeezed initial states yield a large transient advantage that thermalisation eventually erases, establishing squeezing and interrogation time as complementary thermometric resources. At equilibrium, strong coupling replaces the exponential Boltzmann suppression of the low-temperature relative error by a far milder polynomial divergence. As the model maps directly onto circuit quantum electrodynamics, these protocols appear within current experimental reach.
Abstract We address the dephasing dynamics of the quantum Fisher information for the process of quantum thermometry with probes coupled to squeezed thermal baths via the nondemolition interaction. We also calculate the upper bound for the parameter estimation and investigate how the optimal estimation is affected by the initial conditions and decoherence, particularly the squeezing parameters. Moreover, the feasibility of the optimal measurement of the temperature is discussed in detail. Then, the results are generalized for entangled probes and the multi-qubit scenarios for probing the temperature are analysed. Our results show that the squeezing can decrease the number of channel uses for optimal thermometry. Comparing different schemes for multi-qubit estimation, we find that an increase in the number of the qubits, interacting with the channel, does not necessarily vary the precision of estimating the temperature. Besides, we discuss the enhancement of the quantum thermometry using the parallel strategy and starting from the W state.
Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry often requires measuring energy, which demands full accessibility over the entire system as well as a complex entangled measurement basis. In this paper, we take a different approach and show that single-qubit sequential measurements in the computational basis not only allow for precise thermometry of a many-body system, but may also achieve precision beyond the thermometry capacity of the probe at equilibrium, given by the Cram\'er-Rao bound. Thus, using consecutive single-qubit measurements of the probe out of equilibrium is, in most cases, very beneficial, as it achieves lower-temperature uncertainties and avoids demanding energy measurements when compared with probes at thermal equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the nonequilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. To demonstrate the generality of our findings, we consider thermometry in both spin chains and the Jaynes-Cummings model.
The fundamental metrological limits of temperature sensing in open quantum systems remain largely unresolved, particularly regarding the role of non-Gaussian quantum resources. In this letter, we establish analytic bounds on the quantum Fisher information (QFI) for temperature estimation using non-Gaussian states undergoing dissipative bosonic evolution. By focusing on the short-time regime governed by a time-local master equation, we derive precise scaling laws that elucidate when and how non-Gaussian probes decisively outperform Gaussian states under identical energy constraints. Our analysis uncovers a distinct linear-in-time QFI enhancement unique to Fock states, in contrast to the inherently weaker, quadratic scaling of Gaussian probes. These theoretical insights are substantiated through exact numerical simulations and mapped onto experimentally accessible platforms such as circuit QED. Our results not only clarify the quantum thermometric advantage of non-Gaussianity but also chart a realistic pathway toward harnessing it in noisy quantum technologies.