Advances in quantum metrology

Abstract

In classical estimation theory, the central limit theorem implies that the statistical error in a measurement outcome can be reduced by an amount proportional to n −1/2 by repeating the measures n times and then averaging. Using quantum effects, such as entanglement, it is often possible to do better, decreasing the error by an amount proportional to n −1 . Quantum metrology is the study of those quantum techniques that allow one to gain advantages over purely classical approaches.In this review, we analyze some of the most promising recent developments in this research field. Specifically, we deal with the developments of the theory and point out some of the new experiments. Then we look at one of the main new trends of the field, the analysis of how the theory must take into account the presence of noise and experimental imperfections.Any measurement consists in three parts: the preparation of a probe, its interaction with the system to be measured, and the probe readout. This process is often plagued by statistical or systematic errors. The source of the former can be accidental (e.g. deriving from an insufficient control of the probes or of the measured system) or fundamental (e.g. deriving from the Heisenberg uncertainty relations). Whatever their origin, we can reduce their effect by repeating the measurement and averaging the resulting outcomes. This is a consequence of the central-limit theorem: given a large number n of independent measurement results (each having a standard deviation ∆σ), their average will converge to a Gaussian distribution with standard deviation ∆σ/ √ n, so that the error scales as n −1/2 . In quantum mechanics this behavior is referred to as "standard quantum limit" (SQL) and is associated with procedures which do not fully exploit the quantum nature of the system under investigation 1 . Notably it is possible to do better when one employs quantum effects, such as entanglement among the probing devices employed for the measurements, e.g. see Refs. [1][2][3][4][5][6]. Consequently, the SQL is not a fundamental quantum mechanical bound as it can be surpassed by using "non-classical" strategies. Nonetheless, through Heisenberg-like uncertainty relations, quantum mechanics still sets ultimate limits in precision which are typically referred to as "Heisenberg bounds". In Fig. 1 we present a simple example which may be useful to un-1 In quantum optics, the n −1/2 scaling is also indicated as "shot noise", since it is connected to the discrete nature of the radiation that can be heard as "shots" in a photon counter operating in Geiger mode.derstand the quantum enhancement. Part of the emerging field of quantum technology [7], quantum metrology studies these bounds and the (quantum) strategies which allows us to attain them. More generally it deals with measurement and discrimination procedures that receive some kind of enhancement (in precision, efficiency, simplicity of implementation, etc.) through the use of quantum effects. This paper aims to review some of the most recent developmen...

Frequentist and Bayesian Quantum Phase Estimation

“…Strategies involving probe states characterized by squeezed quadratures [13] or entanglement between particles [14][15][16][17][18][19] are able to overcome the shot noise, the ultimate quantum bound being the so-called Heisenberg limit. Quantum noise reduction in phase estimation has been demonstrated in several proof-of-principle experiments with atoms and photons [20,21].…”

Transmission of Photonic Quantum Polarization Entanglement in a Nanoscale Hybrid Plasmonic Waveguide

“…2,3 Quantum entanglement has powerful applications in information processing and communications, as well as in the enhanced precision of measurement. 4 Quantum metrology, 4 as an important application of quantum entanglement, is the measurement of physical parameters with enhanced resolution and sensitivity, enabled by taking advantage of quantum theory, particularly by exploiting quantum entanglement. For example, phase measurement with super-sensitivity beyond the standard quantum limit (SQL) can be realized by using N-particles entangled state (N ≥ 2).…”

Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks

“…Quantum Fisher information (QFI) is the central concept in quantum metrology [1,2,3,4,5,6,7,9,10,11,8,12]. It depicts the theoretical bound for the variance of an estimator [13,14] Var(θ) ≥ 1 F .…”