Optimal quantum estimation of loss in bosonic channels

We address the estimation of the loss parameter of a bosonic channel probed by Gaussian signals. We derive the ultimate quantum bound on precision and show that no improvement may be obtained by having access to the environment degrees of freedom. We found that, for small losses, the variance of the optimal estimator is proportional to the loss parameter itself, a result that represents a qualitative improvement over the shot noise limit. An observable based on the symmetric logarithmic derivative is derived, which attains the ultimate bound and may be implemented using Gaussian operations and photon counting.Comment: 4 pages, 2 figures, replaced with published versio

Spectral unmixing of Multispectral Lidar signals

In this paper, we present a Bayesian approach for spectral unmixing of multispectral Lidar (MSL) data associated with surface reflection from targeted surfaces composed of several known materials. The problem addressed is the estimation of the positions and area distribution of each material. In the Bayesian framework, appropriate prior distributions are assigned to the unknown model parameters and a Markov chain Monte Carlo method is used to sample the resulting posterior distribution. The performance of the proposed algorithm is evaluated using synthetic MSL signals, for which single and multi-layered models are derived. To evaluate the expected estimation performance associated with MSL signal analysis, a Cramer-Rao lower bound associated with model considered is also derived, and compared with the experimental data. Both the theoretical lower bound and the experimental analysis will be of primary assistance in future instrument design

Limits on Parameter Estimation of Quantum Channels

The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel\u27s Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when estimating multiple parameters simultaneously. For the single parameter case, we establish Cramer--Rao bounds for general quantum channels using both the SLD and RLD Fisher information. The task of estimating multiple parameters simultaneously is more involved than the single parameter case. In the multiparameter case, Cramer--Rao bounds take the form of matrix inequalities. We provide a method to obtain scalar Cramer--Rao bounds from the corresponding matrix inequalities. We then establish a scalar Cramer--Rao bound using the RLD Fisher information. Our bounds apply universally and we also show how they are efficiently computable by casting them as optimization problems. In the single parameter case, we recover the so-called Heisenberg scaling\u27\u27 using our SLD-based bound. On the other hand, we provide a no-go condition for Heisenberg scaling using our RLD-based bound for both the single and multiparameter settings. Finally, we apply our bounds to the example of estimating the parameters of a generalized amplitude damping channel

Optimal experiment design revisited: fair, precise and minimal tomography

Given an experimental set-up and a fixed number of measurements, how should one take data in order to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [arXiv:quant-ph/0411093v1]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of 'minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. We find that these two designs are generally similar. Monte-Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.Comment: 16 pages, 7 figure

Squeezed vacuum as a universal quantum probe

We address local quantum estimation of bilinear Hamiltonians probed by Gaussian states. We evaluate the relevant quantum Fisher information (QFI) and derive the ultimate bound on precision. Upon maximizing the QFI we found that single- and two-mode squeezed vacuum represent an optimal and universal class of probe states, achieving the so-called Heisenberg limit to precision in terms of the overall energy of the probe. We explicitly obtain the optimal observable based on the symmetric logarithmic derivative and also found that homodyne detection assisted by Bayesian analysis may achieve estimation of squeezing with near-optimal sensitivity in any working regime. Besides, by comparison of our results with those coming from global optimization of the measurement we found that Gaussian states are effective resources, which allow to achieve the ultimate bound on precision imposed by quantum mechanics using measurement schemes feasible with current technology.Comment: revised version, 2 figure

COMPARISON ON DIFFERENT DISCRETE FRACTIONAL FOURIER TRANSFORM (DFRFT) APPROACHES

As an extension of conventional Fourier transform and a time-frequency signal analysis tool, the fractional Fourier transforms (FRFT) are suitable for dealing with various types of non-stationary signals. Taking advantage of the properties and non-stationary features of linear chirp signals in the Fourier transform domain, several methods of extraction and parameter estimation for chirp signals are proposed, and a comparative study has been done on chirp signal estimation. Computation of the discrete fractional Fourier transform (DFRFT) and its chirp concentration properties are dependent on the basis of DFT eigenvectors used in the computation. Several DFT-eigenvector bases have been proposed for the transform, and there is no common framework for comparing them. In this thesis, we compare several different approaches from a conceptual viewpoint and point out the differences between them. We discuss five different approaches, namely: (1) the bilinear transformation method, (2) the Grunbaum method, (3) the Dickenson-Steiglitz method, also known as the S-matrix method, (4) the quantum mechanics in finite dimension( QMFD) method, and (5) the higher order S-matrix method, to find centered DFT (CDFT) commuting matrices and the various properties of these commuting matrices. We study the nature of eigenvalues and eigenvectors of these commuting matrices to determine whether they resemble those of corresponding continuous Gauss-Hermite operator. We also measure the performance of these five approaches in terms of mailobe-to-sidelobe ratio, 10-dB bandwidth, quality factor, linearity of eigenvalues, parameter estimation error, and, finally peak-to-parameter mapping regions. We compare the five approaches using these several parameters and point out the best approach for chirp signal applications

Time-frequency represetation of radar signals using Doppler-Lag block searching Wigner-Ville distribution

Radar signals are time-varying signals where the signal parameters change over time. For these signals, Quadratic Time-Frequency Distribution (QTFD) offers advantages over classical spectrum estimation in terms of frequency and time resolution but it suffers heavily from cross-terms. In generating accurate Time-Frequency Representation (TFR), a kernel function must be able to suppress cross-terms while maintaining auto-terms energy especially in a non-cooperative environment where the parameters of the actual signal are unknown. Thus, a new signal-dependent QTFD is proposed that adaptively estimates the kernel parameters for a wide class of radar signals. The adaptive procedure, Doppler-Lag Block Searching (DLBS) kernel estimation was developed to serve this purpose. Accurate TFRs produced for all simulated radar signals with Instantaneous Frequency (IF) estimation performance are verified using Monte Carlo simulation meeting the requirements of the Cramer-Rao Lower Bound (CRLB) at SNR > 6 dB