DYNAMICS AND CHARACTERIZATION OF QUANTUM SYSTEMS INTERACTING WITH CLASSICAL NOISE

The goal of research in quantum information is to investigate how quantum systems can be used to store, transmit and elaborate information and how the non-classical nature of their correlations allows defining protocols that outperform their classical counterparts. Despite of the many progresses, both theoretical and experimental, made in this field in the latest decades, many challenges lie ahead for practical implementations of quantum technologies. One of the most important ones is caused by the unavoidable interaction of quantum systems with their surroundings: The coupling to the environment is generally detrimental to the quantum information contained in the system as the system undergoes decoherence. In the quest for quantum technologies, it is fundamental to overcome the problem of decoherence and loss of information. Different physical implementations of qubits, such as superconducting and solid-state devices, are affected by the interaction with the environment in a way that can be described in terms of classical stochastic noise. The classical noise model can also be used to give an approximate, sometimes equivalent, description of full quantum models of system-environment interaction. This thesis contains my personal contribution to the study of the dynamics of discrete-variable quantum systems affected by classical noise. It covers in particular single- and two-qubit systems affected by Gaussian and non-Gaussian noise. It also discusses the dynamics of a quantum walk affected by spatially correlated classical noise. Analytical solutions for particular forms of noise and interactions, and a general numerical method for simulation of the dynamics are presented. Moreover, the thesis presents the experimental implementation of a quantum optical simulator of noisy dynamics of single-qubit systems. Finally, the use of quantum systems as probes of the spectral properties of large classical environments is discussed, showing that entanglement is a resource for improvements in the precision of the estimation

Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Bayesian algorithms for mobile terminal positioning in outdoor wireless environments

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Statistical modelling of algorithms for signal processing in systems based on environment perception

One cornerstone for realising automated driving systems is an appropriate handling of uncertainties in the environment perception and situation interpretation. Uncertainties arise due to noisy sensor measurements or the unknown future evolution of a traffic situation. This work contributes to the understanding of these uncertainties by modelling and propagating them with parametric probability distributions

Approximate Gaussian Conjugacy: Parametric Recursive Filtering Under Nonlinearity, Multimodal, Uncertainty, and Constraint, and Beyond

This is a post-peer-review, pre-copyedit version of an article published in Frontiers of Information Technology & Electronic Engineering. The final authenticated version is available online at: https://doi.org/10.1631/FITEE.1700379Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Unraveling the Thousand Word Picture: An Introduction to Super-Resolution Data Analysis

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light’s diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we’ve termed the interpretation problem

Kwantowa metrologia z atomami i światłem

The primary objective of this dissertation is to propose methods of generating nonclassical states of matter or light and examine the possibility of using such states in precise measurements of physical quantities. The first part of this objective is realised by using quantum-mechanical formalism with an emphasis on the theory of ultra-cold atomic gases and cavity quantum electrodynamics, and the second part is realised with methods of the theory of estimation with the Fisher information playing the pivotal role. The fusion of these methods is generally known as quantum metrology. In recent years, a lot of theoretical and experimental effort was put in the field of quantum metrology since it not only promises to develop measurement techniques that give better precision than the same measurements performed in a classical framework but also can be used to study the most fundamental aspects of quantum theory, like quantum entanglement. The first method which we consider is based on the mechanism of creating spinsqueezed states known as the one-axis twisting, which can be realised, for instance, in a Bose-Einstein condensate trapped in a double-well potential forming effectively a two-mode system. We show that the spin-squeezed states are just a small family of entangled states that can be generated by one-axis twisting Hamiltonian. This vast family of twisted states includes even the highest entangled state known as the Schrödinger’s cat. We also show how to exploit this quantum resource in a measurement of an unknown parameter with imperfect atomic detectors and when the strength of the interaction between the atoms is not precisely known. The second scheme for creating non-classical states is based on the quantum non-demolition measurement. This method involves an atom passing through an optical cavity which entangles with the photons inside the cavity and a subsequent measurement on the atom that collapses the combined matter-light state to a nonclassical state of light. To take into account photon losses in the cavity, we harness the master equation in Lindblad form. We show how such non-classical states can be extracted from the cavity and used later in a Mach-Zehnder interferometer. Based on the Wigner function, we also explain what features of this kind of states give rise to a high sensitivity of an interferometer. Finally, we show how a system that exhibits chaotic properties can be studied from the metrological perspective with the help of quantum Fisher information. Classical chaotic systems are systems that are highly sensitive to initial conditions. However, quantum systems can never exhibit this type of dynamics since the Schrödinger’s equation is linear. Therefore, one often says about quantum signatures of chaos. First, we show a textbook example of classical chaos, which is a double-rod pendulum, and, subsequently, we show how quantum Fisher information can serve to investigate characteristic time-scales of chaotic systems and the transition from integrable to chaotic dynamics. This could open a new possibility to study the relationship between the classical and quantum chaos.Głównym celem tej dysertacji jest zaproponowanie metod tworzenia kwantowych stanów materii oraz ´swiatła i sprawdzenie mozliwo´sci wykorzystania tych stanów ˙ do precyzyjnych pomiarów wielko´sci fizycznych. Pierwsza cz ˛e´s´c tego celu realizowana jest przy pomocy formalizmu kwantowo-mechanicznego w kontek´scie teorii ultra-zimnych gazów atomowych oraz kwantowej elektrodynamiki we wn ˛ece, natomiast druga cz ˛e´s´c realizowana jest za pomoc ˛a metod teorii estymacji z informacj ˛a Fishera w roli głównej. Poł ˛aczenie powyzszych metod jest znane ogólnie pod poj ˛eciem ˙ kwantowej metrologii. W ostatnich latach wiele teoretycznego i eksperymentalnego wysiłku zostało włozonego w dziedzin ˛e kwantowej metrologii, poniewa ˙ z dzi ˛eki niej ˙ mozliwy b ˛edzie nie tylko rozwój technik pomiarowych daj ˛acych lepsz ˛a precyzj ˛e ni ˙ z˙ te same pomiary wykonane w ramach klasycznej teorii, ale takze mo ˙ ze by´c u ˙ zyta do ˙ badania fundamentalnych aspektów mechaniki kwantowej takich jak spl ˛atanie. Pierwsz ˛a metod ˛a, któr ˛a rozwazamy to mechanizm tworzenia tworzenia stanów ˙ spinowo-´sci´sni ˛etych znany jako one-axis twisting, który moze by´c zastosowany na ˙ przykład w kondensacie Bosego-Einsteina uwi ˛ezionego w podwójnej studni potencjału tworz ˛ac efektywnie kondensat dwu składnikowy. Pokazujemy, ze stany spinowo- ˙ ´sci´sni ˛ete stanowi ˛a tylko mał ˛a rodzin ˛e stanów spl ˛atanych, które mog ˛a by´c wytworzone przez Hamiltonian one-axis twisting. Ta duza rodzina stanów typu ˙ twisted zawiera nawet najbardziej spl ˛atany stan znany jako kot Schroödingera. Pokazujemy równiez jak wykorzysta´c te kwantowe zasoby w pomiarze nieznanego parametru, ˙ wykorzystuj ˛ac nieidealne detektory atomowe oraz w przypadku kiedy oddziaływanie pomi ˛edzy atomami nie jest dokładnie znane. Drugi schemat tworzenia kwantowo-skorelowanych stanów jest oparty na quantum non-demolition measurement. W metodzie tej atom przelatuj ˛acy przez wn ˛ek ˛e optyczn ˛a zostaje spl ˛atany z obecnymi w niej fotonami, a w wyniku pomiaru wykonanego na atomie nast ˛epuje kolaps funkcji falowej ł ˛acznego stanu materii i ´swiatła do nieklasycznego stanu ´swiatła. W celu uwzgl ˛ednienia strat fotonów we wn ˛ece uzywamy równania ˙ master w formie Lindblada. Pokazujemy jak takie nieklasyczne stany mog ˛a zosta´c wydobyte z wn ˛eki oraz uzyte pó ´zniej w interferometrze Macha- ˙ Zehndera. Bazuj ˛ac na funkcji Wignera wyja´sniamy równiez jakie cechy tego rodzaju ˙ stanów przyczyniaj ˛a si ˛e do niezwykle wysokiej czuło´sci interferometru. Na koniec pokazujemy, jak układ wykazuj ˛acy wła´sciwo´sci chaotyczne moze zo- ˙ sta´c badany z perspektywy metrologicznej za pomoc ˛a kwantowej informacji Fishera. Klasyczne układy chaotyczne to układy, które s ˛a bardzo czułe na warunki pocz ˛atkowe. Jednakze, kwantowe uk ˙ łady nie mog ˛a wykazywa´c takiego rodzaju dynamiki, poniewaz równanie Schrödingera jest liniowe w funkcji falowej. Mo ˙ zna jed- ˙ nak mówi´c o tak zwanych kwantowych sygnaturach chaosu. Na pocz ˛atku pokazujemy podr˛ecznikowy przykład klasycznego chaosu, jakim jest podwójne wahadło, a nast ˛epnie pokazujemy jak kwantowa informacja Fishera moze pos ˙ łuzy´c do badania ˙ charakterystycznych skal czasowych układów chaotycznych i przej´scia pomi ˛edzy porz ˛adkiem a chaosem. Takie podej´scie otwiera nowe mozliwo´sci badania zwi ˛azku ˙ pomi ˛edzy kwantowym chaosem a porz ˛adkiem