Abstract The heat capacity C of a given probe is a fundamental quantity that determines, among other properties, the maximum precision in temperature estimation. In turn, C is limited by a quadratic scaling with the number of constituents of the probe, which provides a fundamental limit in quantum thermometry. Achieving this fundamental bound with realistic probes, i.e. experimentally amenable, remains an open problem. In this work, we tackle the problem of engineering optimal thermometers by using networks of spins. Restricting ourselves to two-body interactions, we derive general properties of the optimal configurations and exploit machine-learning techniques to find the optimal couplings. This leads to simple architectures, which we show analytically to approximate the theoretical maximal value of C and maintain the optimal scaling for short- and long-range interactions. Our models can be encoded in currently available quantum annealers, and find application in other tasks requiring Hamiltonian engineering, ranging from quantum heat engines to adiabatic Grover’s search.
Abstract The heat capacity C of a given probe is a fundamental quantity that determines, among other properties, the maximum precision in temperature estimation. In turn, C is limited by a quadratic scaling with the number of constituents of the probe, which provides a fundamental limit in quantum thermometry. Achieving this fundamental bound with realistic probes, i.e. experimentally amenable, remains an open problem. In this work, we tackle the problem of engineering optimal thermometers by using networks of spins. Restricting ourselves to two-body interactions, we derive general properties of the optimal configurations and exploit machine-learning techniques to find the optimal couplings. This leads to simple architectures, which we show analytically to approximate the theoretical maximal value of C and maintain the optimal scaling for short- and long-range interactions. Our models can be encoded in currently available quantum annealers, and find application in other tasks requiring Hamiltonian engineering, ranging from quantum heat engines to adiabatic Grover’s search.
The precision of a quantum thermometer is fundamentally constrained by the spectral structure of the probe itself, and a systematic mapping between the configurations of energy levels and thermometric performance provides relevant information to design optimized devices. In this work, we establish such a mapping by analyzing a broad class of quantum systems, ranging from finite spin ensembles and degenerate atoms to confining potentials, quantum walks, and continuous-spectrum models. We derive exact scaling laws for the quantum Fisher information, revealing two distinct high-temperature universality classes: finite-spectrum probes exhibit a T−4 decay, while unbounded or continuous spectra yield a slower T−2 decay. At low temperatures, we show that sensitivity, though universally exponentially suppressed, can be enhanced arbitrarily by engineering degenerate excited states or a quantum walk on a fully connected topology. By contrast, specific quantum walk topologies provide a distinct enhancement mechanism based on gap engineering, whereby an optimal network size yields an optimized T−2 low-temperature scaling. Furthermore, power-law spectra enable tunable scaling of thermometric performance with system size, offering a design principle for optimal probes in specific temperature windows. Our results contribute to transform spectral information into a resource for quantum thermometry, providing both fundamental bounds and practical guidelines to tailored temperature sensing.
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 investigate the fundamental and practical precision limits of thermometry in bosonic and fermionic environments by coupling an N -level probe to them and continuously monitoring it. Our findings show that the ultimate precision limit, quantified by the Fisher information, scales linearly with N , offering an exponential improvement over equilibrium thermometry, where the scaling is only log 2 N . For a fixed Hamiltonian structure, we develop a maximum-likelihood estimation strategy that maps the observed continuously monitored trajectories of the probe into temperature estimates with minimal error. By optimizing over all possible Hamiltonian structures, we discover that the optimal configuration is an effective two-level system, with both levels exhibiting degeneracy that increases with N —a stark contrast to equilibrium thermometry, where the ground state remains nondegenerate. Our results have practical implications. First, continuous monitoring is experimentally feasible on several platforms and accounts for the preparation time of the probe, which is often overlooked in other approaches such as prepare and reset. Second, the linear scaling is robust against deviations from the effective two-level structure of the optimal Hamiltonian. Additionally, this robustness extends to cases of initial ignorance about the temperature. Thus, in global estimation problems, the linear scaling remains intact even without adaptive strategies.
Equilibrium quantum thermometry seeks to enhance temperature estimation for reservoirs at thermal equilibrium using quantum resources, yet achieving the Heisenberg scaling in a physical protocol remains a significant challenge. Moreover, quantum measurements face an intrinsic limitation arising from an exponential suppression of thermometric sensitivity as the temperature approaches zero. This limitation severely hinders access to high precision in the low-temperature regime. Here, we demonstrate a simple but powerful quantum engineering protocol to overcome these limitations and reach high precision for temperature estimation. By engineering a small subset of central probes and their linear coupling to a larger array of adjacent probes, we show that the composite partition function can be tailored to yield Heisenberg-limited precision for temperature estimation. Critically, we show that this protocol can simultaneously saturate the Landau bound at low temperatures, an inverse quadratic scaling with the temperature, thereby fully exploiting the potential of equilibrium thermometry and circumventing the exponential suppression of thermometric sensitivity in the low-temperature regime. This scheme relies solely on linear interactions, avoiding complicated many-body couplings and ensuring scalability with the number of probes. Moreover, we find that the comparable precision can be achieved using only local measurements on the central probes, substantially reducing the implementation complexity compared to global measurement schemes and offering a promising approach toward high-precision equilibrium thermometry in physical systems.
Abstract Quantum thermodynamics has emerged as a separate sub-discipline, revising the concepts and laws of thermodynamics, at the quantum scale. In particular, there has been a disruptive shift in the way thermometry, and thermometers are perceived and designed. Currently, we face two major challenges in quantum thermometry. First, all of the existing optimally precise temperature probes are local, meaning their operation is optimal only for a narrow range of temperatures. Second, aforesaid optimal local probes mandate complex energy spectrum with immense degeneracy, rendering them impractical. Here, we address these challenges by formalizing the notion of global thermometry leading to the development of optimal temperature sensors over a wide range of temperatures. We observe the emergence of different phases for such optimal probes as the temperature interval is increased. In addition, we show how the best approximation of optimal global probes can be realized in spin chains, implementable in ion traps and quantum dots.
What is the minimum time required to take the temperature? In this paper, we solve this question for any process where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime (more precisely, the measurement uncertainty decays polynomially with the temperature as T→0, in contrast to the usual exponential decay with T−1). We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shit is highlighted, and (ii) a collective superradiant coupling between a N-qubit probe and a sample, which enables a quadratic decay with N2 of the measurement uncertainty.