Approximate Quantum Markov Chains

124 Pages·2018·1.84 MB·English

by  David Sutter

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 28
David Sutter
Approximate
Quantum Markov
Chains
123
SpringerBriefs in Mathematical Physics
Volume 28
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David Sutter
Approximate Quantum
Markov Chains
123
DavidSutter
Institute for Theoretical Physics
ETHZurich
Zürich
Switzerland
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
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Acknowledgements
First and foremost, I would like to thank my advisor Renato Renner for his
encouragement and support. His never-ending enthusiasm, optimism, and persis-
tencyindoingresearchaswellashisprecisioninthoughtandcommunicationwere
highlyinspiringandclearlysharpenedmymind.Hisguidanceduringthelastyears
was outstanding. I was entirely free to work on what I like most, at the same time
alwaysknowingthathewouldimmediatelyinterruptincaseIdriftofftoanalyzing
meaningless problems.
IamespeciallygratefultoJürgFröhlichforintroducingmetotheexcitingfield
of mathematical physics and for carefully listening to my oftentimes vague new
ideas.Jürg’simmenseknowledge about physicsisextraordinaryandI enormously
enjoyed our regular meetings. Furthermore, my sincere thanks go to Emre Telatar
for investing his time in studying my work and for being my co-examiner. I also
would like to thank Manfred Sigrist for interesting discussions in the early morn-
ings at the institute and for representing the physics department at my defense.
During the last couple of years, I was extremely lucky to collaborate with
various brilliant people including Mario Berta, Frédéric Dupuis, Omar Fawzi,
Aram W. Harrow, Hamed Hassani, Raban Iten, Marius Junge, John Lygeros,
PeymanMohajerinEsfahani,RenatoRenner,JosephM.Renes,VolkherB.Scholz,
Tobias Sutter, Marco Tomamichel, Mark M. Wilde, and Andreas Winter. I would
like to thank all of them for their patience and effort to work with me.
IamverygratefultoallmembersoftheInstituteforTheoreticalPhysicsatETH
Zurich and in particular the Quantum Information Theory Group people for their
support and the friendly atmosphere. It was an exciting and truly wonderful time.
Finally and most importantly, I would like to thank my twin brother, my sister,
and my parents for their constant support and encouragement.
Zürich, Switzerland David Sutter
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Classical Markov Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Robustness of Classical Markov Chains . . . . . . . . . . . . . . 3
1.2 Quantum Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Robustness of Quantum Markov Chains . . . . . . . . . . . . . . 5
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Schatten Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Functions on Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Entropy Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.2 Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.3 Measured Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.4 Rényi Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Background and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Tools for Non-commuting Operators . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Spectral Pinching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.2 Smooth Spectral Pinching . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.3 Asymptotic Spectral Pinching. . . . . . . . . . . . . . . . . . . . . . 52
3.2 Complex Interpolation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Background and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii
viii Contents
4 Multivariate Trace Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Multivariate Araki-Lieb-Thirring Inequality . . . . . . . . . . . . . . . . . 65
4.3 Multivariate Golden-Thompson Inequality . . . . . . . . . . . . . . . . . . 66
4.4 Multivariate Logarithmic Trace Inequality . . . . . . . . . . . . . . . . . . 70
4.5 Background and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Approximate Quantum Markov Chains . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Quantum Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Sufficient Criterion for Approximate Recoverability . . . . . . . . . . . 79
5.2.1 Approximate Markov Chains are not Necessarily
Close to Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Necessary Criterion for Approximate Recoverability. . . . . . . . . . . 82
5.3.1 Tightness of the Necessary Criterion. . . . . . . . . . . . . . . . . 86
5.4 Strengthened Entropy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.1 Data Processing Inequality . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.2 Concavity of Conditional Entropy. . . . . . . . . . . . . . . . . . . 94
5.4.3 Joint Convexity of Relative Entropy . . . . . . . . . . . . . . . . . 95
5.5 Background and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Appendix A: A Large Conditional Mutual Information Does Not
Imply Bad Recovery.. .... .... .... .... .... ..... .... 101
Appendix B: Example Showing the Optimality of the K -Term . .... 105
max
Appendix C: Solutions to Exercises. .... .... .... .... .... ..... .... 109
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 117
Chapter 1
Introduction
Markov chains are named after the Russian mathematician Andrei Markov
(1856–1922),whointroducedthemin1907.Supposewehaveasequenceofrandom
variables (Xn)n≥1. The simplest model is the case where the random variables are
assumedtobepairwiseindependent.Forthisscenariomanyniceresults,suchasthe
lawoflargenumberorthecentrallimittheorem,areknown.Atthesametimethe
pairwiseindependenceassumptionmakesthemodelratherrestrictive.
Markov’sideawastoconsideramoregeneraldependencestructurethathowever
isstillsimpleenoughthatitcanbeanalyzedrigorously.Informally,hisideawasto
assumethattherandomvariables(Xn)n≥1 areorderedinaveryspecificway.1 This
orderingimpliesthatalltheinformationthattherandomvariables(X1,··· ,Xk−1)
couldhaveabout Xk foranyk >1iscontainedin Xk−1.Moreprecisely,werequire
thatthecollectiveentirepast(X1,...,Xk−2)isindependentofthecollectiveentire
future (Xk,...) conditioned on the present Xk−1. This model has the advantage
thatinordertodescribe Xk weonlyneedtoremember Xk−1 andcanforgetabout
the past (X1,...,Xk−2). This makes the model simple enough that we can prove
precisepropertiesanddescribeitsbehaviorforlargevaluesofn.Atthesametime,
themodelisconsiderablymoregeneralthanthepairwiseindependenceassumption
whichmakesitsuitableformanysituations(see,e.g.,[1–4]).
Markovchainsareintensivelystudiedandhavebeengeneralizedtothequantum
mechanicalsetup[5]whererandomvariablesarereplacedbydensityoperatorsona
Hilbertspace.2Naturalquestionsthatariseare:
WhatarethemaindifferencesbetweenclassicalandquantumMarkovchains?
What do we know about sequences of random variables that approximately
form a Markov chain? Do they approximately behave as (exact) Markov
chains?
1Wethensay(Xn)n≥1formsaMarkovchaininorderX1↔ X2↔ X3↔... .
2InSect.1.2andinparticularinSect.5.1weintroducetheconceptofaquantumMarkovchain.
©TheAuthor(s)2018 1
D.Sutter,ApproximateQuantumMarkovChains,SpringerBriefsinMathematical
Physics,https://doi.org/10.1007/978-3-319-78732-9_1
2 1 Introduction
Thisbookwillanswerthesequestions.Wewillfirstintroducethereadertoquan-
tumMarkovchainsandexplainhowtodefinearobustversionofthisconceptthat
willbecalledapproximatequantumMarkovchains.
Intheliteraturethereexiststheterm“shortMarkovchains”whichshoulddistin-
guishtheMarkovchainbetweenthreerandomvariablesfrominfinitechains.Since
weonlyconsiderMarkovchainsdefinedforthreerandomvariablesinthisbookwe
droptheterm“short”.
1.1 ClassicalMarkovChains
ThreerandomvariablesX,Y,Z withjointdistributionP formaMarkovchainin
XYZ
order X ↔Y ↔ Z if X and Z areindependentconditionedonY.Inmathematical
termsthiscanbeexpressedas
PXYZ isaMarkovchain ⇐⇒ PXZ|Y = PX|YPZ|Y , (1.1)
where PX|Y denotes the probability distribution of X conditioned on Y. Bayes’
theorem directly implies that the right-hand side of (1.1) can be rewritten as
PXYZ = PXYPZ|Y. Operationally, the Markov chain condition tells us that all the
informationthepair(X,Y)hasabout Z iscontainedinY.Inotherwords,thereis
noneedtoremember X inordertodetermine Z ifwealreadyknowY.Supposewe
loosetherandomvariable Z.TheMarkovchainconditionensuresthatitispossible
toreconstruct Z byonlyactingonY withastochasticmap.3 Moreprecisely,
PXYZ isaMarkovchain ⇐⇒ ∃ stochasticmatrixWZ|Y suchthatPXYZ = PXYWZ|Y.
(1.2)
Bayes’theoremdirectlyimpliesthatWZ|Y canbealwayschosenasWZ|Y = PZ|Y.A
thirdcharacterizationof P beingaMarkovchainisthattheconditionalmutual
XYZ
informationvanishes,i.e.,
P isaMarkovchain ⇐⇒ I(X : Z|Y) =0, (1.3)
XYZ P
where
I(X : Z|Y) := H(XY) +H(YZ) −H(XYZ) −H(Y) (1.4)
P P P P P
(cid:2)
denotes the conditional mutual information and H(X) :=− P (x)
P x∈X X
logP (x)istheShannonentropy.
X
3Thereconstructionreferstoastochasticallyindistinguishablecopywhichmeansthatifwedenote
thereconstructedrandomvariableby Z(cid:8) werequirethattheprobabilitylawof(X,Y,Z(cid:8))isthe
sameas(X,Y,Z).