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arXiv:1902.02599 · Physical Review Letters

Probing Bayesian Credible Regions Intrinsically: A Feasible Error Certification for Physical Systems

Changhun Oh, Yong Siah Teo, Hyunseok Jeong

Published Jul 26, 2019• 6 citations
arXivHTMLPDF

Abstract

Standard computation of size and credibility of a Bayesian credible region for certifying any point estimator of an unknown parameter (such as a quantum state, channel, phase, etc.) requires selecting points that are in the region from a finite parameter-space sample, which is infeasible for a large dataset or dimension as the region would then be extremely small. We solve this problem by introducing the in-region sampling theory to compute both region qualities just by sampling appropriate functions over the region itself using any Monte Carlo sampling method. We take in-region sampling to the next level by understanding the credible-region capacity (an alternative description for the region content to size) as the average l_{p}-norm distance (p>0) between a random region point and the estimator, and present analytical formulas for p=2 to estimate both the capacity and credibility for any dimension and a sufficiently large dataset without Monte Carlo sampling, thereby providing a quick alternative to Bayesian certification. All results are discussed in the context of quantum-state tomography.

DOI
10.1103/physrevlett.123.040602

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Standard computation of size and credibility of a Bayesian credible region for certifying any point estimator of an unknown parameter (such as a quantum state, channel, phase, etc.) requires selecting points that are in the region from a finite parameter-space sample, which is infeasible for a large dataset or dimension as the region would then be extremely small. We solve this problem by introducing the in-region sampling theory to compute both region qualities just by sampling appropriate functions over the region itself using any Monte Carlo sampling method. We take in-region sampling to the next level by understanding the credible-region capacity (an alternative description for the region content to size) as the average l_{p}-norm distance (p>0) between a random region point and the estimator, and present analytical formulas for p=2 to estimate both the capacity and credibility for any dimension and a sufficiently large dataset without Monte Carlo sampling, thereby providing a quick alternative to Bayesian certification. All results are discussed in the context of quantum-state tomography.
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Bayesian error analysis paves the way to the construction of credible and plausible error regions for a point estimator obtained from a given dataset. We introduce the concept of region accuracy for error regions (a generalization of the point-estimator mean squared error) to quantify the average statistical accuracy of all region points with respect to the unknown true parameter. We show that the increase in region accuracy is closely related to the Bayesian region dual operations in Shang et al (2013 New J. Phys. 15 123026). Next with only the given dataset as viable evidence, we establish various adaptive methods to maximize the region accuracy relative to the true parameter subject to the type of reported Bayesian region for a given point estimator. We highlight the performance of these adaptive methods by comparing them with non-adaptive procedures in three quantum-parameter estimation examples. The results of and mechanisms behind the adaptive schemes can be understood as the region analog of adaptive approaches to achieving the quantum Cramér–Rao bound for point estimators.
Precise measurement of physical quantities plays a crucial role in the development of science and technology.The main purpose of the dissertation is two-fold: to investigate the ultimate precision for estimation of physical quantities using Gaussian states and to propose an efficient method for certification of Bayesian error region in general quantum parameter estimation.In the first part, we begin with analyzing sensitivity for estimating a phase difference in an optical interferometer.Optical interferometry is widely used in science and industry for measuring small displacements.Recently, a large-scale optical interferometer socalled the Laser interferometer Gravitational-Wave Observatory (LIGO) has succeeded in detecting a gravitational wave, the signal of which is extremely small.On the other hand, it has been shown that a non-classical feature of quantum states can improve the sensitivity of estimation, such as in optical interferometer, including the LIGO.From a practical point of view, we inspect the practically achievable precision using non-classical Gaussian states in Mach-Zehnder interferometer with feasible measurements and realistic photon loss.We then investigate the precision of single-mode phase estimation using Gaussian measurement, which can be realized by using homodyne detection, and i show that non-Gaussian measurement is necessary to utilize the power of Gaussian input probes optimally.Finally, we find the optimal measurement for general Gaussian quantum metrology and identify three distinct optimal measurements corresponding to different circumstances.In the second part, we study the Bayesian error region, which is a crucial concept for a general estimation process.When estimating a physical quantity, one has to supply the error interval (single-parameter) or error region (multi-parameter) as well as the estimate.However, it has been shown that as the dimension of quantum systems of interest grows, it becomes intractable to calculate the size and credibility of Bayesian error regions.As an alternative, we derive an analytical expression for the properties, the size and credibility, of Bayesian error regions, in an asymptotic regime.We then propose an efficient numerical method to calculate them for high-dimensional quantum systems even in a non-asymptotic regime.
An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region—the region of largest likelihood among all regions of the same size—as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region—the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies.
High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the standard strategies of rejection sampling, importance sampling, and Markov-chain sampling can be adapted to this context, where the samples must obey the constraints imposed by the positivity of the statistical operator. For illustration, we generate sample points in the probability space of qubits, qutrits, and qubit pairs, both for tomographically complete and incomplete measurements. We use these samples for various purposes: establish the marginal distribution of the purity; compute the fractional volume of separable two-qubit states; and calculate the size of regions with bounded likelihood.