Abstract Quantum systems can be used as probes in the context of metrology for enhanced parameter estimation. In particular, the delicacy of critical systems to perturbations can make them ideal sensors. Arguably the simplest realistic probe system is a spin- 1 2 impurity, which can be manipulated and measured in-situ when embedded in a fermionic environment. Although entanglement between a single impurity probe and its environment produces nontrivial many-body effects, criticality cannot be leveraged for sensing. Here we introduce instead the two-impurity Kondo model as a novel paradigm for critical quantum metrology, and examine the multiparameter estimation scenario at finite temperature. We explore the full metrological phase diagram numerically and obtain exact analytic results near criticality. Enhanced sensitivity to the inter-impurity coupling driving a second-order phase transition is evidenced by diverging quantum Fisher information (QFI) and quantum signal-to-noise ratio (QSNR). However, with uncertainty in both coupling strength and temperature, the multiparameter QFI matrix becomes singular—even though the parameters to be estimated are independent—resulting in vanishing QSNRs. We demonstrate that by applying a known control field, the singularity can be removed and measurement sensitivity restored. For general systems, we show that the degradation in the QSNR due to uncertainties in another parameter is controlled by the degree of correlation between the unknown parameters.
Abstract Quantum systems can be used as probes in the context of metrology for enhanced parameter estimation. In particular, the delicacy of critical systems to perturbations can make them ideal sensors. Arguably the simplest realistic probe system is a spin- 1 2 impurity, which can be manipulated and measured in-situ when embedded in a fermionic environment. Although entanglement between a single impurity probe and its environment produces nontrivial many-body effects, criticality cannot be leveraged for sensing. Here we introduce instead the two-impurity Kondo model as a novel paradigm for critical quantum metrology, and examine the multiparameter estimation scenario at finite temperature. We explore the full metrological phase diagram numerically and obtain exact analytic results near criticality. Enhanced sensitivity to the inter-impurity coupling driving a second-order phase transition is evidenced by diverging quantum Fisher information (QFI) and quantum signal-to-noise ratio (QSNR). However, with uncertainty in both coupling strength and temperature, the multiparameter QFI matrix becomes singular—even though the parameters to be estimated are independent—resulting in vanishing QSNRs. We demonstrate that by applying a known control field, the singularity can be removed and measurement sensitivity restored. For general systems, we show that the degradation in the QSNR due to uncertainties in another parameter is controlled by the degree of correlation between the unknown parameters.
Fundamental scientific advances are often facilitated by breakthroughs in calibration and readout techniques that increase the measurement precision of sensors. The betterment of sensing technologies enables the validation of existing scientific theories and congruently, may also pave the way for identifying breakdowns in these frameworks, signalling the necessity for new theoretical developments. A significant facet of the growing interest in quantum metrology is that it represents a prime example in which quantum features such as entanglement, squeezing, and criticality, can be harnessed to enhance the precision of parameter estimation well beyond the capacity of their classical counterparts. We begin by exploring the paradigmatic problem of temperature estimation using impurity probes embedded in a host environment. The physical setup considered provides a natural mechanism for the probe-environment unit to equilibrate and thermalize. Furthermore the nature of probe-environment interactions allows us to investigate the role that entanglement plays in equilibrium temperature estimation problems. Our analysis reveal that spectral features of the environment are imprinted on the probe system due to established entanglement and uncover a low temperature regime with a universal thermometric response that is independent of the microscopic details of the environment. Physical incarnations of nanoelectronic devices are operated in the linear-response regime with transport properties representing the measurement of choice. Motivated by this, we next propose nanoelectronic devices operated in this way as a novel paradigm for quantum sensing. We derive general expressions for the quantum Fisher information --- allowing us to provide a precision benchmark for quantum transport coefficients in the linear-response regime. The latter half of this thesis is concerned with aspects of quantum critical metrology and multiparameter estimation. Our first contribution is to propose an alternative platform for critical quantum metrology, the two-impurity Kondo model --- which does not rely on the preparation of exotic critical states, but instead can be passively tuned to criticality. Employing local measurements of the in situ probe at finite temperature, we find enhanced precision at criticality in estimating inter-impurity coupling, but low precision when utilized as a thermometer. Considering uncertainty in both of these parameters, we find the quantum Fisher information matrix to be singular, despite these parameters being physically independent. We demonstrate that application of a known control field can lift the singularity. In practical settings, an external parameter is tuned to bring a system to its quantum critical point. Therefore, we expect that experimental uncertainties in this control parameter to play a significant role in fully exploiting criticality as a resource for sensing. Motivated by this, we next propose a general framework interpolating between the single and multi-parameter settings. Applied to the Ising and Lipkin-Meshkov-Glick models, we establish a fundamental trade-off between the amount of uncertainty a critical system can withstand while still maintaining a quantum advantage for sensing. In the final chapter of the thesis, we investigate the physical origin of quantum fisher information matrix singularities. A singularity signals a fundamental breakdown in our ability to estimate parameters and bound their precision. We attribute these situations to emergent metrological symmetries whereby the same set of measurement outcomes are obtained for different combinations of the system parameters. We demonstrate that by using Bayesian estimation, deep insights can be revealed and estimation of effective parameters is possible by employing a parameter transformation of the Bayesian posterior distribution.
Critical quantum metrology exploits the hypersensitivity of quantum systems near phase transitions to achieve enhanced precision in parameter estimation. While single-parameter estimation near critical points is well established, the simultaneous estimation of multiple parameters, which is essential for practical sensing applications, remains challenging. This difficulty arises from sloppiness, a phenomenon that typically renders the quantum Fisher information matrix (QFIM) singular or nearly singular. In this work, we demonstrate that multiparameter critical metrology is not only feasible but can also retain divergent precision scaling, provided one accepts a trade-off in the scaling exponent. Using the ground state of the single-cavity Dicke model (DM), we show that two Hamiltonian parameters can be simultaneously estimated with a scalar variance bound scaling as the square root of the critical parameter. This overcomes the inherent sloppiness by leveraging higher-order contributions to the QFIM. To recover the optimal quadratic scaling, we introduce the Dicke dimer (DD) with photon hopping. In this extended model, a triple point in the phase diagram enables the simultaneous closure of two excitation gaps, which effectively increases the rank of the QFIM and restores the ideal single-parameter scaling for specific parameter pairs. Furthermore, we extend our analysis to dissipative settings subject to photon loss. Finally, we establish a connection between the derived critical scalings and the fundamental state preparation time, providing a unified framework to operationally compare different sensing strategies. Our results demonstrate that critical quantum metrology can be made robust against dissipation and scalable to multiparameter scenarios, paving the way for practical quantum sensors operating near phase transitions.
Critical quantum metrology relies on the extreme sensitivity of a system's eigenstates near the critical point of a quantum phase transition to Hamiltonian perturbations. This means that these eigenstates are extremely sensitive to all the parameters of the Hamiltonian. In realistic settings, there is always some degree of uncertainty in the control parameters used to tune the system to criticality. These uncertainties, while not the target of estimation, can significantly affect the attainable precision, effectively acting as nuisance parameters in the estimation process. Despite being a practically relevant source of noise, their impact on critical metrology has been largely overlooked. In this work we present a general framework that interpolates between single- and multiparameter estimation settings, enabling a systematic analysis of how such uncertainties influence sensitivity. We apply this framework to the paradigmatic transverse field Ising and Lipkin-Meshkov-Glick models, explicitly demonstrating how uncertainty in control parameters affects the metrological performance of critical sensors. For finite-size systems, we identify a fundamental trade-off between robustness to uncertainty and the ability to retain a quantum advantage at the critical point. Our results contribute to a deeper understanding of the practical limitations of critical quantum metrology and provide a route toward its more resilient implementation.
We study quantum-impurity models as a platform for quantum thermometry. A single quantum spin-21 impurity is coupled to an explicit, structured, fermionic thermal sample system, which we refer to as the environment or bath. We critically assess the thermometric capabilities of the impurity as a probe, when its coupling to the environment is of Ising or Kondo exchange type. In the Ising case, we find sensitivity equivalent to that of an idealized two-level system, with peak thermometric performance obtained at a temperature that scales linearly in the applied control field, independent of the coupling strength and environment spectral features. By contrast, a richer thermometric response can be realized for Kondo impurities, since strong probe-environment entanglement can then develop. At low temperatures, we uncover a regime with a universal thermometric response that is independent of microscopic details, controlled only by the low-energy spectral features of the environment. The many-body entanglement that develops in this regime means that low-temperature thermometry with a weakly applied control field is inherently less sensitive, while optimal sensitivity is recovered by suppressing the entanglement with stronger fields.
Quantum metrology aims to enhance measurement precision beyond the classical limit by leveraging quantum resources. Unlike multi-parameter dynamic quantum metrology, many questions regarding multiparameter quantum metrology at thermal equilibrium remain elusive. In particular, the ultimate precision limits achievable in this equilibrium setting are not yet well understood. In this work, we examine the fundamental limits of estimating multiple parameters with a quantum probe at thermal equilibrium. We first show that the Heisenberg limit with respect to the number of probes can be achieved, and our bound coincides with the known single-parameter bound when only one parameter is estimated. We then consider the low temperature limit, revealing a qualitatively different behavior compared to the finite temperature case. We give an example to illustrate the usage of our main results. Finally, we show the conditions under which the sensitivity bound can be attained and the optimal measurements to achieve it.
Temperature estimation of interacting quantum many-body systems is both a challenging task and topic of interest in quantum metrology, given that critical behavior at phase transitions can boost the metrological sensitivity. Here we study noninvasive quantum thermometry of a finite, two-dimensional Ising spin lattice based on measuring the dephasing dynamics of a spin probe coupled to the lattice. We demonstrate a strong critical enhancement of the achievable precision in terms of the quantum Fisher information, which depends on the coupling range and the interrogation time. Our numerical simulations are compared to instructive analytic results for the critical scaling of the sensitivity in the Curie-Weiss model of a fully connected lattice and to the mean-field description in the thermodynamic limit, both of which fail to describe the critical spin fluctuations on the lattice the spin probe is sensitive to. Phase metrology could thus help to investigate the critical behavior of finite many-body systems beyond the validity of mean-field models. Published by the American Physical Society 2024