Quantum information: an overview

291 Pages·2007·1.479 MB·English

by  Gregg Jaeger

Quantum Information
Gregg Jaeger
Quantum Information
An Overview
Dr.GreggJaeger
QuantumImagingLaboratory
DepartmentofElectricalandComputerEngineering
BostonUniversity
8SaintMary’sStreet
Boston,MA02215
USA
and
DepartmentofNaturalSciences
CollegeofGeneralStudies
871CommonwealthAvenue
Boston,MA02215
USA
[email protected]
LibraryofCongressControlNumber:2006926441
ISBN-10:0-387-35725-4 e-ISBN-10:0-387-36944-9
ISBN-13:978-0-387-35725-6 e-ISBN-13:978-0-387-36944-0
Printedonacid-freepaper.
(cid:1)C 2007SpringerScience+BusinessMedia,LLC
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,
NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
9 8 7 6 5 4 3 2 1
springer.com
To my daughter Alia and all those who have inspired, supported,
and/or prodded me
Foreword
In one word, this is a responsible book; the rest is commentary.
Around 1992 a few of us were led by Charles Bennett into a Garden of
Eden of quantum information, communication, and computation. No sooner
had we started exploring our surroundings and naming the birds and the
beasts, than Peter Shor put an end to that apparent innocence by showing
thatfactoringcouldbeturned—bymeansofquantumhardware—intoapoly-
nomial task. Fast factoring meant business; everybody seemed to be awfully
interested in factoring. Not that anyone had any use for factoring per se, but
itseemedthatalltheworld’ssecretswereprotectedbyfactor-keyedpadlocks.
Think of all the power and the glory (and something else) that you might get
by acting as a consultant to big businesses and government agencies, helping
them pick everyone else’s locks and at the same time build unpickable ones
(well, nearly unpickable) for themselves. And if one can get an exponential
advantage in factoring, wouldn’t an exponential advantage be lying around
the corner for practically any other computational task? Quantum informa-
tion “and all that” has indeed blossomed in a few years into a wonderful new
chapter ofphysics, comparablein flavor and scopetothermodynamics. Ithas
alsoturnedintoaveritable“industry”—producingpapers,conferences,exper-
iments, effects, devices—even proposals for quantum computer architectures.
Dutifully,alsoentirebooksonthesubjecthavebeenappearingwithacertain
regularity. Every time I see a new one, my first reaction tends to be, “Who
ordered that?” meaning, What needs does this book fill? What market does
it address?
I’m convinced that a bona fide academic book (as contrasted to a com-
mercial book) is first and foremost a knowledge-structuring exercise, a taut
“clothesline” (to use an image by Herbert Wilf) on which to neatly pin one’s
thoughtsandfindthemstillthereinthemorning.Inthisrespect,thepresent
work is no exception. But one doesn’t have to go through all the labor of
producing a real book just for that. A second, also quite honorable motive, is
to let your colleagues know that you’ve been there yourself; that you’ve seen
a few things that they may have missed; that from a certain angle you get
viii Foreword
a much better view; and so on; that, in other words, by your meisterstu¨ck
you claim full membership in the guild. In the meantime your colleagues had
of course been looking at what you were doing all along, and probably had
already made you member in pectore. I have no doubts that this applies to
the present case.
But in many, perhaps most, of the books I’ve seen I believe I detected in
an obstinate bass line, some sort of rumbling or blurbing that has no words
toitbutIwouldbetemptedtointerpretlikethis:“Yes,thisbookofourswill
be good for an advanced undergraduate course in Quantum Computation,
Quantum Cryptography, or Entanglement Distillation (or any other permu-
tation of a number of similar sexy terms). But the real reason you must want
it, for yourself and for your students, is because we are nearing the moment
when a Quantium—rumored to be able to do all computations exponentially
faster—is to be commercially available as a drop-in replacement for the Pen-
tium! You need this book because you cannot afford not to be yourself one
of the very designers of the Quantium, or at least one of the first to design
it in!” I may just be hearing voices. But Quantum Information, Communica-
tion, and Computation is too rich a conceptual discipline to need debasing
with the subliminal lure of “universal exponential speedup.” For the moment
such a promise should be kept in the same class as “Energy so cheap it won’t
be worth metering;” it doesn’t even have to be false to be irresponsible. This
book steers clear of all that.
IrecallthetitleofE.T.Jaynes’sbookoninformationtheory,Probability—
The Logic of Physics, and paraphrase it as “the logic of incomplete informa-
tion,”thusstressingthatphysics,eventhoughcentralformotivation,is,from
a conceptual viewpoint, merely incidental to information theory. An incom-
plete description is just that, namely, one that is not sufficiently detailed to
identify a single individual: several individuals may fit it. The art of proba-
bility is nothing more than doing ordinary Aristotelian logicin parallel on all
those “several” (as often as not 1024) individuals, and in the end lumping the
results into “bins” according to whatever traits are relevant to our question
of the moment. Introducing (that is, making up) a probability distribution is
inessenceequivalenttodoingsomeofthatbinningbefore puttingthesystem
through the “logic engine” rather than after. In fact, if this is done prop-
erly, the two approaches commute, and the second, of course, may save much
computational effort.
Quantumbehaviorhasconfrontedphysicswithmanynovelties.Whatmat-
tershereisthatithasintroducedformerlyunsuspectedwaysforadescription
to be incomplete. Introduced where? into physics? My gut feeling is that in
this business physics per se is largely irrelevant. (Think, for example, of how
entropy, introduced for very good reasons by physicists, is in fact the funda-
mental quantitative parameter of any probability distribution—and thus, as
we’ve seen—an essential aspect of any incomplete description.) Be that as it
may, one of the duties of information theory is to acknowledge these new as-
pects of incompleteness whose prototype is found in physics and incorporate
Foreword ix
them by adding the necessary new modules to the underlying “logic engine”
mentioned above. There is no need to attempt to “interpret” or “explain”
quantum mechanics before setting about this task; on the contrary, having
in hand the resulting formal “quantum-enhanced” information theory may in
the end make it easier to address the interpretation problem itself.
Jaeger’s book seems to me consistent with the above strategy. That is, it
takes quantum mechanics as a premise. It doesn’t waste time arguing about,
or changing, the premise itself, and concentrates instead on developing an
inferenceenginecapableofhandlingpremisesofthatkind.That’showweget
Quantum Information Theory: An Overview rather than one more book on
“Quantum information communication computation and all that.”
Tommaso Toffoli
Preface
Quantum information science is a rapidly developing area of interdisciplinary
investigation at the nexus of quantum mechanics and information theory. It
now plays a significant role in a number of subdisciplines of physics, informa-
tiontechnology,andengineering.Anumberofbooksonquantuminformation
areavailablebutarebecomingoutdatedand/ordiffersignificantlyinapproach
from this book, or cover only particular aspects of the subject. Historically,
lecture notes for the first general course on quantum information, given at
Hewlett-Packard and edited by Lo, Popescu, and Spiller, were published in
1998 [289]. Physicists have also long benefited from the generously provided
on-line lecture notes of Preskill [341]. At least one comprehensive monograph
on quantum information science was published at the turn of the century,
namely, the meticulous nearly 700-page book of Nielsen and Chuang [315].
More recent books by Paviˇci´c [325] and Stenholm and Suominen [404] are
noteworthyfortheirutility.Amonographdetailingthemathematicalfounda-
tions of quantum information theory by Hayashi, which originally appeared
inJapanesein2003,hasjustappearedinEnglish[208].ThebooksofBenenti,
Casati, and Strini [33] and Gruska [201] are valuable textbooks for teaching
thesubject.Quantumkeydistributionandquantumcomputingarecurrently
the most exciting applications of quantum information science that are suf-
ficiently well developed that a number of books are specifically dedicated to
one or the other of them: in the case of quantum cryptography and com-
munication, comprehensive collections have been edited by Alber et al. [7],
Beth and Leuchs [59], Bouwmeester, Ekert, and Zeilinger [72], Braunstein et
al. [79], and Sergienko [375]; in the case of quantum computation, the books
of Brylinski and Chen [89], Hirvensalo [216], Kitaev, Shen, and Vyalyi [252],
and Pittenger [335], and that edited by Lomonaco [287] focus on quantum
algorithms and/or associated mathematics. A number of popular books on
quantum computing have also been published, for example [83, 121].
Inourinformationage,electronicaccesstoprimarysourcesiswidelyavail-
able,allowingonetolocatethefinestdetailsoforiginalinvestigationsonceone
iswellorientedwithtoolsandreferencesinhand.Therefore,nowwhatoneof-
xii Preface
tenneedsmostwhenapproachingthesubjectofquantuminformationscience
is an overview that efficiently yet rigorously presents the fundamentals and
that provides a detailed weblinked bibliography to take one further [1]. This
bookisintendedtobesuchahandyreferenceforpractitionersandstudentsof
quantum physics and computer science that also treats foundational aspects
of quantum mechanics connected with quantum information science, includ-
ingthoseassociatedwithquantummeasurementwhichplaysanessentialrole
inrelatingclassicalandquantuminformation.Mostoftheexamplesprovided
here are quantum-optical ones as a pragmatic matter, arising from the fact
thatinterferometryiscentraltoquantuminformationprocessingandthefact
that interferometry has primarily progressed through optical physics. How-
ever, exciting innovations have been made by experimental groups working
with a range of physical systems. Hopefully, workers in areas of experimental
physics and engineering other than optics will soon provide comprehensive
and detailed overviews of each of the experimental methods of manipulat-
ing quantum information. For the time being, discussions of various devices
for quantum information processing can be found [19, 74, 407]. Particularly
noteworthy are the books edited by Everitt [165] and Leggett et al. [274].
In the twentieth century, the formalism introduced in Dirac’s The Princi-
ples of Quantum Mechanics and von Neumann’s Mathematische Grundlagen
derQuantenmechanik wasbroughttobearonabroadrangeofphysicalprob-
lems. Elements of this formalism and related mathematics are outlined in the
appendices, together with standard quantum postulates. During the last two
decades of the twentieth century, investigations of the foundational problems
of quantum mechanics and the physics of computation were pivotal in giving
rise to quantum information science as a subject in its own right, providing
a conceptual basis for the development of quantum protocols and algorithms.
In turn, the investigation of foundational problems has benefited from the
work of those seeking solutions to central issues in quantum information sci-
ence, such as those of communication complexity. Aspects of this important
interplay have been addressed here.
It is my hope that, in addition to its serving as a practical tool for re-
searchers and students, this book will assist those seeking to understand the
subject to appreciate the many decades of work back to which the origins of
this exciting, relatively new field can be traced. Although the aim in includ-
ing this material is not to present a history of the exploration of foundations
of quantum mechanics or its philosophical underpinnings, a number of perti-
nent such results from earlier decades of the twentieth century are included
because they will likely prove important to future progress in both quantum
mechanics and information theory. The discussion of early work is here often
in the language of quantum information so as to facilitate access to earlier,
foundational work in quantum mechanics by those approaching fundamental
issues from a twenty-first century perspective.
Gregg Jaeger Cambridge, MA, August 2006