Physical Review A

Linear combination estimator of multiple-outcome detections with discrete measurement outcomes

Likun Zhou, Jia-Min Xu, Weiyang Zhang, Jiong Cheng, Tai-Shuang Yin, Youbin Yu, Rui‐Pin Chen, Aixi Chen, Guangri Jin, Wen Yang

Published Apr 2, 2021• 8 citations

Publisher

Abstract

Finding the optimal estimator capable of saturating the Cram\'er-Rao bound is important for quantum parameter estimation. The maximum likelihood estimator (MLE) is well known to be asymptotically optimal, but it is usually nonanalytic and hence requires extensive numerical computation. On the other hand, the inversion estimator is widely used in experiments due to its simplicity, but it is usually suboptimal. Here we derive an estimator based on the linear combination of multiple inversion estimators associated with the occurrence probability of different measurement outcomes. This linear combination estimator shares the asymptotic optimality of the MLE and the simplicity of the inversion estimator and is applicable to general multi-outcome measurements, irrespective of the specific system and the noise in the system. We demonstrate this estimator for the intensity-difference measurement to a six-photon twin-Fock input state.

DOI: 10.1103/physreva.103.043702

Related papers

Multi-outcome homodyne detection in a coherent-state light interferometer

J. Z. Wang, Z. Q. Yang, A. X. Chen +2 more

The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.

arXiv:1904.00557Optics ExpressMar 29, 20198 citations

Quantum phase estimation with a general binary-outcome measurement

H.L. Zheng, Xuexin Ren, P. Liu +1 more

As the simplest phase estimation protocol, the inversion estimator has been widely adopted in the quantum metrology, since its performance can be simply quantified by the error-propagation formula. For a general binary-outcome measurement, we show that both the inversion estimator and the maximum likelihood estimator are the same and can asymptotically saturate the Cramér-Rao lower bound. These observations are applied to the parity detection and the zero-nonzero photon counting that have achieved super-resolved phase measurement near the shot-noise phase sensitivity in a coherent-state light Mach–Zehnder interferometer. We numerically show that the phase uncertainty almost follows the theoretical prediction of the lower bound. At the optimal working point, the best sensitivities also reach their associated analytic results for the two binary-outcome measurement schemes.

Results in PhysicsOct 18, 20223 citations

Optimal Experiment Design for Quantum State and Process Tomography and Hamiltonian Parameter Estimation

Robert L. Kosut, Ian A. Walmsley, HERSCHEL A. RABITZ

A number of problems in quantum state and system identification are addressed. Specifically, it is shown that the maximum likelihood estimation (MLE) approach, already known to apply to quantum state tomography, is also applicable to quantum process tomography (estimating the Kraus operator sum representation (OSR)), Hamiltonian parameter estimation, and the related problems of state and process (OSR) distribution estimation. Except for Hamiltonian parameter estimation, the other MLE problems are formally of the same type of convex optimization problem and therefore can be solved very efficiently to within any desired accuracy. Associated with each of these estimation problems, and the focus of the paper, is an optimal experiment design (OED) problem invoked by the Cramer-Rao Inequality: find the number of experiments to be performed in a particular system configuration to maximize estimation accuracy; a configuration being any number of combinations of sample times, hardware settings, prepared initial states, etc. We show that in all of the estimation problems, including Hamiltonian parameter estimation, the optimal experiment design can be obtained by solving a convex optimization problem. Software to solve the MLE and OED convex optimization problems is available upon request from the first author.

arXiv:quant-ph/0411093Nov 12, 200430 citations

Quantum non demolition measurements: Parameter estimation for mixtures of multinomials

Tristan Benoist, Fabrice Gamboa, Clément Pellegrini

In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cramér Rao bound.

arXiv:1707.06074Electronic Journal of StatisticsJan 1, 20186 citations

Optimal phase estimation with photon-number difference measurement using twin-Fock states of light*

Jia-Min Xu, J. Z. Wang, A. X. Chen +2 more

Quantum phase measurement with multiphoton twin-Fock states has been shown to be optimal for detecting equal numbers of photons at the output ports of a Mach–Zehnder interferometer (i.e., the so-called single-fringe detection), since the phase sensitivity can saturate the quantum Cramér–Rao lower bound at certain values of phase shift. Here we report a further step to achieve a global phase estimation at the Heisenberg limit by detecting the particle-number difference (i.e., the J ^ z measurement). We show the role of experimental imperfections on the ultimate estimation precision with the six-photon twin-Fock state of light. Our results show that both the precision and the sensing region of the J ^ z measurement are better than those of the single-fringe detection, due to combined contributions of the measurement outcomes. We numerically simulate the phase estimation protocol using an asymptotically unbiased maximum likelihood estimator.

Chinese Physics BOct 17, 20198 citations

Tutorial: Maximum likelihood estimation in the context of an optical measurement

Anthony Vella

The method of maximum likelihood estimation (MLE) is a widely used statistical approach for estimating the values of one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, I briefly review the mathematical foundations of MLE, then reformulate the problem for the measurement of a spatially-varying optical intensity distribution. In this context, the detection of each individual photon is treated as a random event, the outcome being the photon's location. A typical measurement consists of many detected photons, which accumulate to form a spatial intensity profile. Here, I show a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement of multiple photons incident on a detector comprised of a discrete array of pixels. An estimate for the parameter(s) of interest may then be obtained by maximizing the likelihood function, while the FIM determines the uncertainty of the estimate. To illustrate these concepts, several simple examples are presented for the one- and two-parameter cases, revealing many interesting properties of the MLE formalism, as well as some practical considerations for optical experiments. Throughout these examples, connections are also drawn to optical applications of quantum weak measurements, including off-null ellipsometry and scatterometry.

arXiv:1806.04503Jun 8, 20184 citations