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arXiv:2002.00553 · Physical Review Research

Dissipative adiabatic measurements: Beating the quantum Cramér-Rao bound

Da-Jian Zhang, Jiangbin Gong

Published Jun 30, 2020• 22 citations
arXivHTMLPDF

Abstract

It is challenged only recently that the precision attainable in any measurement of a physical parameter is fundamentally limited by the quantum Cramér-Rao Bound (QCRB).Here, targeting at measuring parameters in strongly dissipative systems, we propose an innovative measurement scheme called dissipative adiabatic measurement and theoretically show that it can beat the QCRB.Unlike projective measurements, our measurement scheme, though consuming more time, does not collapse the measured state and, more importantly, yields the expectation value of an observable as its measurement outcome, which is directly connected to the parameter of interest.Such a direct connection allows us to extract the value of the parameter from the measurement outcomes in a straightforward manner, with no fundamental limitation on precision, in principle.Our findings not only provide a marked insight into quantum metrology but also are highly useful in dissipative quantum information processing.

DOI
10.1103/physrevresearch.2.023418

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It is challenged only recently that the precision attainable in any measurement of a physical parameter is fundamentally limited by the quantum Cramér-Rao Bound (QCRB).Here, targeting at measuring parameters in strongly dissipative systems, we propose an innovative measurement scheme called dissipative adiabatic measurement and theoretically show that it can beat the QCRB.Unlike projective measurements, our measurement scheme, though consuming more time, does not collapse the measured state and, more importantly, yields the expectation value of an observable as its measurement outcome, which is directly connected to the parameter of interest.Such a direct connection allows us to extract the value of the parameter from the measurement outcomes in a straightforward manner, with no fundamental limitation on precision, in principle.Our findings not only provide a marked insight into quantum metrology but also are highly useful in dissipative quantum information processing.
Conventional strategies of quantum metrology are built upon POVMs, thereby possessing several general features, including the demolition of the state to be measured, the need of performing a number of measurements, and the degradation of performance under decoherence and dissipation. Here, we propose an innovative measurement scheme, called dissipative adiabatic measurements (DAMs), based on which, we further develop an approach to estimation of parameters characterizing dissipative processes. Unlike a POVM, whose outcome is one of the eigenvalues of an observable, a DAM yields the expectation value of the observable as its outcome, without collapsing the state to be measured. By virtue of the very nature of DAMs, our approach is capable of solving the estimation problem in a state-protective fashion with only MMM measurements, where MMM is the number of parameters to be estimated. More importantly, contrary to the common wisdom, it embraces decoherence and dissipation as beneficial effects and offers a Heisenberg-like scaling of precision, thus outperforming conventional strategies. Our DAM-based approach is direct, efficient, and expected to be immensely useful in the context of dissipative quantum information processing.
Multiparameter quantum metrology is ubiquitous in physics, yet it remains fundamentally challenging. As reflected by the uncertainty principle, distinct unknown parameters generally cannot be estimated simultaneously at their individual ultimate precision limits. The quantum Cramér--Rao bound (QCRB) and the Holevo Cramér--Rao bound (HCRB) represent the ultimate precision bounds in multiparameter quantum metrology. Here we introduce new formulations of these bounds based on quantum state purification. We show that, for any mixed state, their values are directly related to those of its purification, provided that nuisance parameters are introduced on the environmental system. Leveraging this connection, we develop a remarkably simple method for asymptotically attaining either the HCRB or twice the QCRB for arbitrary mixed states, using random purification channels and individual measurements.
The power of quantum sensing rests on its ultimate precision limit, quantified by the quantum Cramer-Rao bound (QCRB), which can surpass classical bounds. In multi-parameter estimation, the QCRB is not always saturated as the quantum nature of associated observables may lead to their incompatibility. Here we explore the precision limits of multi-parameter estimation through the lens of quantum geometry, enabling us to experimentally evaluate the QCRB via quantum geometry measurements. Focusing on two- and three-parameter estimation, we elucidate how fundamental quantum uncertainty principles prevent the saturation of the bound. By linking a metric of "quantumness" to the system geometric properties, we investigate and experimentally extract the attainable QCRB for three-parameter estimations.
The quantum Cramér-Rao bound (QCRB) stands as a cornerstone of quantum metrology. Yet, akin to its classical counterpart, it provides only local information and overlooks higher-order details. We leverage the theory of higher-order asymptotics to circumvent these issues, providing corrections to the performance of estimators beyond the QCRB. While the QCRB often yields a whole family of optimal states and several optimal measurements, our approach allows us to identify optimal states and measurements within the family that are otherwise equivalent according to the QCRB alone; these are requisite for optimal metrology before reaching the asymptotic limit. These results are especially pertinent when dealing with unitary processes.
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.
Estimation of parameters is a pivotal task throughout science and technology. The quantum Cramér-Rao bound provides a fundamental limit of precision allowed to be achieved under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a Cramér-Rao bound for a fairly large class of open system dynamics, which is governed by a (time-dependent) dynamical semigroup map. We illustrate the utility of this scenario through three examples.