Table Of ContentTheory of Quantum Information
John Watrous
InstituteforQuantumComputing
UniversityofWaterloo
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0
Unported License. Visit http://creativecommons.org/licenses/by-sa/3.0/ to
viewacopyofthislicense.
Contents
1 Mathematicalpreliminaries 1
1.1 Linearalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 ComplexEuclideanspaces . . . . . . . . . . . . . . . . . . 1
1.1.2 Linearoperators . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Operatordecompositionsandnorms . . . . . . . . . . . . 26
1.2 Analysis,convexity,andprobabilitytheory . . . . . . . . . . . . . 37
1.2.1 Analysisandconvexity . . . . . . . . . . . . . . . . . . . . 37
1.2.2 Semidefiniteprogramming . . . . . . . . . . . . . . . . . . 50
1.2.3 Probabilitytheory . . . . . . . . . . . . . . . . . . . . . . . 54
1.3 Suggestedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 Basicnotionsofquantuminformation 63
2.1 Registersandstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1.1 Registersandclassicalstatesets . . . . . . . . . . . . . . . 63
2.1.2 Quantumstatesofregisters . . . . . . . . . . . . . . . . . . 66
2.1.3 Reductionsandpurificationsofquantumstates . . . . . . 73
2.2 Quantumchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.2.1 Definitionsandbasicnotionsconcerningchannels . . . . . 79
2.2.2 Representationsandcharacterizationsofchannels . . . . . 84
2.2.3 Examplesofchannelsandothermappings . . . . . . . . . 99
2.2.4 Extremalchannels . . . . . . . . . . . . . . . . . . . . . . . 105
2.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.3.1 Twoequivalentdefinitionsofmeasurements . . . . . . . . 109
2.3.2 Basicnotionsconcerningmeasurements . . . . . . . . . . . 114
2.3.3 Extremalmeasurementsandensembles . . . . . . . . . . . 122
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 132
iii
3 Similarityanddistanceamongstatesandchannels 135
3.1 Quantumstatediscrimination . . . . . . . . . . . . . . . . . . . . . 135
3.1.1 Discriminatingbetweenpairsofquantumstates . . . . . . 136
3.1.2 Discriminatingquantumstatesofanensemble . . . . . . . 144
3.2 Thefidelityfunction . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.2.1 Elementarypropertiesofthefidelityfunction . . . . . . . 152
3.2.2 Alternativecharacterizationsofthefidelityfunction . . . . 156
3.2.3 Furtherpropertiesofthefidelityfunction . . . . . . . . . . 168
3.3 Channeldistancesanddiscrimination . . . . . . . . . . . . . . . . 178
3.3.1 Channeldiscrimination . . . . . . . . . . . . . . . . . . . . 178
3.3.2 Thecompletelyboundedtracenorm . . . . . . . . . . . . . 181
3.3.3 Distancesbetweenchannels . . . . . . . . . . . . . . . . . . 190
3.3.4 Propertiesofthecompletelyboundedtracenorm . . . . . 200
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
3.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 214
4 Unitalchannelsandmajorization 219
4.1 Subclassesofunitalchannels . . . . . . . . . . . . . . . . . . . . . 219
4.1.1 Mixed-unitarychannels . . . . . . . . . . . . . . . . . . . . 220
4.1.2 Weyl-covariantchannels . . . . . . . . . . . . . . . . . . . . 231
4.1.3 Schurchannels . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.2 Generalpropertiesofunitalchannels . . . . . . . . . . . . . . . . . 243
4.2.1 Extremepointsofthesetofunitalchannels . . . . . . . . . 243
4.2.2 Fixed-points,spectra,andnormsofunitalchannels . . . . 249
4.3 Majorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4.3.1 Majorizationforrealvectors . . . . . . . . . . . . . . . . . 254
4.3.2 MajorizationforHermitianoperators . . . . . . . . . . . . 263
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
4.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 270
5 Quantumentropyandsourcecoding 273
5.1 Classicalentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.1.1 Definitionsofclassicalentropicfunctions . . . . . . . . . . 273
5.1.2 Propertiesofclassicalentropicfunctions . . . . . . . . . . 276
5.2 Quantumentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
5.2.1 Definitionsofquantumentropicfunctions . . . . . . . . . 289
5.2.2 Elementarypropertiesofquantumentropicfunctions . . . 292
5.2.3 Jointconvexityofquantumrelativeentropy . . . . . . . . 300
5.3 Sourcecoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
5.3.1 Classicalsourcecoding . . . . . . . . . . . . . . . . . . . . 309
5.3.2 Quantumsourcecoding . . . . . . . . . . . . . . . . . . . . 315
5.3.3 Encodingclassicalinformationintoquantumstates . . . . 320
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
5.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 335
6 Bipartiteentanglement 339
6.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
6.1.1 Separableoperatorsandstates . . . . . . . . . . . . . . . . 340
6.1.2 SeparablemapsandtheLOCCparadigm . . . . . . . . . . 355
6.1.3 SeparableandLOCCmeasurements . . . . . . . . . . . . . 363
6.2 Manipulationofentanglement . . . . . . . . . . . . . . . . . . . . 371
6.2.1 Entanglementtransformation . . . . . . . . . . . . . . . . . 371
6.2.2 Distillableentanglementandentanglementcost . . . . . . 378
6.2.3 Boundentanglementandpartialtransposition . . . . . . . 385
6.3 Phenomenaassociatedwithentanglement . . . . . . . . . . . . . . 392
6.3.1 Teleportationanddensecoding . . . . . . . . . . . . . . . . 392
6.3.2 Non-classicalcorrelations . . . . . . . . . . . . . . . . . . . 406
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
6.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 422
7 Permutationinvarianceandunitarilyinvariantmeasures 427
7.1 Permutation-invariantvectorsandoperators . . . . . . . . . . . . 427
7.1.1 Thesubspaceofpermutation-invariantvectors . . . . . . . 428
7.1.2 Thealgebraofpermutation-invariantoperators . . . . . . 438
7.2 Unitarilyinvariantprobabilitymeasures . . . . . . . . . . . . . . . 447
7.2.1 UniformsphericalmeasureandHaarmeasurebasics . . . 447
7.2.2 Applicationsofunitarilyinvariantmeasures . . . . . . . . 460
7.3 Measureconcentrationanditapplications . . . . . . . . . . . . . . 470
7.3.1 Lévy’slemmaandDvoretzky’stheorem . . . . . . . . . . . 470
7.3.2 Applicationsofmeasureconcentration . . . . . . . . . . . 489
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
7.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 505
8 Quantumchannelcapacities 507
8.1 Classicalinformationoverquantumchannels . . . . . . . . . . . . 507
8.1.1 Classicalcapacitiesofquantumchannels . . . . . . . . . . 508
8.1.2 TheHolevo–Schumacher–Westmorelandtheorem . . . . . 521
8.1.3 Theentanglement-assistedclassicalcapacitytheorem . . . 539
8.2 Quantuminformationoverquantumchannels . . . . . . . . . . . 559
8.2.1 Definitionsofquantumcapacityandrelatednotions . . . 559
8.2.2 Thequantumcapacitytheorem . . . . . . . . . . . . . . . . 568
8.3 Non-additivityandsuper-activation . . . . . . . . . . . . . . . . . 587
8.3.1 Non-additivityoftheHolevocapacity . . . . . . . . . . . . 588
8.3.2 Super-activationofquantumchannelcapacity . . . . . . . 594
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
8.5 Bibliographicremarks . . . . . . . . . . . . . . . . . . . . . . . . . 608
Preface
This is a draft of a book that began as a set of course notes for a graduate
course on the theory of quantum information that I have taught several
timesattheUniversityofWaterloo.
The book is primarily intended for graduate students and researchers
having some familiarity with quantum information and computation, such
as would be covered in an introductory-level undergraduate or graduate
course on the subject. The focus of the book is on the mathematical aspects
of quantum information, with an emphasis on proofs. No attention is paid
tomotivesforstudyingthetheoryofquantuminformation,asitisassumed
that the reader has already been motivated—and is perhaps interested in
provingnewtheoremsonquantuminformationofhisorherown.Itshould
also be said that this is not a physics book: the Schrödinger equation will
not be found herein, and the difficult technological challenge of building
quantuminformationprocessingdevicesisblissfullyignored.
The selection of topics covered in this book is not intended to be fully
representativeofthediversesubjectofquantuminformationscience.There
is, for example, no discussion of quantum cryptography, quantum error
correcting codes and fault-tolerance, quantum algorithms and complexity
theory, or topological quantum computing, which are among the topics
within the theoretical branches of quantum information science having
fundamental importance. Nevertheless, one is likely to encounter some of
the core mathematical notions discussed in this book when studying these
andothertopics.
As the students who have taken my course on the theory of quantum
information will attest, I sometimes choose to deviate from the standard
conventions of quantum information and computation, particularly with
respect to notation and terminology. I have exhibited this behavior once
again when writing this book. For example, I have avoided the use of the
vii
commonlyusedDiracnotation,andinsomecasesIhavechangedthenames
andsymbols associatedwithconceptsas Ihaveseen fit.Ihopethat readers
who have previously grown familiar with the notation and conventions of
quantum information that I have chosen not to follow will excuse me for
this,andhopethattheywillfindvalueinthisbooknevertheless.
Each chapter aside from the first includes a collection of exercises, some
of which can reasonably be viewed as straightforward, and some of which
are much more difficult. In some cases, these exercises have been derived
from research papers that clearly reveal their solutions, and I have not
attemptedtodisguisethisfactorhidetheirsource.Whiletheexercisesmay
potentiallybeusefultocourseinstructors,theirtruepurposeistobeuseful
to students of the subject; there is no substitute for the learning experience
tobefoundinwrestlingwith(andideallysolving)adifficultproblem.
IthankDebbieLeung,AshwinNayak,MarcoPiani,andPatrickHayden
for helpful discussions on some of the topics covered in this book, and I
thank Sascha Agne for assisting me with German translations. I also thank
the following people for comments, corrections, and suggestions on my
coursenotesandpreviousversionsofthisbook:
AlessandroCosentino, LeungMingLam, AlexeyRastegin,
MohammadDerakhshani, AlexandreLaplante, JohnRinehart,
OliviaDiMatteo, AnthonyLeverrier, AnsisRosmanis,
EdwardEffros, BenLovitz, VincentRusso,
ChrisFerrie, AbelMolina, FredShultz,
MirmojtabaGharibi, AdamMeikle, YuanSu,
GusGutoski, MarisOzols, LePhucThinh,
GuiyangHan, DanPuzzuoli, ChunhaoWang,
AnirudhKrishna, HammamQassim, NengkunYu.
TheInstituteforQuantumComputingandtheSchoolofComputerScience
at the University of Waterloo have provided me with both the opportunity
to write this book and with an environment in which it was possible, for
whichIamgrateful.IamalsogratefultotheNaturalSciencesandEngineer-
ing Research Council of Canada and the Canadian Institute for Advanced
Researchfortheirfinancialsupportofmyresearchprogram.
Finally, I thank Christiane Lemieux for encouraging my efforts to write
thisbookontoomanyoccasionstocount.
JohnWatrous
[email protected]
Waterloo,September2016
Chapter 1
Mathematical preliminaries
This chapter is intended to serve as a review of the mathematical concepts
to be used throughout this book, and also as a reference to be consulted as
subsequent chapters are studied, if the need should arise. The first section
focuses on linear algebra, and the second on analysis and related topics.
Unliketheotherchaptersinthisbook,thepresentchapterdoesnotinclude
proofs, and is not intended to serve as a primary source for the material it
reviews—a collection of references provided at the end of the chapter may
beconsultedbyreadersinterestedinaproperdevelopmentofthismaterial.
1.1 Linear algebra
Thetheoryofquantuminformationreliesheavilyonlinearalgebrainfinite-
dimensionalspaces.Thesubsectionsthatfollowpresentanoverviewofthe
aspects of this subject that are most relevant within the theory of quantum
information. It will be assumed that the reader is already familiar with the
most basic notions of linear algebra, including those of linear dependence
andindependence,subspaces,spanningsets,bases,anddimension.
1.1.1 Complex Euclidean spaces
ThenotionofacomplexEuclideanspaceisusedthroughoutthisbook.One
associatesacomplexEuclideanspacewitheverydiscreteandfinitesystem;
and fundamental notions such as states and measurements of systems are
representedinlinear-algebraictermsthatrefertothesespaces.
1
DefinitionofcomplexEuclideanspaces
Analphabetisafiniteandnonemptyset,whoseelementsmaybeconsidered
to be symbols. Alphabets will generally be denoted by capital Greek letters,
Σ Γ ∆
including , , and , while lower case Roman letters near the beginning
of the alphabet, including a, b, c, and d, will be used to denote symbols
in alphabets. Examples of alphabets include the binary alphabet {0,1}, the
n-fold Cartesian product {0,1}n of the binary alphabet with itself, and the
alphabet {1,...,n},for n beingafixedpositiveinteger.
Σ CΣ Σ
For any alphabet , one denotes by the set of all functions from
C CΣ
to the complex numbers . The set forms a vector space of dimension
|Σ| over the complex numbers when addition and scalar multiplication are
definedinthefollowingstandardway:
1. Addition: for vectors u,v ∈ CΣ, the vector u+v ∈ CΣ is defined by the
equation (u+v)(a) = u(a)+v(a) forall a ∈ Σ.
2. Scalar multiplication: for a vector u ∈ CΣ and a scalar α ∈ C, the vector
αu ∈ CΣ isdefinedbytheequation (αu)(a) = αu(a) forall a ∈ Σ.
A vector space defined in this way will be called a complex Euclidean space.1
The value u(a) is referred to as the entry of u indexed by a, for each u ∈ CΣ
and a ∈ Σ.Thevectorwhoseentriesareallzeroissimplydenoted0.
Complex Euclidean spaces will be denoted by scripted capital letters
W X Y Z
neartheendofthealphabet,suchas , , ,and .Subsetsofthesespaces
will also be denoted by scripted letters, and when possible this book will
A B C
followaconventiontouseletterssuchas , ,and nearthebeginningof
the alphabet when these subsets are not necessarily vector spaces. Vectors
will be denoted by lowercase Roman letters, again near the end of the
alphabet,suchas u, v, w, x, y,and z.
Whennisapositiveinteger,onetypicallywritesCn ratherthanC{1,...,n},
anditisalsotypicalthatoneviewsavectoru ∈ Cn asann-tupleoftheform
u = (α ,...,α ),orasacolumnvectoroftheform
1 n
α
1
u = ... , (1.1)
α
n
1 Many quantum information theorists prefer to use the term Hilbert space. The term
complex Euclidean space will be preferred in this book, however, as the term Hilbert space
referstoamoregeneralnotionthatallowsthepossibilityofinfiniteindexsets.
2
