The Functional Analysis of Quantum Information Theory: A Collection of Notes Based on Lectures by Gilles Pisier, K. R. Parthasarathy, Vern Paulsen and Andreas Winter

149 Pages·2015·1.39 MB·English

by  Ved Prakash Gupta

Lecture Notes in Physics 902
Ved Prakash Gupta
Prabha Mandayam
V.S. Sunder
The Functional
Analysis of
Quantum
Information Theory
A Collection of Notes Based on
Lectures by Gilles Pisier, K. R.
Parthasarathy, Vern Paulsen and
Andreas Winter
Lecture Notes in Physics
Volume 902
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Ved Prakash Gupta (cid:129) Prabha Mandayam (cid:129)
V.S. Sunder
The Functional Analysis
of Quantum Information
Theory
A Collection of Notes Based on Lectures
by Gilles Pisier, K.R. Parthasarathy,
Vern Paulsen and Andreas Winter
123
VedPrakashGupta PrabhaMandayam
SchoolofPhysicalSciences DepartmentofPhysics
JawaharlalNehruUniversity IndianInstituteofTechnologyMadras
NewDelhi,India Chennai,India
V.S.Sunder
TheInstituteofMathematicalSciences
Chennai,India
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-16717-6 ISBN978-3-319-16718-3 (eBook)
DOI10.1007/978-3-319-16718-3
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Preface
Thesenotesgrewoutofa2-weekworkshoponTheFunctionalAnalysisofQuantum
InformationTheorythatwasheldattheInstituteofMathematicalSciences,Chennai
during 26/12/2011 to 06/01/2012. This was initially the brain-child of Ed Effros,
who was to have been one of the four principal speakers; but as things unfolded,
he had to pull out at the eleventh hour due to unforeseen and unavoidable
circumstances. But he mentored us through the teething stages with suggestions
onpossibleanddesirablesubstitutes.Afterafewhiccups,wearrivedataperfectly
respectable cast of characters, largely owing to Prof. K.R. Parthasarathy agreeing
to fill the breach to the extentthat his somewhatfrail health would permit. While
everybodyelsehadbeenaskedtogivefive90-minlectureseach,heagreedgamely
togivethree60-minlectures,whichwereeachasdelightfulasonehasexpectedhis
lecturestobe.
ThethreeotherspeakerswereGilles Pisier,VernPaulsen,andAndreasWinter.
Given the impeccable clarity in their lectures, it was not surprising that the
workshop had a substantial audience (a mixed bag of mathematicians, physicists,
and theoretical computer scientists) for the entire 2 weeks, and several people
wantedto knowiftheproceedingsoftheworkshopcouldbepublished.Giventhe
widescopeoftheproblemsdiscussedhere,wehopethesenoteswillproveusefulto
studentsandresearchscholarsacrossallthreedisciplines.
The quite non-trivial and ambitious task of trying to put together a readable
accountofthelectureswastakenuponbythethreeofus,withperiodicassistance
fromMadhushreeBasuandIssanPatri.Whenwefinallyfinishedafirstdraftsome
28monthslaterandsolicitedpermissionfromthespeakersforustopublicizethis
account,theyrespondedquite promptlyand positively,with Vern evenofferingto
slightlyeditandexpand“his”partandGillesuncomplaininglyagreeingtoeditthe
lastfewpagesof“his”part.
As even a casual reading of these notes will show, different parts have been
written by different people, and some material has been repeated (the Kraus
decomposition for instance). Further the schism in the notational conventions of
physicists and mathematicians (e.g., in which of its two arguments the inner
product is (conjugate) linear) reflects which convention which speaker followed!
v
vi Preface
Thisschismwaspartlybecausewedidnotwanttointroducenewerrorsbytrying
to adopt a convention other than in the notes taken from the lectures.Thus, this
must be viewed not as a textbook in a polished and complete form, but rather as
a transcription of notes derived from lectures, reflecting the spontaneity of live
classroominteractionsandconveyingthe enthusiasmandintuitionforthe subject.
It goeswithoutsaying that anyshortcomingsin these notesare to be attributed to
thescribes,whileanythingpositiveshouldbecreditedtothespeakers.
NewDelhi,India VedPrakashGupta
Chennai,India PrabhaMandayam
Chennai,India V.S.Sunder
Acknowledgements
Firstly, we express our gratitude to so many people in the institute (IMSc) who
helpedinsomanywaystomaketheworkshopassuccessfulasitwas,notleastour
genial Director Balu for generously funding the whole enterprise. We also thank
oursystemspeoplewhocheerfullyvideographedallthelectures,thesevideosbeing
madeavailableasaYou-tubeplaylistathttp://www.imsc.res.in/~sunder/faqit.html;
andthereaderisstronglyurgedtoviewthesevideosinparallelwithreadingthese
notes,toreallygetthefullbenefitofthem.
PM was a post-doctoral fellow with the quantum information group at IMSc
duringthewritingofthismanuscript.ShewouldliketothankIMScandthequantum
informationgroup in particular for the opportunityto participate in the workshop
andbeapartofthiseffort.
VSS also wishes to explicitly thank Dr. S. Ramasami, the former Secretary
of the Department of Science and Technology, for having gladly, promptly, and
unconditionallypermittedtheuseofthegenerousgrant(theJ.C.BoseFellowship)
hereceivesfromtheminawonderfulexampleofthe“reasonableaccommodation”
that the UN Convention on the Rights of Persons with Disability speaks about,
whichhaspermittedhimtotravelwhenneededandbeafunctionalmathematician
inspiteofhealthconstraints.
Wewillberemissifwedidnotrecordourappreciationoftheextremelypleasant
and civil manner in which we were requested by the Editors of Springer LNP to
permit them to publish these notes. This association/correspondence has been an
altogetherrefreshingandhearteningexperience.
vii

Contents
1 OperatorSpaces ............................................................. 1
1.1 OperatorSpaces......................................................... 1
1.1.1 CompletelyBoundedandCompletelyPositiveMaps......... 2
1.1.2 OperatorSystems............................................... 2
1.1.3 FundamentalFactorisationofCBMaps....................... 3
1.2 MoreonCBandCPMaps.............................................. 18
1.3 Ruan’sTheoremandItsApplications.................................. 27
1.3.1 Ruan’sTheorem ................................................ 27
1.3.2 SomeApplicationsandSomeBasicFacts..................... 29
1.3.3 min and max Operator Space Structures
onaBanachSpace.............................................. 30
1.4 TensorProductsofOperatorSpaces ................................... 31
1.4.1 InjectiveTensorProduct........................................ 31
1.4.2 ProjectiveTensorProduct...................................... 32
1.4.3 GeneralRemarks................................................ 33
1.4.4 APassingRemarkontheHaagerupTensorProduct.......... 34
1.5 TensorProductsofC(cid:2)-Algebras....................................... 34
1.5.1 minandmaxTensorProductsofC(cid:2)-Algebras................ 34
1.5.2 Kirchberg’sTheorem........................................... 35
References..................................................................... 37
2 EntanglementinBipartiteQuantumStates .............................. 39
2.1 QuantumStates,ObservablesandProbabilities....................... 39
2.2 Entanglement............................................................ 42
2.2.1 SchmidtDecomposition........................................ 43
2.2.2 UnitaryBases,EPRStatesandDenseCoding ................ 46
2.3 SchmidtRankofBipartiteEntangledStates........................... 48
2.3.1 SubspacesofMinimalSchmidtRank.......................... 48
ix