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Quantum Information and Computation

Majorization and the interconversion of bipartite states

A NielsenMichael, VidalGuifré

Published Jan 1, 2001• 3 citations
Publisher

Abstract

Majorization is a powerful, easy-to-use and flexible tool which arises frequently in quantum mechanics as a consequence of fundamental connections between unitarity and the majorization relation. E...
DOI
10.5555/2011326.2011331

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The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.
Many authors outlined the students' difficulties in the conceptual understanding of quantum physics. One of the major problems is that students find the mathematical language used to describe these systems (that cannot be seen) generally too obscure. A lot of research has been done to use computer visualizations of quantum mechanical topics and it proved to be a powerful and flexible tool, allowing to see through the mathematics. In this work it is shown how the use of different visualizations regarding bound states (and in particular the hydrogen atom) may be useful to foster a deeper understanding of topics of very general interest for the study of quantum physics.