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Title: Applications of the operator Schmidt decomposition in the quantification and exploitation of correlations
Author: Caiaffa, Matteo
ISNI:       0000 0004 7967 2298
Awarding Body: University of Strathclyde
Current Institution: University of Strathclyde
Date of Award: 2018
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In this thesis we analyse a particular decomposition of the density matrix into tensor product terms - known as the operator Schmidt decomposition (OSD) - showing how it can be used in order to measure and exploit correlations in bipartite quantum systems. Correlations rest at the heart of Quantum Information theory for both their foundational significance and their irreplaceable role in quantum computation and communication. However, because of their difficult characterisation, detecting and measuring correlations are usually believed arduous tasks. This is particularly true in the mixed-state domain, where the diversity of potential correlations represents a further complication. For these reasons, it would be advisable to define a common framework for examining and quantifying correlations of all kinds and degrees, both for pure and mixed states. Here we argue that the OSD is a powerful tool for this purpose, in that it can be used to devise measures of correlations, whether classical or quantum. In turn, these measures can be exploited in order to detect the presence of entanglement and steering, for example. The first part of this work is devoted to the definitions of such measures and the analysis of their properties. In the second part instead we consider the possibility of taking advantage of the OSD in the context of quantum process discrimination and tomography. These tasks are central to the implementation of quantum technologies, since the actual realisation of any application based upon quantum phenomena largely relies on the determination of quantum processes. We provide a set of tools - based on the OSD - that could serve as a means by which enabling or improving certain specific protocols of ancilla-assisted quantum process discrimination (AAPD) and tomography (AAPT). First, we present a quantifier for the performance of bipartite input states in AAPD. We show that the possibility to improve the discrimination power of this protocol - or to enable it altogether - is imprinted in the OSD of the input state. For what AAPT is concerned, we demonstrate how the OSD of the input state can be exploited in order to allow a characterization of an unknown local channel via a relatively small number of local transformations of the input state. More in general, we provide several results which show a sharp connection between the tasks of channel discrimination and tomography, the OSD of the input state, and the degree of correlations carried by the latter. We conclude the thesis with a collection of results of a seemingly different nature, but that were actually inspired by the examination of the previously mentioned ancilla-assisted tasks. In short, we define a family of state-dependent metrics on the space of quantum channels and show that they are deeply connected to the OSD of the state defining them. As a byproduct, the latest results entail a possible generalisation of the Choi-Jamio lkowski isomorphism, thus providing an interesting motivation to the extension of this research project.
Supervisor: Piani, Marco ; Oi, Daniel Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral