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Operational tools for moment characterization, entanglement verification and quantum key distribution [Elektronische Ressource] / vorgelegt von Moroder Tobias

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119 pages
Operational tools for momentcharacterization, entanglementverification and quantum key distributionDer Naturwissenschaftlichen Fakult¨atder Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnbergzurErlangung des Doktorgradesvorgelegt vonMoroder Tobiasaus ErlangenAls Dissertation genehmigt von der Naturwissen-schaftlichen Fakultat der Universitat Erlangen-Nurnberg¨ ¨ ¨Tag der mundlichen Prufung: 31.07.2009¨ ¨Vorsitzender der Promotionskommission: Prof. Dr. Eberhard Bansch¨Erstberichterstatter: Prof. Dr. Norbert Lutkenhaus¨Zweitberichterstatter: Prof. Dr. Hajo LeschkeDrittberichterstatter: Prof. Dr. Maciej LewensteiniSummaryIn this thesis we address several different topics within the field of quantuminformation theory. These results can be classified to either enhance the appli-cability of certain conceptual ideas to be more suited for an actual experimentalsituation or to ease the analysis for further investigation of central problems. Indetail, the present thesis contains the following achievements:We start our discussion with the question under which conditions a givenset of expectation values is compatible with the first and second moments ofthe spin operators of a generic spin j state. The aim lies on an operationaldescription that allows working with the moments directly rather than using adensity operator.
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Operational tools for moment
characterization, entanglement
verification and quantum key distribution
Der Naturwissenschaftlichen Fakult¨at
der Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg
zur
Erlangung des Doktorgrades
vorgelegt von
Moroder Tobias
aus ErlangenAls Dissertation genehmigt von der Naturwissen-
schaftlichen Fakultat der Universitat Erlangen-Nurnberg¨ ¨ ¨
Tag der mundlichen Prufung: 31.07.2009¨ ¨
Vorsitzender der Promotionskommission: Prof. Dr. Eberhard Bansch¨
Erstberichterstatter: Prof. Dr. Norbert Lutkenhaus¨
Zweitberichterstatter: Prof. Dr. Hajo Leschke
Drittberichterstatter: Prof. Dr. Maciej Lewensteini
Summary
In this thesis we address several different topics within the field of quantum
information theory. These results can be classified to either enhance the appli-
cability of certain conceptual ideas to be more suited for an actual experimental
situation or to ease the analysis for further investigation of central problems. In
detail, the present thesis contains the following achievements:
We start our discussion with the question under which conditions a given
set of expectation values is compatible with the first and second moments of
the spin operators of a generic spin j state. The aim lies on an operational
description that allows working with the moments directly rather than using a
density operator. We link this characterization of physical moments to the Bose-
symmetric extension problem for a particular two qubit state that is completely
determined by the given moments. Via this reformulation we can provide opera-
tional sub- and superset approximations in order to identify moments which are
assured to be physical and others which are clearly incompatible with quantum
mechanics. We show that this operational approximate solution becomes more
accurate for increasing total spin numbersj and converges to the exact solution
in the limiting case.
Another part deals with the theoretical concept of entanglement witnesses;
the most common and applicable tool for entanglement detection. In particular,
we concentrate how to improve the detection strength of a linear entanglement
witness by nonlinear terms, i.e., such that one detects more entangled states
at once. The considered improvement method has the advantage that it can be
employed multiple times such that it generates a sequence of nonlinear entan-
glement witnesses with higher nonlinearities and which become stronger in each
step. This developed iteration process can be employed for different tasks and
we analyze two distinguished cases: Either we optimize the iteration method for
a given target state or we try to improve the entanglement witness with respect
to all entangled states equally.
In the remaining parts we discuss different options in order to make already
existing ideas more applicable for actual experiments, since most of the famous
applications in quantum information theory have only been introduced on a
very idealized level and hence are not directly valid for the real experiment. Of
course this is of particular importance for the field of quantum key distribution,
because one naturally likes to conclude unconditional security also for a key
generated in a real experiment and not just for a simplified version of it. With
this goal in mind we first investigate the theoretical concept of a squash model,
that represents an elegant “evaluation trick” to directly apply for instance the
security analysis of an idealized quantum key distribution protocol to the real
experiment. However this concept even helps with other tasks like detecting or
quantifying entanglement. We develop a formalism to check whether a given
realistic measurement device has such a squash model or not and provide rele-
vant detection schemes with and without this particular property. In contrast
to this theoretical model we also address an experimental option which equally
well provides security of a realistic quantum key distribution experiment by just
using the idealized version of it. We exploit the idea that one can combine a
variable beam splitter with a simple click/no-click detector in order to achieve
the statistics of a photon number resolving detector. Via this hardware change it
is straightforward to estimate the crucial parameters for the security statement.ii
Using this technique provides a way to even prove security of realistic quantum
key distribution experiments for which the theoretical squash model idea does
not work. This results for example in a new distance record for a quantum key
distribution experiment with present technology. As a last result along these
lines we focus on experimental entanglement verification. Considering the mere
question of entanglement verification this practicality issue occurs since one of-
ten uses—because of various reasons—an oversimplified model for the performed
measurements. We show that via such a misinterpretation of the measurement
results one can indeed make mistakes, nevertheless we are more interested in
conditions under which such errors can be excluded. For that we introduce and
investigate a similar, but less restrictive, concept of the squash model. As an ap-
plication we show that the usual tomography entanglement test, typically used
in parametric down-conversion or even multipartite photonic experiments, can
easily be made error-free.iii
Zusammenfassung
In der vorliegenden Arbeit behandeln wir verschiedene Themen aus dem Gebiet
der Quanteninformationstheorie. Die unterschiedlichen Ergebnisse sollen entwe-
der die Anwendung theoretischer Konzepte auf reale Experimente erleichtern
oder bei der weiterfu¨hrenden Arbeit an zentralen Fragen helfen. Im Detail wer-
den folgende Punkte genauer untersucht:
Wir wenden uns als Erstes der Fragestellung zu, unter welchen Vorraus-
setzungen eine gegebene Menge an Erwartungswerten mit den ersten beiden
Momenten der Spinoperatoren eines Zustandes von Gesamtspin j kompatible
ist. Hierbei liegt das Hauptaugenmerk auf einer operationellen Beschreibung,
so dass man direkt mit den Momenten arbeiten kann, anstatt den kompletten
Dichteoperator zu verwenden. Wir zeigen, dass diese Charakterisierung physi-
kalischer Momente zu dem Problem der Bose-symmetrischen Erweiterungen fu¨r
speziell bestimmte Zweiqubit Zust¨ande a¨quivalent ist. Mittels dieser Umformu-
¨lierung bestimmen wir operationelle Teil- und Ubermengen der exakten Menge,
um somit auf einfache Art und Weise Momente identifizieren zu ko¨nnen, die
sicher physikalisch sind, als auch andere, die eindeutig inkompatibel mit den
Vorhersagen der Quantenmechanik sind. Diese approximative Lo¨sung des Pro-
blems gewinnt mit steigender Gesamtspinzahlj an Genauigkeit, und konvergiert
im Limes gegen die exakte L¨osung.
Im na¨chsten Abschnitt bescha¨ftigen wir uns mit dem theoretischen Konzept
der Verschr¨ankungszeugen; der gebra¨uchlichsten und anwendungsfreundlichsten
Methode zur Verschr¨ankungsdetektion. Wir untersuchen, wie man die Qualita¨t
eines linearen Verschr¨ankungszeugen durch quadratische, nichtlineare Terme
verbessen kann, so dass man mehr verschr¨ankte Zusta¨nde auf einmal detektiert.
Das betrachtete Verbessungsschema hat den Vorteil dass man es iterative be-
nutzen kann. Dies liefert eine Abfolge von nichtlinearen Verschra¨nkungszeugen
mit Nichtlinearita¨ten immer ho¨herer Ordnung, deren St¨arke in jedem Schritt
zunimmt. Die entwickelte Iterationsmethode kann fu¨r verschiedene Aufgaben
verwendet werden und wir betrachten zwei ausgezeichnete Extremfa¨lle: Ent-
weder optimieren wir die Iterationsmethode fur einen vorgegeben Zielzustand¨
oder wir verbessern den Verschrankungszeugen gleichmassig in Bezug auf alle¨ ¨
moglichen Zustande.¨ ¨
Im restlichen Teil der Arbeit untersuchen wir verschiedene Mo¨glichkeiten,
wie man bereits vorhandene Ideen besser an reale Experimente anpasst. Hier
sei angemerkt dass viele Anwendungen der Quanteninformationstheorie nur fu¨r
stark idealisierte Systeme vorgeschlagen und diskutiert wurden und somit nicht
direkt fu¨r das Experiment anwendbar sind. Dieser Punkt ist fu¨r das Feld der
Quantenschlu¨sselverteilung von besonderer Bedeutung, da man die vorbehalts-
lose Sicherheit eines realen, durch ein Experiment generierten Schlu¨ssels zeigen
mochte und nicht nur fur die einer idealisierten Version davon. Mit diesem Ziel¨ ¨
im Hinterkopf untersuchen wir zuerst die theoretische Idee des sogenannten
Squash Modells. Dieses bietet einen eleganten “Ausrechentrick”, der es erlaubt,
die Sicherheitsanalyse eines idealisierten Quantenschlusselverteilungsprotokolls¨
direkt fur das wirkliche Experiment zu verwenden. Dennoch sei angemerkt, dass¨
dieses Modell auch bei anderen Fragestellungen wie zum Beispiel der Verifizie-
rung oder der Quantifizierung von Verschrankung hilft. Wir entwickeln einen¨
geeigneten Formalismus zur Prufung dieses Squash Modells fur einen gegebe-¨ ¨
nen Messaufbau und betrachten physikalisch relevante Messschemen mit undiv
ohne dieser Eigenschaft. Im Gegensatz zu diesem theoretischen Modell behand-
len wir auch eine experimentelle Moglichkeit, die es uns erlaubt, die Sicher-¨
heitsanalyse des idealisierten Systems direkt fur das realistische Experiment zu¨
verwenden. Dazu benutzen wir die Idee, dass man durch die Kombination ei-
nes variablen Strahlteilers mit einem einfachen “Click/No-Click” Detektor, die
Statistik von photonauflosenden Detektoren erhalten kann. Durch diese Hard-¨
wareanderung ist es moglich, die sicherheitsrelevanten Parameter des Quan-¨ ¨
tenschlusselverteilungsexperiments direkt abzuschatzen. Diese Methode bietet¨ ¨
somit auch einen Weg, die Sicherheit eines realen Quantenschlusselverteilungs-¨
experiments zu beweisen, fu¨r das die theoretische Squash Modell Idee nicht
funktioniert. Als Ergebnis erha¨lt man unter anderem einen neuen Distanzre-
kord fu¨r ein reales Quantenschlu¨sselverteilungsexperiment mit derzeitiger Tech-
nologie. Als letzten Punkt betrachten wir das Problem der experimentellen Ver-
schra¨nkungsverifizierung. Bei dieser Verifizierung hat man oftmals ein Prakti-
kabilita¨tsproblem, weil man—ob verschiedener Gru¨nde—meistens nur ein sehr
vereinfachtes Modell fu¨r die verwendenten Messapparate benutzt. Wir zeigen,
dass man in diesem Fall der Fehlinterpretation der beobachteten Messausga¨nge
Fehler machen kann, allerdings untersuchen wir in erster Linie Bedingungen, un-
ter denen ein f¨alschlicher Verschr¨ankungsnachweis ausgeschlossen werden kann.
Zu diesem Zweck verwenden wir ein dem Squash Modell a¨hnliches, aber weniger
restriktives Konzept. Eine interessante Anwendung liefern die ublicherweise mit-¨
tels Tomographie durchgefuhrten Verschrankungstests, die man typischerweise¨ ¨
fur parametrische Fluoreszenz Quellen oder fur photonische Mehrparteienexpe-¨ ¨
rimente verwendet. Wir zeigen, dass diese sehr einfach fehlerfrei gemacht werden
konnen.¨Contents
1 Introduction 1
2 Truncated moment problem 5
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Expectation value matrix . . . . . . . . . . . . . . . . . . 8
2.2.3 Standard form . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Different representations . . . . . . . . . . . . . . . . . . . 10
2.3.2 Two qubit reduction . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 First moment problem . . . . . . . . . . . . . . . . . . . . 13
2.4 Approximation techniques . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Approximating subset . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Approximating superset . . . . . . . . . . . . . . . . . . . 15
2.5 Discussion of generic problem . . . . . . . . . . . . . . . . . . . . 17
2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Additional comments . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Nonlinear entanglement witnesses 23
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Iteration process . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Definition, properties and an example iteration . . . . . . 28
3.3 Optimized iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Averaged iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5.1 Optimized iteration . . . . . . . . . . . . . . . . . . . . . 35
3.5.2 Random iteration . . . . . . . . . . . . . . . . . . . . . . . 38
3.5.3 Averaged iteration . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Squash model 41
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Squash model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . 42
vvi CONTENTS
4.2.2 QKD application . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 QKD measurements . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Polarization measurement . . . . . . . . . . . . . . . . . . 47
4.3.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.3 BB84 and 6-state results . . . . . . . . . . . . . . . . . . . 50
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Entanglement verification 55
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Ion trap example . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Positive squashing operation . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . 58
5.3.2 Entanglement verification . . . . . . . . . . . . . . . . . . 59
5.3.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 Example: Polarization measurement . . . . . . . . . . . . . . . . 62
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Detector decoy quantum key distribution 67
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Estimating photon number statistics . . . . . . . . . . . . . . . . 69
6.2.1 Main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2.2 Finite settings . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.3 Realistic detectors . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Entanglement based QKD schemes . . . . . . . . . . . . . . . . . 74
6.3.1 Security analysis . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Detector decoy estimation . . . . . . . . . . . . . . . . . . . . . . 79
6.4.1 Simple detector decoy setup . . . . . . . . . . . . . . . . . 79
6.4.2 Refined detector decoy setup . . . . . . . . . . . . . . . . 81
6.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5.1 Data simulation . . . . . . . . . . . . . . . . . . . . . . . 82
6.5.2 Resulting key rates . . . . . . . . . . . . . . . . . . . . . . 84
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 Final remarks 91
Bibliography 93
Acknowledgments 105
Publication list 107
Curriculum Vitae 109Chapter 1
Introduction
Quantum information theory represents a recent interdisciplinary research field.
Motivated by a few key ideas like Shor’s factoring algorithm or the idea of quan-
tum key distribution for secure communication, this field has drawn interest by
the insight that quantum mechanics offers, rather than being a disturbance effect
in devices of always shrinking size, the potential for fascinating new applications
which are impossible otherwise. Hence the present time can be compared with
the end of the nineteenth century when it was realized that electromagnetism
is not just a mere physical sandbox but instead represents a theory from which
one can benefit greatly, and in fact our past century was dominated by all the
applications which emerged from this theory.
Because of its interdisciplinary setting there are many different research areas
which are all combined under the term of quantum information theory; clearly
some parts are more suited for physicists than others. In this introduction we
would like to motivate some of those research topics which are discussed more
thoroughly within the subsequent chapters of this thesis.
Like any other theory of physics quantum information rests on a few basic
principles and a consequent formalism how to describe, explain and predict the
observations from an experiment. From this underlying formalism one typically
develops different working tools in order to simplify the investigation and analy-
sis of further problems. Obviously each of these working tools has its advantage
for particular applications, but in general it is always good to have various op-
tions to choose from. Consider for instance the problem how to describe the
state of a system. Here one can use for example the density operator or a
quasiprobability function to describe the state in a phase-space representation.
Although both tools describe the same physical state, the density operator has
its virtue for low-dimensional systems whereas a quasiprobability function pro-
vides an easier description for certain infinite dimensional states. Let us point
out that each working tool should also be operational, i.e., the advantage should
be of an useable form. As an example of such a particular working tool we ad-
dress the question how to operationally describe the information from a system.
In many occasions it happens that one is only interested in some properties of
the system, like the expectation values of just a handful of different operators
which are actually measured in the experiment. Consequently one also likes to
have a formalism which describes such partial information of the entire system
more compactly than just given by the usual density operator, which includes,
12 CHAPTER1. INTRODUCTION
in this case, lots of redundant and unknown information that also increases with
the dimension of the system. An example of such a compact tool is given by the
covariance matrix that provides an operational information description if one is
only interested in the first and second moments of the position and momentum
operator. Inspired by this remarkable example we examine in Chap. 2 the anal-
ogous problem how to operationally describe the first two moments of the spin
operators of a generic spin-j system. We link this information characterization
to a well-known extension problem for low-dimensional quantum states and con-
sequently can formulate operational approximations. This description becomes
in particular convenient for very large spin numbersj and indeed becomes exact
in the limiting case of an infinite total spin number. For very small systems we
present an alternative reformulation of the problem, which can easily be tackled
by numerical means, and compare it with the operational approximations.
As for any other theory there are still some open questions in quantum
information theory about very basic concepts. It is obvious that one first needs
to attain knowledge of the underlying principles in order to see how one is able to
exploit them in applications. This is even more important for the fundamental
resources. Clearly entanglement represents one, if not the most important, of
these resources of quantum mechanics and indeed it has already been employed
for various different ideas like teleportation or superdense coding. Nevertheless,
very basic properties of this resource are not fully explored yet. Already the
quite simple sounding question whether a given state is entangled or not, known
as the separability problem, causes severe problems and an operational solution
is only known for particular cases. More specific questions how to quantify,
distill or protect entanglement are even less investigated, but are clearly equally
important for further studies. In Chap. 3 we investigate one of the most powerful
and applied tools for the separability problem, known as entanglement witnesses.
These are just linear operators that “witness” entanglement of the underlying
state if their expectation value is beyond a certain threshold. However a single
witness is unfortunately only capable to detect a rather small fraction of all
entangled states, and consequently one needs to evaluate much more than one
of theses operators. In order to overcome this drawback we describe a generic
way how to improve the detection strength of an entanglement witness by adding
a nonlinear, quadratic term. This method can be iterated and hence generates
a whole sequence of nonlinear entanglement witnesses that become stronger
in each step. We optimize this iteration according to various objectives and
investigate which fraction of entangled states can be detected by this method
in the end.
Another working area in the field of quantum information deals with the
task how to make proposed applications more suited for an actual experiment.
At this point it is important to stress that most of these applications, like quan-
tum key distribution as a prominent example, have been introduced on a rather
abstract level. In this more conceptual context it is often easier to discuss its
advantages and to perform the necessary analysis, e.g., for quantum key dis-
tribution this analysis refers to the precise method to bound the eavesdroppers
information on the raw key from the observed data. However, in any real exper-
iment the situation is often far from this idealized description. Any real device
suffers from various types of imperfections and noise while certain requirements,
like a single photon source, are simply beyond our present experimental capa-
bilities. This is by far not a minor point, since these differences between theory