Decompositions of Hilbert Spaces, Stability Analysis and Convergence Probabilities for Discrete-Time Quantum Dynamical Semigroups

We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces.Comment: 14 pages, no figures, to appear in Journal of Physics A, 201

Nonlinear resonance and excitability in interconnected systems

Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications. The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods. The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity

The geometry of rest–spike bistability

Funder: Qualcomm; doi: http://dx.doi.org/10.13039/100005144Abstract: Morris–Lecar model is arguably the simplest dynamical model that retains both the slow–fast geometry of excitable phase portraits and the physiological interpretation of a conductance-based model. We augment this model with one slow inward current to capture the additional property of bistability between a resting state and a spiking limit cycle for a range of input current. The resulting dynamical system is a core structure for many dynamical phenomena such as slow spiking and bursting. We show how the proposed model combines physiological interpretation and mathematical tractability and we discuss the benefits of the proposed approach with respect to alternative models in the literature

The geometry of rest–spike bistability

Funder: Qualcomm; doi: http://dx.doi.org/10.13039/100005144Abstract: Morris–Lecar model is arguably the simplest dynamical model that retains both the slow–fast geometry of excitable phase portraits and the physiological interpretation of a conductance-based model. We augment this model with one slow inward current to capture the additional property of bistability between a resting state and a spiking limit cycle for a range of input current. The resulting dynamical system is a core structure for many dynamical phenomena such as slow spiking and bursting. We show how the proposed model combines physiological interpretation and mathematical tractability and we discuss the benefits of the proposed approach with respect to alternative models in the literature

Convergence Analysis for Discrete-Time Quantum Semigroup

In this work the asymptotic behavior of completely-positive trace-precerving maps is analyzed. First, the probabilities of converging to invariant subspaces, in the limit of infinite iteration, are studied. Next, two different decompositions of the quantum system's Hilbert space are introduced, both aimed to analyze the convergence behavior and speed. Finally the possibilities that the dynamics converges to a subspace, after a finite amount of time, is investigate

Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups

We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces

Home Automation Oriented Gesture Classification From Inertial Measurements

In this paper, a machine learning (ML) approach is presented that exploits accelerometers data to deal with gesture recognition (GR) problems. The proposed methodology aims at providing high accuracy classification for home automation systems, which are generally user independent, device independent, and device orientation independent, an heterogeneous scenario that has not been fully investigated in previous GR literature. The approach illustrated in this paper is composed of three main steps: event identification; feature extraction; and ML-based classification. The elements of the novelty of the proposed approach are 1) a preprocessing phase based on principal component analysis to increase the performance in real-world scenario conditions and 2) the development of parsimonious novel classification techniques based on sparse Bayesian learning. This methodology is tested on two datasets of four gesture classes (horizontal, vertical, circles, and eight-shaped movements) and on a further dataset with eight classes. In order to authentically describe a real-world home automation environment, the gesture movements are collected from more than 30 people who freely perform any gesture. It results in a dictionary of 12 and 20 different movements, respectively, in the case of the four-class and the eight-class databases

A machine learning based approach for gesture recognition from inertial measurements

The interaction based on gestures has become a prominent approach to interact with electronic devices. In this paper a Machine Learning (ML) based approach to gesture recognition (GR) is illustrated; the proposed tool is freestanding from user, device and device orientation. The tool has been tested on a heterogeneous dataset representative of a typical application of gesture recognition. In the present work two novel ML algorithms based on Sparse Bayesian Learning are tested versus other classification approaches already employed in literature (Support Vector Machine, Relevance Vector Machine, k-Nearest Neighbor, Discriminant Analysis). A second element of novelty is represented by a Principal Component Analysisbased approach, called Pre-PCA, that is shown to enhance gesture recognition with heterogeneous working conditions. Feature extraction techniques are also investigated: a Principal Component Analysis based approach is compared to Frame-Based Description methods