Geometry of Quantum States

Abstract

A geometric framework for quantum mechanics arose during the mid1970s when authors such as Cantoni explored thenotion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantumstates provides a natural quantum mechanical analogue for classicalphase space. This central idea can be seen easily since theprojection of Schrödinger's equation from a Hilbert space into thespace of pure spaces is a set of Hamilton's equations. Over theintervening years considerable work has been carried out by avariety of authors and a mature description of quantum mechanics ingeometric terms has emerged with many applications. This currentoffering would seem ideally placed to review the last thirty yearsof progress and relate this to the most recent work in quantumentanglement.Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on)attempts to cover considerable ground in its 466 pages. Its topicsrange from colour theory in Chapter 1 to quantum entanglement inChapter 15—to say that this is a whirlwind tour is, perhaps, nounderstatement. The use of the work 'introduction' in the subtitleof GQS, might suggest to the reader that this work be viewed as atextbook and I think that this interpretation would be incorrect.The authors have chosen to present a survey of different topics withthe specific aim to introduce entanglement in geometric terms—thebook is not intended as a pedagogical introduction to the geometricapproach to quantum mechanics. Each of the fifteen chapters is ashort, and mostly self-contained, essay on a particular aspect orapplication of geometry in the context of quantum mechanics withentanglement being addressed specifically in the final chapter.The chapters fall into three classifications: those concerned withthe mathematical background, those which discuss quantum theory andthe foundational aspects of the geometric framework, andapplications of the geometric approach.The first four chapters contain the standard mathematics required tounderstand the rest of the material presented: specific areas incolour theory, set theory, probability theory, differential geometryand projective geometry are all covered with an eye to the materialthat follows. Chapter 5 starts the first real discussion of quantumtheory in GQS and serves as an elegant, succinctintroduction to the geometry which underlies quantum theory. Thismay be the most worthwhile chapter for the casual reader who wantsto understand the key ideas in this field.Chapter 6 builds on the discussion in Chapter 5, introducing a grouptheoretic approach to understand coherent states and Chapter 7describes a geometric tool in the form of an approach to complexprojective geometry called 'the stellar representation'. Chapter 8returns to a more purely quantum mechanical discussion as theauthors turn to study the space of density matrices. This chaptercompletes the discussion which started in Chapter 5.Chapter 9 begins the part of the book concerned with applications ofthe geometric approach. From this point on the book aims,specifically, to prepare the reader for the material in Chapter 15beginning with a discussion on the purification of mixed quantumstates. In the succeeding chapters a definite choice has been madeto present a geometric approach to certain quantum informationproblems. For example, Chapter 10 contains an extremely wellformulated discussion of measurement and positive operator-valuedmeasures with several well illustrated examples and Chapter 11reopens the discussion of density matrices. Entropy and majorizationare again revisited in Chapter 12 in much greater detail than inprevious chapters. Chapters 13 and 14 concern themselves with adiscussion of various metrics and their relation to the problem ofdistinguishing between probability distributions and theirsuitability as probability measures.