Quantum field tomography

We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of unknown quantum fields based on data of correlation functions. At the basis of the analysis is the concept of continuous matrix product states, a complete set of variational states grasping states in quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used in the context of compressed sensing such as Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation functions. In the absence of a phase reference, we highlight how specific higher order correlation functions can still be predicted. We exemplify the functioning of the approach by reconstructing randomised continuous matrix product states from their correlation data and study the robustness of the reconstruction for different noise models. We also apply the method to data generated by simulations based on continuous matrix product states and using the time-dependent variational principle. The presented approach is expected to open up a new window into experimentally studying continuous quantum systems, such as encountered in experiments with ultra-cold atoms on top of atom chips. By virtue of the analogy with the input-output formalism in quantum optics, it also allows for studying open quantum systems.Comment: 31 pages, 5 figures, minor change

A Compact Formulation for the ℓ2,1\ell_{2,1} Mixed-Norm Minimization Problem

Parameter estimation from multiple measurement vectors (MMVs) is a fundamental problem in many signal processing applications, e.g., spectral analysis and direction-of- arrival estimation. Recently, this problem has been address using prior information in form of a jointly sparse signal structure. A prominent approach for exploiting joint sparsity considers mixed-norm minimization in which, however, the problem size grows with the number of measurements and the desired resolution, respectively. In this work we derive an equivalent, compact reformulation of the $\ell_{2,1}$ mixed-norm minimization problem which provides new insights on the relation between different existing approaches for jointly sparse signal reconstruction. The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the jointly sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the $\ell_{2,1}$ mixed-norm minimization and the atomic-norm minimization. Additionally, for the case of irregular sampling or a large number of samples, we present a low complexity, grid-based implementation based on the coordinate descent method

Maximum Likelihood Estimation of Exponentials in Unknown Colored Noise for Target Identification in Synthetic Aperture Radar Images

This dissertation develops techniques for estimating exponential signals in unknown colored noise. The Maximum Likelihood (ML) estimators of the exponential parameters are developed. Techniques are developed for one and two dimensional exponentials, for both the deterministic and stochastic ML model. The techniques are applied to Synthetic Aperture Radar (SAR) data whose point scatterers are modeled as damped exponentials. These estimated scatterer locations (exponentials frequencies) are potential features for model-based target recognition. The estimators developed in this dissertation may be applied with any parametrically modeled noise having a zero mean and a consistent estimator of the noise covariance matrix. ML techniques are developed for a single instance of data in colored noise which is modeled in one dimension as (1) stationary noise, (2) autoregressive (AR) noise and (3) autoregressive moving-average (ARMA) noise and in two dimensions as (1) stationary noise, and (2) white noise driving an exponential filter. The classical ML approach is used to solve for parameters which can be decoupled from the estimation problem. The remaining nonlinear optimization to find the exponential frequencies is then solved by extending white noise ML techniques to colored noise. In the case of deterministic ML, the computationally efficient, one and two-dimensional Iterative Quadratic Maximum Likelihood (IQML) methods are extended to colored noise. In the case of stochastic ML, the one and two-dimensional Method of Direction Estimation (MODE) techniques are extended to colored noise. Simulations show that the techniques perform close to the Cramer-Rao bound when the model matches the observed noise

Maximum Likelihood Estimation of Exponentials in Unknown Colored Noise for Target in Identification Synthetic Aperture Radar Images

This dissertation develops techniques for estimating exponential signals in unknown colored noise. The Maximum Likelihood ML estimators of the exponential parameters are developed. Techniques are developed for one and two dimensional exponentials, for both the deterministic and stochastic ML model. The techniques are applied to Synthetic Aperture Radar SAR data whose point scatterers are modeled as damped exponentials. These estimated scatterer locations exponentials frequencies are potential features for model-based target recognition. The estimators developed in this dissertation may be applied with any parametrically modeled noise having a zero mean and a consistent estimator of the noise covariance matrix. ML techniques are developed for a single instance of data in colored noise which is modeled in one dimension as 1 stationary noise, 2 autoregressive AR noise and 3 autoregressive moving-average ARMA noise and in two dimensions as 1 stationary noise, and 2 white noise driving an exponential filter. The classical ML approach is used to solve for parameters which can be decoupled from the estimation problem. The remaining nonlinear optimization to find the exponential frequencies is then solved by extending white noise ML techniques to colored noise. In the case of deterministic ML, the computationally efficient, one and two-dimensional Iterative Quadratic Maximum Likelihood IQML methods are extended to colored noise. In the case of stochastic ML, the one and two-dimensional Method of Direction Estimation MODE techniques are extended to colored noise. Simulations show that the techniques perform close to the Cramer-Rao bound when the model matches the observed noise

Parallel Factor-Based Model for Two-Dimensional Direction Estimation

Two-dimensional (2D) Direction-of-Arrivals (DOA) estimation for elevation and azimuth angles assuming noncoherent, mixture of coherent and noncoherent, and coherent sources using extended three parallel uniform linear arrays (ULAs) is proposed. Most of the existing schemes have drawbacks in estimating 2D DOA for multiple narrowband incident sources as follows: use of large number of snapshots, estimation failure problem for elevation and azimuth angles in the range of typical mobile communication, and estimation of coherent sources. Moreover, the DOA estimation for multiple sources requires complex pair-matching methods. The algorithm proposed in this paper is based on first-order data matrix to overcome these problems. The main contributions of the proposed method are as follows: (1) it avoids estimation failure problem using a new antenna configuration and estimates elevation and azimuth angles for coherent sources; (2) it reduces the estimation complexity by constructing Toeplitz data matrices, which are based on a single or few snapshots; (3) it derives parallel factor (PARAFAC) model to avoid pair-matching problems between multiple sources. Simulation results demonstrate the effectiveness of the proposed algorithm

Blind Signal Separation for Digital Communication Data

to appear in EURASIP E-reference in Signal Processing, invited paper.International audienceBlind source separation, often called independent component analysis , is a main field of research in signal processing since the eightees. It consists in retrieving the components, up to certain indeterminacies, of a mixture involving statistically independent signals. Solid theoretical results are known; besides, they have given rise to performent algorithms. There are numerous applications of blind source separation. In this contribution, we particularize the separation of telecommunication sources. In this context, the sources stem from telecommunication devices transmitting at the same time in a given band of frequencies. The received data is a mixed version of all these sources. The aim of the receiver is to isolate (separate) the different contributions prior to estimating the unknown parameters associated with a transmitter. The context of telecommunication signals has the particularity that the sources are not stationary but cyclo-stationary. Now, in general, the standard methods of blind source separation assume the stationarity of the sources. In this contribution , we hence make a survey of the well-known methods and show how the results extend to cyclo-stationary sources

ESPRIT for multidimensional general grids

We present a new method for complex frequency estimation in several variables, extending the classical (1d) ESPRIT-algorithm. We also consider how to work with data sampled on non-standard domains (i.e going beyond multi-rectangles)