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arXiv:2111.05285 · Quantum Science and Technology

Cost-effective estimation of single-mode thermal states by probabilistic quantum metrology

Massimo Frigerio, Stefano Olivares, Matteo G. A. Paris

Published May 19, 2022• 3 citations
arXivHTMLPDF

Abstract

Abstract In probabilistic quantum metrology, one aims at finding measurements that concentrate the Fisher information on the resulting conditional quantum states, i.e. those states post-selected according to the outcome of the measurement. The quantum Cramér–Rao (QCR) bound itself cannot be surpassed this way, but it could be possible to improve the information-cost ratio (where cost refers to the experimental implementation), or even the total Fisher information. In this paper, we propose a post-selection protocol achieving this goal using single-photon subtraction and thermal states of radiation. In particular, we achieve a larger information-cost ratio for the estimation of temperature than the strategy attaining the QCR bound. Our scheme involves only classical states of radiation and this contrasts with (but does not contradict) recent results proving that for unitary quantum statistical models, post-selection strategies can outperform direct protocols only exploiting nonclassical states.

DOI
10.1088/2058-9565/ac6dfe

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Abstract In probabilistic quantum metrology, one aims at finding measurements that concentrate the Fisher information on the resulting conditional quantum states, i.e. those states post-selected according to the outcome of the measurement. The quantum Cramér–Rao (QCR) bound itself cannot be surpassed this way, but it could be possible to improve the information-cost ratio (where cost refers to the experimental implementation), or even the total Fisher information. In this paper, we propose a post-selection protocol achieving this goal using single-photon subtraction and thermal states of radiation. In particular, we achieve a larger information-cost ratio for the estimation of temperature than the strategy attaining the QCR bound. Our scheme involves only classical states of radiation and this contrasts with (but does not contradict) recent results proving that for unitary quantum statistical models, post-selection strategies can outperform direct protocols only exploiting nonclassical states.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
Quantum-enhanced (i.e., higher performance by quantum effects than any classical methods) mean value estimation of observables is a fundamental task in various quantum technologies; in particular, it is an essential subroutine in quantum computing algorithms. Notably, the quantum estimation theory identifies the ultimate precision of such an estimator, which is referred to as the quantum Cramér-Rao (QCR) lower bound or equivalently the inverse of the quantum Fisher information. Because the estimation precision directly determines the performance of those quantum technological systems, it is highly demanded to develop a generic and practically implementable estimation method that achieves the QCR bound. Under imperfect conditions, however, such an ultimate and implementable estimator for quantum mean values has not been developed. In this paper, we propose a quantum-enhanced mean value estimation method in a depolarizing noisy environment that asymptotically achieves the QCR bound in the limit of a large number of qubits. To approach the QCR bound in a practical setting, the method adaptively optimizes the amplitude amplification and a specific measurement that can be implemented without any knowledge of state preparation. We provide a rigorous analysis for the statistical properties of the proposed adaptive estimator such as consistency and asymptotic normality. Furthermore, several numerical simulations are provided to demonstrate the effectiveness of the method, particularly showing that the estimator needs only a modest number of measurements to almost saturate the QCR bound.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
Abstract In quantum metrology, it is widely believed that the quantum Cramér–Rao bound is an attainable bound. However, this is not always true. In order to clarify this point, we explain why the quantum Cramér–Rao bound cannot be attained. Here, we investigate noiseless channel estimation under energy constraint for states, using a physically reasonable error function, and present a state achieving the optimal rate as well as the attainable bound. We propose an experimental generation of the optimal states for enhanced metrology using squeezing transformations. This makes phase/unitary estimation physically implementable.
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.