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arXiv:2312.04176

Temperature-Enhanced Critical Quantum Metrology

Laurin Ostermann, Karol Gietka

Published Dec 7, 2023• 0 citations
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Abstract

We show that the performance of critical quantum metrology protocols, counter-intuitively, can be enhanced by finite temperature. We consider a toy-model squeezing Hamiltonian, the Lipkin-Meshkov-Glick model and the paradigmatic Ising model. We show that the temperature enhancement of the quantum Fisher information can be achieved by adiabatic preparation of the critical state and by preparing it directly in the proximity of the critical point. We also find a relatively simple, however, non-optimal measurement capable of harnessing finite temperature to increase the parameter estimation sensitivity. Therefore, we argue that temperature can be considered as a resource in critical quantum metrology.

DOI
10.48550/arxiv.2312.04176

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We show that the performance of critical quantum metrology protocols, counter-intuitively, can be enhanced by finite temperature. We consider a toy-model squeezing Hamiltonian, the Lipkin-Meshkov-Glick model and the paradigmatic Ising model. We show that the temperature enhancement of the quantum Fisher information can be achieved by adiabatic preparation of the critical state and by preparing it directly in the proximity of the critical point. We also find a relatively simple, however, non-optimal measurement capable of harnessing finite temperature to increase the parameter estimation sensitivity. Therefore, we argue that temperature can be considered as a resource in critical quantum metrology.
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Quantum criticality is a resource for quantum-enhanced metrology, but existing schemes face intrinsic limitations. These arise because using criticality directly in the encoding dynamics restricts the accessible parameters to those explicitly supported by the critical Hamiltonian, and the requirement for critical conditions narrows the effective estimation range. To solve this, we introduce a general framework termed criticality-assisted noncommutative preparation (CANP). In this approach, critical evolution is employed as a state-preparation resource. We establish the underlying algebraic conditions and show that the intrinsic noncommutativity between the preparation and encoding operations leads to a genuine enhancement of the quantum Fisher information (QFI). Remarkably, this enhancement may be achieved at fixed total sensing time and energy cost. The effect is quantified by the Wigner-Yanase skew information, which measures noncommutativity and exhibits the same critical scaling as the QFI. We demonstrate effective use of CANP in the quantum Rabi and Lipkin-Meshkov-Glick models. Our results establish CANP as a robust technique to effectively implement criticality-enhanced quantum metrology.
Critical metrology relies on the precise preparation of a system in its ground state near a quantum phase transition point where quantum correlations get very strong. Typically, this increases the quantum Fisher information with respect to changes in system parameters and thus improves the optimally possible measurement precision limited by the Cramér-Rao bound. Hence critical metrology involves encoding information about the unknown parameter in changes of the system's ground state. Conversely, in conventional metrology methods like Ramsey interferometry, the eigenstates of the system remain unchanged, and information about the unknown parameter is encoded in the relative phases that excited system states accumulate during their time evolution. Here we introduce an approach combining these two methodologies into a unified protocol applicable to closed and driven-dissipative systems. We show that the quantum Fisher information in this case exhibits an additional interference term originating from the interplay between eigenstate and relative phase changes. We provide analytical expressions for the quantum and classical Fisher information in such a setup, elucidating as well a straightforward measurement approach that nearly attains the maximum precision permissible under the Cramér-Rao bound. We showcase these results by focusing on the squeezing Hamiltonian, which characterizes the thermodynamic limit of Dicke and Lipkin-Meshkov-Glick Hamiltonians.
Quantum metrology fundamentally relies upon the efficient management of quantum uncertainties. We show that under equilibrium conditions the management of quantum noise becomes extremely flexible around the quantum critical point of a quantum many-body system: this is due to the critical divergence of quantum fluctuations of the order parameter, which, via Heisenberg's inequalities, may lead to the critical suppression of the fluctuations in conjugate observables. Taking the quantum Ising model as the paradigmatic incarnation of quantum phase transitions, we show that it exhibits quantum critical squeezing of one spin component, providing a scaling for the precision of interferometric parameter estimation which, in dimensions d>2, lies in between the standard quantum limit and the Heisenberg limit. Quantum critical squeezing saturates the maximum metrological gain allowed by the quantum Fisher information in d=∞ (or with infinite-range interactions) at all temperatures, and it approaches closely the bound in a broad range of temperatures in d=2 and 3. This demonstrates the immediate metrological potential of equilibrium many-body states close to quantum criticality, which are accessible, e.g., to atomic quantum simulators via elementary adiabatic protocols.
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.
Critical quantum metrology exploits the dramatic growth of the quantum Fisher information near quantum phase transitions to enhance the precision of parameter estimation. This enhancement is commonly associated with a closing energy gap, which causes the characteristic timescales for adiabatic preparation or relaxation to diverge with increasing system size. As a consequence, the apparent growth of the quantum Fisher information largely reflects the increasing evolution time induced by critical slowing down rather than a genuine improvement in metrological performance, thereby limiting the practical usefulness of such protocols. Here we show that the relationship between energy gaps, quantum correlations, and achievable precision in interacting quantum systems can be far more subtle. In particular, quantum-enhanced sensitivity can also emerge when the energy gap increases, eliminating critical slowing down and enabling substantially faster relaxation dynamics. Although the corresponding quantum Fisher information may decrease due to the shorter evolution time, the resulting precision can nevertheless remain quantum-enhanced. Building on this insight, we introduce an anti-critical quantum metrology scheme in which quantum-enhanced precision arises while the energy gap grows. We illustrate this mechanism using the quantum Rabi model, thereby identifying a route to metrological advantage that avoids the slow dynamics associated with conventional criticality.