Roads to objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong quantum Darwinism – a review

The problem of objectivity, i.e. how to explain on quantum grounds the objective character of the macroscopic world, is one of the aspects of the celebrated quantum-to-classical transition. Initiated by W. H. Zurek and collaborators, this problem gained some attention recently with several approaches being developed. The aim of this work is to compare three of them: quantum Darwinism, Spectrum Broadcast Structures, and strong quantum Darwinism. The paper is concentrated on foundations, providing a synthetic analysis of how the three approaches realize the idea of objectivity and how they are related to each other. As a byproduct of this analysis, a proof of a generalized Spectrum Broadcast Structure theorem is presented. Recent quantum Darwinism experiments are also briefly discussed.

The pseudo-spectrum of operator pencils

In this paper, we propose a new definition of the pseudo-spectrum for operator pencils, which is associated with two bounded operators [Formula: see text] and [Formula: see text] defined in a Hilbert space. Unlike other definitions available in the literature, we prove, under specific conditions on [Formula: see text] and [Formula: see text], that the pseudo-spectrum for operator pencils is equal to an [Formula: see text]-neighborhood of the generalized spectrum. Moreover, we demonstrate how to use this concept to redefine the pseudo-spectrum of an unbounded operator. We illustrate its usefulness through a numerical example dealing with the Schrödinger operator.

A class of strongly stable approximation for unbounded operators

We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schr¨odinger’s operator where the discretization process based upon the Kantorovich’s projection.

NEW SUFFICIENT CONDITIONS IN THE GENERALIZED SPECTRUM APPROACH TO DEAL WITH SPECTRAL POLLUTION

In this work, we propose new sufficient conditions to solve the spectralpollution problem by using the generalized spectrum method. We give the theoretical foundation of the generalized spectral approach, as well as illustrate its effectivenessby numerical results.