Modern tools for quantum technology : non-Gaussianity, local quantum estimation theory and their applications to quantum optical systems

Abstract

In any protocol aimed at manipulating or transmitting information, symbols are encoded in states of some physical system. If this system is quantum mechanical rather than classical, as is bound to become common if the miniaturisation of processing units persists at the current rate, the laws of information processing are different than the ones governing ordinary classical devices. Quantum Information (QI) science investigates these laws, and explores the novel, potentially revolutionary, possibilities offered by quantum mechanical systems across the whole spectrum of information technologies. In the last two decades, we have witnessed the rise of QI science from the realm of theoretical conjecture to that of actual technological application, with the prominent example of operating quantum cryptographic systems. One of the strengths of QI is the versatility of its theoretical framework, which is applicable to a number of technological substrates. Quantum information has thus been encoded in the degrees of freedom of different physical systems: the first examples have been the polarisation of light and the excitation of a two-level atom, both abstractly described as qubits, that is quantum states living in a two-dimensional Hilbert space. Successively, discrete variable (DV) physical systems with higher dimensions (qudits) have been considered and, more recently, great advances have been achieved for infinite-dimensional systems, the so-called continuous-variable (CV) systems, such as light modes or motional degrees of freedom of trapped particles. The aim of this thesis is to characterize at a quantum level physical systems and operations as a resources for quantum information processing. In particular, we will deal with two different main topics in quantum information: the analysis of the role of non-Gaussianity in CV quantum information and the study of different quantum estimation problems for quantum technology purposes, with a main attention on quantum optical implementation. The first part of the thesis is then focused on the concept of non-Gaussianity (nonG). In particular we will address the quantification of the nonG of quantum states and operations in the quantum information framework. We will illustrate in details the properties and the relationships of the two measures of non-Gaussianity we recently proposed, based on the Hilbert-Schmidt distance and on the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We will then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behaviour on most of the examples taken into account. However, we also show that they introduce a different relation of order, i.e. they are not strictly monotone each other. We exploit the non-Gaussianity measures for states in order to introduce a measure of non-Gaussianity for quantum operations, to assess Gaussification and de-Gaussification processes, and to investigate in details the role played by non-Gaussianity in entanglement distillation protocols. Besides, we will exploit the QRE-based non-Gaussianity measure to provide new insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information and the Holevo bound and in the framework of the quantum central limit theorem. We will also deal with parameter estimation and present a theorem connecting the QRE nonG to the quantum Fisher information. Finally, we will firstly derive some experimentally friendly lower bounds to the QRE based nonG for some class of states and by considering the possibility to perform on the states only certain efficient or inefficient measurements, and then we will present and characterize some experimental protocols that generate non-Gaussian states by means of photon-subtraction and photon-addition operations. If one wants to completely exploit a resource, one should be able to estimate its value. However, many quantities of interest in physics are not always directly accessible and this is particularly true for quantum mechanical systems where several relevant quantities do not correspond to proper quantum observables. In these situations one should resort to indirect measurements and infer the value of the quantity of interest by inspecting a set of data coming from the measurement of different observables. This is a parameter estimation problem which may be properly addressed in the framework of quantum estimation theory. The goal of an estimation problem is not only retrieve the actual value of the unknown parameter, but obtain this information with the minimum uncertainty. The second part of the thesis is thus dedicated to this topic, and we will focus in particular on the estimation of resources for quantum technology and of noise parameters affecting the resources themselves, i.e. we will evaluate the corresponding Quantum Fisher Information (QFI) and thus the ultimate precision posed by quantum mechanics in the estimation of these parameters, looking for optimal probes and optimal measurements protocols able to attain these bounds. We will start by considering, both in DV and CV systems, the estimation of the key-ingredient for quantum information, i.e. the entanglement. Then we will move to the CV realm, addressing the estimation of quantities characterizing single-mode Gaussian operations, as the displacement and squeezing parameters. We will also highlight the non-Gaussianity induced by Kerr interaction as a resource for the estimation of these parameters. Finally we will consider the paradigmatic problem in quantum estimation theory: the estimation of the quantum phase. Here, both in the qubit and in the CV case, we will consider quantum states affected by a phase-diffusive noise. We will look for the ultimate bounds on the estimation, finding the optimal probe states and the optimal measurements, and we will also show an experimental verification of these results, demonstrating an optimal estimation protocol for optical qubits. The thesis is thus organized in three main chapters. In the first chapter we will introduce all the preliminary notions that are necessary to deal with the arguments treated through the thesis. Then, in the following chapters we will dedicate our study to the non-Gaussianity and to quantum estimation problems, respectively as a resource and a tool for quantum technology.