Quantum estimation of unknown parameters

Martínez-Vargas, Esteban; Pineda, Carlos; Leyvraz, F.; Barberis-Blostein, Pablo

doi:10.1103/physreva.95.012136

Cited by 22 publications

(18 citation statements)

References 15 publications

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“…And while this does not guarantee that our solution will be optimal for a few observations (an adaptive scheme may be better than repeating the same measurement in that case), we will see that having an error that is a function of the number of repetitions where the first point is already tight, and that also tends towards the asymptotically optimal solution as the number of shots grows, is enough to draw conclusions to important questions such as the role of photon number correlations or the performance of experimentally feasible measurements in the regime of limited data. For instance, we have found an example where the correlations between the paths of the Mach-Zehnder interferometer appear to be particularly useful in this regime, and we have demonstrated that while measuring quadratures and counting photons after the action of a beam splitter are asymptotically equivalent in an ideal scenario, the former measurement scheme is better for a low number of repeated experiments.It is interesting to note that a related approach was recently discussed in [35], where the authors presented a modification of the quantum Van Trees inequality and used it to construct an adaptive strategy based on an optimal parameter-independent single-shot measurement scheme. Therefore, our work and [35] are complementary, since we will mainly focus on repeated measurements to connect the optimal single-shot and asymptotic regimes and to 2 If the probability model belongs to the exponential family and the estimator is unbiased, then it is possible to saturate the Cramér-Rao bound exactly even for a single shot [4,23,24].…”

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confidence: 99%

Quantum metrology in the presence of limited data

Rubio

Dunningham

2019

New J. Phys.

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Quantum metrology protocols are typically designed around the assumption that we have an abundance of measurement data, but recent practical applications are increasingly driving interest in cases with very limited data. In this regime the best approach involves an interesting interplay between the amount of data and the prior information. Here we propose a new way of optimising these schemes based on the practically-motivated assumption that we have a sequence of identical and independent measurements. For a given probe state we take our measurement to be the best one for a single shot and we use this sequentially to study the performance of different practical states in a Mach-Zehnder interferometer when we have moderate prior knowledge of the underlying parameter. We find that we recover the quantum Cramér-Rao bound asymptotically, but for low data counts we find a completely different structure. Despite the fact that intra-mode correlations are known to be the key to increasing the asymptotic precision, we find evidence that these could be detrimental in the low data regime and that entanglement between the paths of the interferometer may play a more important role. Finally, we analyse how close realistic measurements can get to the bound and find that measuring quadratures can improve upon counting photons, though both strategies converge asymptotically. These results may prove to be important in the development of quantum enhanced metrology applications where practical considerations mean that we are limited to a small number of trials.the experiment enough times and that we have certain prior knowledge about the unknown parameter 2 [4,5,18,19] (see footnote 2), and this simplifies the optimisation of the error considerably. Furthermore, the Fisher information has a certain fundamental character. In particular, it can be seen as a distinguishability metric [20] that arises in the expansion of the Bures distance between two infinitesimally close states [3,18]. Moreover, its reciprocal gives us the asymptotic limit for the Bayesian mean square error as a function of the number of repetitions under some fairly general assumptions 3 [5] (see footnote 3), and this is also the case for other approaches that are more conservative than the Cramér-Rao bound too [21,22].Nevertheless, the fact that this technique normally requires many repetitions to be useful is an important drawback to study realistic physical systems such as those previously mentioned. This problem has already been acknowledged in the literature (e.g. in [4, 5, 18, 27]), and several solutions have been proposed. A conceptually simple and straightforward approach consists in using a general measure of uncertainty and estimating how many measurements are needed such that the results predicted by the asymptotic theory are valid, which can always be done numerically [5,28]. In addition, we can rely on numerical techniques such as Monte Carlo simulations [29] or machine learning [30] to perform the optimisation directly, or can simply examine the behavi...

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confidence: 99%

Quantum metrology in the presence of limited data

Rubio

Dunningham

2019

New J. Phys.

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“…We assume a coherent probe state |α⟩, and, without loss of generality, that the displacement is positive and real, i.e., α = |α| > 0. This probe state is acted upon by the rotation operator R(θ) given by (5). The action of the rotation operator R(θ) on the probe state |α⟩ gives the encoded state e −iθ α .…”

Section: Phase Estimationmentioning

confidence: 99%

“…Bayesian approach they are treated as random variables with an associated prior distribution function. The application of the Bayesian estimation scheme for various quantum parameter estimation problems has previously been studied in [5], [8], [17], [18]. The Bayesian estimation approach utilizes knowledge of the prior distribution of the unknown parameter θ and the likelihood of the observed data to find the posterior distribution of θ.…”

Section: Introductionmentioning

confidence: 99%

Machine-Learning-Based Parameter Estimation of Gaussian Quantum States

Kundu¹,

McKay²,

Mallik³

2022

IEEE Trans. Quantum Eng.

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We propose a machine learning framework for parameter estimation of single mode Gaussian quantum states. Under a Bayesian framework, our approach estimates parameters of suitable prior distributions from measured data. For phase-space displacement and squeezing parameter estimation, this is achieved by introducing Expectation-Maximization (EM) based algorithms, while for phase parameter estimation an empirical Bayes method is applied. The estimated prior distribution parameters along with the observed data are used for finding the optimal Bayesian estimate of the unknown displacement, squeezing and phase parameters. Our simulation results show that the proposed algorithms have estimation performance that is very close to that of 'Genie Aided' Bayesian estimators, that assume perfect knowledge of the prior parameters. In practical scenarios, when numerical values of the prior distribution parameters are not known beforehand, our proposed methods can be used to find optimal Bayesian estimates from the observed measurement data.

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“…Optimality of a quantum sensing protocol is defined via a cost function C which must be identified in context of a specific measurement task. In our study of variational N -atom Ramsey interferometry, we wish to optimize for phase estimation accuracy, defined as the mean square error (φ) relative to the actual phase φ averaged with respect to a certain prior distribution P δφ (φ) over a finite dynamic range δφ of the interferometer [20,22,[63][64][65][66][67]. We contrast this to approaches optimizing accuracy for a specific value of the phase, corresponding to δφ → 0, as is done in a Fisher information approach [68,69], and underlies the discussion of Ramsey interferometry with squeezed spin states (SSS) [70,71] or GHZ states [72].…”

Section: Introductionmentioning

confidence: 99%

Quantum Variational Optimization of Ramsey Interferometry and Atomic Clocks

Kaubruegger,

Vasilyev,

Schulte

et al. 2021

Preprint

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Quantum estimation of unknown parameters

Cited by 22 publications

References 15 publications

Quantum metrology in the presence of limited data

Quantum metrology in the presence of limited data

Machine-Learning-Based Parameter Estimation of Gaussian Quantum States

Quantum Variational Optimization of Ramsey Interferometry and Atomic Clocks

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