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Purification and distillation of continuous variable entanglement [Elektronische Ressource] / Boris Hage

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PurificationandDistillationofContinuousVariableEntanglementVonderFakultätfürMathematikundPhysikderGottfriedWilhelmLeibnizUniversitätHannoverzurErlangungdesGradeseinesDoktorsderNaturwissenschaften–Dr. rer. nat. –genehmigteDissertationvonDipl.-Phys. BorisHagegeborenam19. Juni1979inWolfsburg2010Referent: Prof. Dr. RomanSchnabelKorreferent: Prof. Dr. KarstenDanzmannTagderPromotion: 25.01.2010AbstractQuantum communication and quantum computing to a large extent are based onthe distribution and the processing of quantum entanglement. The implementationof the two is demanding because entanglement inherently is highly susceptible to de-coherence, i.e. the uncontrollable loss of information to the environment. If the exist-ingmethodsfailtominimize thedecoherencesufficiently,entanglementdistillation cansolve this problem. Such a procedure extracts a smaller number of more strongly en-tangled states from a larger supply of weaker entangled states. In order to eliminatethedecoherencecompletelyortorealizequantumcommunicationoveralongdistancethisprocedurehasto beapplied iteratively, i.e.thedistillation isapplied repeatedlyonentangledstates,whichalreadyhavebeendistilledpreviously.Withinthescopeofthisthesistheexperimentalimplementationofacontinuousvari-able entanglement distillation protocol was conducted. The underlying entanglementwas prepared in the quadrature amplitudes of monochromatic continuous-wave laserfields.

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PurificationandDistillation
ofContinuousVariableEntanglement
VonderFakultätfürMathematikundPhysik
derGottfriedWilhelmLeibnizUniversitätHannover
zurErlangungdesGradeseines
DoktorsderNaturwissenschaften
–Dr. rer. nat. –
genehmigteDissertation
von
Dipl.-Phys. BorisHage
geborenam19. Juni1979inWolfsburg
2010Referent: Prof. Dr. RomanSchnabel
Korreferent: Prof. Dr. KarstenDanzmann
TagderPromotion: 25.01.2010Abstract
Quantum communication and quantum computing to a large extent are based on
the distribution and the processing of quantum entanglement. The implementation
of the two is demanding because entanglement inherently is highly susceptible to de-
coherence, i.e. the uncontrollable loss of information to the environment. If the exist-
ingmethodsfailtominimize thedecoherencesufficiently,entanglementdistillation can
solve this problem. Such a procedure extracts a smaller number of more strongly en-
tangled states from a larger supply of weaker entangled states. In order to eliminate
thedecoherencecompletelyortorealizequantumcommunicationoveralongdistance
thisprocedurehasto beapplied iteratively, i.e.thedistillation isapplied repeatedlyon
entangledstates,whichalreadyhavebeendistilledpreviously.
Withinthescopeofthisthesistheexperimentalimplementationofacontinuousvari-
able entanglement distillation protocol was conducted. The underlying entanglement
was prepared in the quadrature amplitudes of monochromatic continuous-wave laser
fields. The special decoherence process of phase diffusion was considered, which re-
sulted in non-Gaussian probability distributions of the corresponding variables. The
non-classicalityofsqueezedstatessufferingfromthisdecoherenceprocesswasinvesti-
gatedusingthecharacteristicfunction. Thesimultaneousdistillation, purification and
Gaussification of phase-diffused squeezedstateswas demonstrated. For the first time
distilled entangled states were actually prepared for a downstream application. Fur-
thermore, for the first time the iterative (multi-step) preparation of distilled entangle-
ment was realized. Complete evidence was provided by the first implementation of a
full,unbiasedtwo-modequantumstatetomographyinthecontinuous-variableregime.
Keywords: Quantumcommunication,entangledstates,entanglementdistillation.Zusammenfassung
Die Konzepte der Quantenkommunikation und der Quantencomputer beruhen wei-
testgehendaufderVerteilungundVerarbeitungvonQuantenverschränkung.EineUm-
setzung gestaltet sich als schwierig, denn Verschränkung weist eine starke Anfällig-
keitfürDekohärenz,einenunkontrolliertenInformationsverlustandieUmgebung,auf.
Wenn die vorhandenen Methoden zur Minimierung der Dekohärenz unzureichend
sind, kann die Destillation von Verschränkung dieses Problem lösen. Im Destillations-
verfahren wird eine kleinere Anzahl stark verschränkter aus einer größeren Menge
schwachverschränkerZuständeextrahiert.UmeinevollständigeEliminierungderDe-
kohärenz zu erreichen oder Quantenkommunikation auf große Distanz zu realisieren,
mussdieDestillationaufiterativeWeisevorgenommenwerden;diesbedeutet,dassdas
Destillationsverfahren wiederholt auch auf das Resultat einer vorherigen Destillation
angewendetwird.
Im Rahmen der vorliegenden Arbeit wurde ein Verschränkungsdestillationsproto-
kollinkontinuierlichenVariablenexperimentellumgesetzt.DiezugrundeliegendeVer-
schränkungbestandindenQuadraturamplitudenvonmonochromatischenDauerstrich-
Laserstrahlen. Der spezielle Dekohärenzprozess der Phasendiffusion, der nicht-Gauß-
förmige Wahrscheinlichkeitsverteilungen zur Folge hat, wurde betrachtet. Die nicht-
Klassizität von phasendiffundierten gequetschen Zuständen wurde untersucht. Die
gleichzeitigeDestillation,PurifikationundGaussifikationvonphasendiffundiertenge-
quetschtenZuständenkonntegezeigtwerden.ZumerstenMalwurdendestilliertever-
schränkteZuständetatsächlichpräpariert,waseinenachfolgendeweitereVerwendung
ermöglichte.Darüberhinauswurdeerstmaligeineiterative DestillationvonVerschrän-
kungrealisiert.AuchhierwurdederpräparierteAusgangszustandfüreineweitereVer-
wendungzurVerfügunggestellt.DerNachweiswurdeaufGrundlageeinervollständi-
gen,unvoreingenommenenzwei-ModenQuantenzustandstomographieerbracht.Dies
stellte die erste Implementierung dieser Methode im Bereich der kontinuierlichen Va-
riablendar.
Schlüsselwörter: Quanteninformation, verschränkte Zustände, Verschränkungsde-
stillation.Acknowledgements
First of all I would like to thank my advisor, Prof. Dr. Roman Schnabel, for being an
excellent mentor. He and the head of our department, Prof. Dr. Karsten Danzmann,
provided a vivid, fruitful and efficient scientific environment. It was an honor for me
tobeapartofourinstitute.
I was delighted to interact with Dr. Jaromír Fiurášek, Associate Professor at the
PalackýUniverstityOlomouc,CzechRepublic. JFprovidedthetheoreticalbackground
forthemain partofthisthesis. Hisenthusiasmwasalwaysstimulatingand hisability
to anticipate an experimentalist’s point of view helped a lot during our productive
discussions.
My gratitude goes to Prof. Dr. Werner Vogel from the University of Rostock, Ger-
many,especiallyforourcollaborationbutalsoforthepleasantlunchbreaksonseveral
conferences.
I am indebtedtomany of my colleagues for their assistance in numerousways. Es-
peciallymyofficematesAli,Daniel,HenningandStefangavemeagreattime.
This thesis would at last not have been possible without the loving support of my
family. Words cannot express my gratitude to Anke, whose love and confidence took
the load off my shoulders. My parents, Bärbel and Herbert, also deserve my greatest
gratitudeforalltheirsupportduringthelongyearsofmyeducation.Contents
Contents 4
1 Introduction 7
2 Theory 11
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 ContinuousVariables/PhaseSpace . . . . . . . . . . . . . . . . . 12
2.1.2 WignerFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Glauber-SudarshanP-function . . . . . . . . . . . . . . . . . . . . 15
2.2 PurificationandDistillation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 PurityofQuantumStates . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Phase-DiffusedSqueezedStates . . . . . . . . . . . . . . . . . . . 16
2.2.3 DistillationofPhase-DiffusedSqueezedStates . . . . . . . . . . . 18
2.2.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.5 GaussianEntanglement–TwoModeSqueezing . . . . . . . . . . 21
2.2.6 Phase-DiffusedEntangledStates . . . . . . . . . . . . . . . . . . . 24
2.2.7 DistillationofPhase-DiffusedEntangledStates . . . . . . . . . . . 25
2.3 QuantumStateMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 HomodyneDetection . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 QuantumStateTomography . . . . . . . . . . . . . . . . . . . . . 31
3 KeyComponentsoftheExperiments 43
3.1 LaserSources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 MainLaser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 AuxiliaryLaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Non-classicalLightSources/Squeezer . . . . . . . . . . . . . . . . . . . 48
3.2.1 OpticalandMechanicalLayout . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 BalancedHomodyneDetector . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 QuantumStateTomograph . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 DataAcquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 RandomPhaseDiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Experiments 75
4.1 PreparationandCharacterizationofPhase-DiffusedSqueezedStates . . 75
40.0 Contents 5
4.2 DistillationandPurificationofPhase-DiffusedSqueezedStates . . . . . 79
4.3 DistillationandPurificationofEntangledStates . . . . . . . . . . . . . . 87
4.4 IterativeDistillation andPurificationofEntanglement . . . . . . . . . . . 95
5 DiscussionandConclusion 103
A Hardware 105
A.1 HomodyneLockingScheme . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1.1 AuxiliaryPhase-LockedLaser . . . . . . . . . . . . . . . . . . . . 107
A.1.2 TheMixerBox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.2.1 BroadbandPhotoDetector . . . . . . . . . . . . . . . . . . . . . . 110
A.2.2 ResonantPhotoDetector. . . . . . . . . . . . . . . . . . . . . . . . 111
A.2.3 AmplifierandMixerfortheMixerBox . . . . . . . . . . . . . . . 112
A.2.4 Subtractor/AdderforBHDs . . . . . . . . . . . . . . . . . . . . . 113
A.2.5 ImprovedSHGElectronics . . . . . . . . . . . . . . . . . . . . . . 114
B CalculusForFun 115
B.1 ErrorSignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.1.1 LinearSetPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.1.2 ExtremalSetPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.1.3 ExtremalNotModulatableSetPoint,PDHMethod . . . . . . . . 116
B.2 ModulationOfLightFields . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.2.1 AmplitudeModulation . . . . . . . . . . . . . . . . . . . . . . . . 119
B.2.2 PhaseModulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
B.3 TheNaiveOpticalResonator . . . . . . . . . . . . . . . . . . . . . . . . . 122
Bibliography 127CHAPTER 1
Introduction
Since the quantum mechanical descriptionof physical systemswas formulated over a
hundredyearsagoithassuccessfullyexplainedmanyphysicalphenomenawhichwere
mystical totheclassicaltheory. Forexample,thephotoelectriceffectwasunexplainable
withoutquantummechanics. Alsothespectrumofsunlightdidnotfitintotheclassical
descriptionoftheinteractionoflightandmatter. Butquantummechanicsnotonlypro-
vided explanations for existing problems but also predicted new phenomena. Maybe
themosttremendousaspectis theexistenceofentangled states of twoor morephysical
systems. In contrast to classical physics the individual system in such a state – even
when space-likeseparated–cannot be fully describedindividually by local quantities.
This led Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) in 1935 to the con-
clusion, that the theory of quantum mechanics must be incomplete, i.e. missing some
hidden variables [1]. However, it turned out that – rather than quantum mechanics
– EPR were wrong about their assumption of local reality. In 1965 John Steward Bell
formulatedthefamousBellinequality [2],basedontheassumptionofhiddenvariables.
In 1982 Alain Aspectand his coworkersconductedthefirst experiment[3] with many
more to follow, which violated the Bell inequality. In this sense quantum mechanics
indeedisavalidandcompletetheory.
Measurementson thesubsystemsofan entangledstatecan exhibit astrongcorrela-
tion exceedingany classical approach. Quantum mechanics also states that in general
itisimpossibletomeasurethequantumpropertiesofasystemwithoutchangingthem.
Thevivid research field ofquantum information makes useof thesespecial properties
to improve the quality of communication and information processing tasks. In gen-
eral, aquantumfieldcan bedescribedbythenumber-operatororalternativelybytwo
non-commuting position and momentum-like operators. It depends on the quantum
pictureofthemeasurementapparatus,whichofthetwoisappropriateforthedescrip-
tion of the experiment. The corresponding measurement results have either discrete
or continuous spectra and form the basis of discrete-variable or continuous-variable
quantuminformation, respectively. Withintheframeofthisthesisweworkinthecon-
tinuous variable regime. Here, entangled states of light can be generatedin a reliable8 INTRODUCTION 1.0
and deterministic way by optical parametric amplifiers (OPAs). The statescan be pre-
cisely manipulated with linear optics. Thefinal measurementscan bebased onhighly
efficient balanced homodyne detectors. These entangled two-mode squeezed states
show Gaussian probability distributions and were used for quantum teleportation [4]
and entanglement swapping [5], [6]. In analogy to two-mode squeezed states of the
light field, entangled states of the collective spins of two atomic ensembles have been
generated [7]. Linking light and matter, the storage of quantum states of light in an
atomic memory has been demonstrated[8]. The teleportationfrom a light-based state
onto an atomic ensemble has been reported [9]. High-speed quantum cryptography
with coherent light beams and homodyne detection has been demonstrated [10]. All
thesespectacular achievements revealthegreatpotentialof thisapproach toquantum
informationprocessing.
Thedistributionofentangledstatesoflightoverlongdistancesisamajorchallenge
in the field of quantum information. However, due to the fragile nature of entangled
states, environmental interactions such as optical losses, phase diffusion and mixing
with thermal states lead to decoherence and destroy the non-classical properties after
some finite transmission-line length. Obviously the first strategy to keep these effects
small is to prevent the environmental interaction by using for example optical fibres
withasmallabsorptioncoefficient. Yet,thisstrategyisnotsufficientinmanycases. For
arbitrarydistances,analogoustoclassicalcommunication,repeaterscanbeinsertedinto
thetransmissionline. Quantumrepeater protocols[11],[12]wereproposedtoovercome
this problem. These devices combine quantum memory [8], entanglement distillation
[13], [14] and entanglement swapping [15]. The longer the distance the more repeater
stageshavetobeused. Amissingpieceinthistoolboxhasbeenafeasible protocolfor
entanglementdistillationandpurification.
Entanglement distillation [13, 14] extracts from several shared copies of weakly en-
tangledmixedstatesasinglecopyofahighlyentangledstateusingonlylocalquantum
operations and classical communication (LOCC) between the two parties sharing the
states. This procedure has to be applicable in an iterative way for a long distance sce-
narioorinordertocompletelycounteractthedecoherence. Inthecontinuous-variable
regime this turned out to be a very challenging task. This regime is mainly based on
linear optics,parametricamplification andhomodynedetection. Allthesedevicesper-
form Gaussian operations. This characterizes the class of operations, which preserve
theGaussianpropertiesofastate. ItwasprovedthatitisimpossibletodistilGaussian
entangledstatesbymeansoftheexperimentallyaccessibleGaussianoperations[16,17].
Though, a whole class of important decoherence processes give rise to non-Gaussian
noise and therefore produce non-Gaussian entangled states. It has been shown [18]
thatinthiscasetheentanglementdistillationcanbecarriedoutusingonlyinterference
onbeamsplitters,balancedhomodynedetectionandconditioningonthemeasurement
outcomes. Precursor experiments confirmed this by successful demonstrations of dis-
tillation and purification protocols for single squeezed modes that suffered from de-
Gaussifyingnoise[19,20,21,22].
Withinthisthesistheresultsofthefirstexperimentalimplementationofaniterative