Pedro W. Lamberti

Metric character of the quantum Jensen-Shannon divergence

In a recent paper, the generalization of the Jensen-Shannon divergence in the context of quantum theory has been studied [Majtey et al., Phys. Rev. A 72, 052310 (2005)]. This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of a numerical procedure for mixed states, that its square root verifies the triangle inequality.

Non-logarithmic Jensen–Shannon divergence

The Jensen–Shannon divergence is a symmetrized and smoothed version of the Kullback–Leibler divergence. Recently it has been widely applied to the analysis and characterization of symbolic sequences. In this paper we investigate a generalization of the Jensen–Shannon divergence. This generalization is done in the framework of the non-extensive Tsallis statistics. We study its basic properties and we investigate its applicability as a tool for segmentating symbolic sequences.

Wootters’ distance revisited: a new distinguishability criterium

Abstract.The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics’ Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states.

Global aspects of light cone cuts

The global behavior of light cone cuts at null infinity, i.e., the intersection of light cones of interior point, with null infinity, is investigated. In particular, it is shown that the topological structure of these cuts, for the case of an asymptotically flat space‐time, is very simple. The nature of the light cone cuts singularities is also analyzed. Generically, only two different types of singularities occur and their local description is given.

Permutation Entropy Applied to the Characterization of the Clinical Evolution of Epileptic Patients under PharmacologicalTreatment

Different techniques originated in information theory and tools from nonlinear systems theory have been applied to the analysis of electro-physiological time series. Several clinically relevant results have emerged from the use of concepts, such as entropy, chaos and complexity, in analyzing electrocardiograms and electroencephalographic (EEG) records. In this work, we develop a method based on permutation entropy (PE) to characterize EEG records from different stages in the treatment of a chronic epileptic patient. Our results show that the PE is useful for clearly quantifying the evolution of the patient along a certain lapse of time and allows visualizing in a very convenient way the effects of the pharmacotherapy.

Measure of similarity between images based on the codispersion coefficient

We propose to use the codispersion coefficient to define a measure of similarity between images. This coefficient has been widely used in spatial statistics to quantify the association between two spatial processes, and here we explore its capabilities in an image processing context is mathematically simple to compute and possesses good statistical properties. The new measure takes into account the spatial association in a specific direction h between a degraded image and the original unmodified image. Three applications are developed to illustrate the capabilities of our proposal. The defined measure captures the spatial association produced by fitting AR-2D processes with different window sizes. It is able to distinguish the levels of similarity between two images for specific directions in two-dimensional space. Finally, it detects stochastic resonance when an image is transmitted by a nonlinear device.

NATURAL METRIC FOR QUANTUM INFORMATION THEORY

We study in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation to other distances is investigated. As an illustrative application, the proposed metric is evaluated for one-qubit mixed states.

Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-state sequences

In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or arises from embedding into a sequence of permutation vectors, where the components are the positions of the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this series of ‘symbols’, as part of a discrete finite-size alphabet. On the one hand, the permutation entropy of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these methods. The potential from such a combined approach – of a permutation procedure and a complexity analysis – is evaluated through the illustration of some simulated data and some real data. In both cases, we compare the individual approaches and the combined approach.

A simple numerical method for solving problems in electrostatics

A very simple numerical method for solving boundary problems in electrostatics is presented. The approximation scheme is based on the substitution of a series expansion of the electrostatic potential by a finite sum and a discretization of the boundary where the conditions are given. These boundary conditions may be of the mixed type. In this way the original problem is reduced to the solution of a system of linear equations. As an application of the method proposed, the problem of a hollow cap of a sphere set to a constant potential is treated.

Detecting dynamical changes in time series by using the Jensen Shannon divergence

Most of the time series in nature are a mixture of signals with deterministic and random dynamics. Thus the distinction between these two characteristics becomes important. Distinguishing between chaotic and aleatory signals is difficult because they have a common wide band power spectrum, a delta like autocorrelation function, and share other features as well. In general, signals are presented as continuous records and require to be discretized for being analyzed. In this work, we introduce different schemes for discretizing and for detecting dynamical changes in time series. One of the main motivations is to detect transitions between the chaotic and random regime. The tools here used here originate from the Information Theory. The schemes proposed are applied to simulated and real life signals, showing in all cases a high proficiency for detecting changes in the dynamics of the associated time series.