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Experimental Analysis of Einstein-Podolsky-Rosen Steering

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Experimental Analysis of Einstein-Podolsky-Rosen Steering EXPERIMENTALANALYSISOF EINSTEIN-PODOLSKY-ROSENSTEERINGFOR QUANTUMINFORMATIONAPPLICATIONS Von der QUEST-Leibniz-Forschungsschule der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften -Dr. rer. nat.- genehmigte Dissertation von DIPL.-PHYS.VITUSHÄNDCHEN 2016 Referent : Prof. Dr. rer. nat. Roman Schnabel Institut für Laserphysik, Hamburg Korreferent : Prof. Dr. rer. nat. Reinhard F. Werner Institut für Theoretische Physik, Hannover Tag der Promotion : 04.03.2016 ABSTRACT The first description of quantum steering goes back to Schrödinger. In 1935 he named it a necessary and indispensable feature of quantum mechanics in response to Einstein’s concerns about the completeness of quantum theory. In recent years investigations of the effect have ex- perienced a revival, as it turned out that steering is a distinct subclass of entanglement. It is strictly stronger than genuine entanglement, meaning that all steerable states are entangled but not vice versa, and strictly weaker than the violation of Bell inequalities. Additionally, the violation of a steering inequality has an intrinsic asymmetry due to the directional construction. One party seemingly steers the other and their roles are in general not interchangeable. The first direct ob- servation of this asymmetry is presented in this thesis. It was demon- strated that certain Gaussian quantum optical states show steering from one party to the other but not in the opposite direction. This is a profound result, as the experiment proves that the class of steerable states itself divides into distinct subclasses. Furthermore, steering found major applications in quantum infor- mation. For example in the field of quantum key distribution (QKD) the non-classical correlations can be exploited to establish a secret, cryptographic key. Thereby, steering allows one-sided device inde- pendent security, since it violates a description with certain classical models. This thesis reports on the final results of the QKD project at the Leibniz University Hannover that aimed at the generation of a secure quantum key based on mutually entangled conjugate pairs of continuous variables. The protocol of a recent security proof was im- plemented in a table-top setup and nearly 100 MBit of usable key with composable and one-sided device independent security against coher- ent attacks were generated. The security analysis included finite size effects, and a newly developed hybrid reconciliation algorithm was used for efficient error correction. Furthermore, a loss study on the entanglement distribution showed that separation of the two detec- tors by a 5 km optical fibre would be possible. Therefore, the results of this work are a major step towards continuous variable QKD with state-of-the-art security in local area networks. Key words: steering, squeezed light, quantum key distribution iii KURZFASSUNG Die erste Beschreibung von Quanten-Steering (z.dt. Quantenlenkung) geht auf Schrödinger zurück. Im Jahr 1935 nannte er es ein notwendi- ges und unverzichtbares Merkmal der Quantenmechanik in Antwort auf Einsteins Bedenken bezüglich der Vollständigkeit der Quanten- theorie. In den letzten Jahren erlebten die Untersuchungen des Ef- fekts einen Aufschwung, da es sich herausstellte, dass Steering eine ausgeprägte Unterklasse der Verschränkung darstellt. Es ist strikt stärker als originäre Verschränkung, d.h. alle Steering Zustände sind verschränkt aber nicht umgekehrt, und strinkt schwächer als die Ver- letzung Bellscher Ungleichungen. Zusätzlich hat die Verletzung einer Steering Ungleichung eine intrinsische Asymmetrie aufgrund der rich- tungsabhängigen Konstruktion. Eine Partei lenkt scheinbar die an- dere und ihre Rollen sind im Allgemeinen nicht vertauschbar. In dieser Arbeit wird die erste direkte Beobachtung dieser Asymmetrie präsentiert. Es wurde gezeigt, dass bestimmte Gaußsche Zustände in der Quantenoptik Steering von einer Partei zur anderen aufweisen, aber nicht in umgekehrter Richtung. Dies ist ein tiefgreifendes Re- sultat, da das Experiment beweist, dass sich die Klasse der Steering Zustände selbst in unterscheidbare Unterklassen aufteilt. Darüber hinaus hat Steering wichtige Anwendungen in der Quan- teninformation gefunden. Zum Beispiel können im Bereich der Quan- tenschlüsselverteilung (QKD) die nicht-klassischen Korrelationen aus- genutzt werden um einen geheimen, kryptographischen Schlüssel zu erzeugen. Dabei erlaubt Steering die einseitige Geräteunabhängigkeit der Sicherheit, da es die Beschreibung durch bestimmte klassische Modelle verletzt. In dieser Arbeit werden die finalen Ergebnisse des QKD Projekts an der Leibniz Universität Hannover vorgestellt, wel- ches die Erzeugung eines sicheren Quantenschlüssels auf Basis ge- meinsam verschränkter, konjugierter Paare kontinuierlicher Variablen zum Ziel hatte. Das Protokoll eines aktuellen Sicherheitsbeweises wur- de in einem Tischaufbau umgesetzt und fast 100 MBit verwendbarer Schlüssel mit zusammensetzbarer und einseitig geräteunabhängiger Sicherheit gegen kohärente Angriffe wurden erzeugt. Die Sicherheits- analyse schloss Effekte der endlichen Größe des Schlüssels mit ein und ein neu entwickelter, hybrider Abgleichalgorithmus wurde zur effizienten Fehlerkorrektur verwendet. Des Weiteren zeigte eine Ver- luststudie an der Verschränkungsverteilung, dass eine Separierung der beiden Detektoren durch 5 km Glasfaserkabel möglich wäre. Die Ergebnisse dieser Arbeit sind daher ein bedeutender Schritt für die Anwendung von QKD mit kontinuierlichen Variablen und modern- ster Sicherheit in lokalen Netzwerken. Schlagworte: Quantenlenkung, gequetschtes Licht, Quantenschlüssel- verteilung v CONTENTS abstract iii kurzfassung v list of figures x list of acronyms xi 1 introduction1 2 quantum optics5 2.1 Foundations of Quantum Mechanics 5 2.1.1 Planck’s Law and the Photo-Electric Effect 5 2.1.2 Description and Interpretation of Quantum States 7 2.1.3 Heisenberg’s Uncertainty Principle 11 2.2 Quantization of the Electromagnetic Field 14 2.2.1 The Classical Electromagnetic Field 14 2.2.2 The Quantum-Mechanical Harmonic Oscillator 16 2.3 Hilbert Spaces for the Description of Light 18 2.3.1 Number States 18 2.3.2 Coherent States 21 2.3.3 Squeezed States 24 2.4 The Quadrature Phase Space 30 2.4.1 Quadrature Operators 30 2.4.2 Statistical Moments and the Covariance Matrix 34 2.4.3 Properties of the Covariance Matrix for Physical States 39 3 detection of optical quantum states 43 3.1 Gaussian Optics 43 3.1.1 Fundamental Solution of the Helmholtz Equa- tion 43 3.1.2 Higher Order Transversal Modes 44 3.1.3 Implications for Experiments 45 3.2 Mixing of Different Modes 48 3.2.1 The Classical Beam Splitter 48 3.2.2 The Quantum Optical Beam Splitter 49 3.2.3 Symplectic Beam Splitter Transformations 49 3.3 Balanced Homodyne Detection 50 3.3.1 Description of the Fields at the Homodyne Beam Splitter 51 3.3.2 Detection of Quadrature Amplitudes 52 vii viii contents 3.4 Expectation Values and Variances of Homodyne Sig- nals 54 3.4.1 Measurement on the Vacuum State 54 3.4.2 Measurement on the Squeezed Vacuum State 55 3.4.3 Measurement on the Coherent State 58 3.5 Reconstruction of the Covariance Matrix 60 4 the notion of entanglement 63 4.1 The Einstein-Podolsky-Rosen Paradox 63 4.2 Classification of Entanglement 65 4.2.1 Genuine Entanglement 65 4.2.2 Violation of a Bell Inequality 66 4.2.3 Einstein-Podolsky-Rosen Steering 68 4.3 Directionality of Gaussian Steering 70 4.4 Two-Mode Squeezed States and the EPR-Reid Crite- rion 73 5 demonstration of one-way einstein-podolsky-rosen steering 81 5.1 Experimental Preliminaries 81 5.1.1 Generation of Squeezed States 81 5.1.2 Generation of Two-Mode Squeezed States 85 5.2 Demonstration of One-Way Steering 87 5.3 Connection to Other Entanglement Applications 89 5.3.1 Tripartite and Multipartite Steering 89 5.3.2 Entanglement Distribution by Separable States 91 5.3.3 Data Reconciliation in Quantum Key Distribu- tion 91 6 application of two-mode squeezed states for qkd 93 6.1 A Brief Introduction to QKD 93 6.1.1 Classification of QKD Schemes 95 6.1.2 The Continuous Variable QKD Project at the Leib- niz University Hannover 97 6.2 CV-QKD with 1sDI Security Against Coherent Attacks 98 6.2.1 The QKD Protocol 99 6.2.2 Implementation of the Protocol 102 6.2.3 Secret Key Distribution Results 108 6.3 Implementation of a 1 km Fiber Based Quantum Chan- nel 111 6.3.1 Concepts for a Remote Homodyne Detector 112 6.3.2 Scheme for Dynamic Polarization Control 114 6.3.3 Distribution of a Squeezed Field Through a 1 km Fiber 118 6.3.4 Distribution of Entanglement Through a 1 km Fiber 121 contents ix 7 conclusion and prospect 125 a mathematical and computational details 129 a.1 Influence of the Electronic Demodulation Phase on the Homodyne Signals 129 a.2 Derivation of Analytic Functions for Gaussian Steer- ing 130 a.3 Mathematica Notebook to Investigate the DPC 131 bibliography 141 curiculum vitae 157 list of publications 159 LISTOFFIGURES Figure 2.1 Coherent state in the quadrature phase space 32 Figure 2.2 Squeezed state in the quadrature phase space 33 Figure 3.1 Intensity distribution of a Ψ12 Hermite-Gaussian mode 45 Figure 3.2 Intensity distribution of a Ψ22 Laguerre-Gaussian mode 46 Figure 3.3 Schematic of a generic mode matching setup 47 Figure 3.4 Schematic of a balanced homodyne detector 50 Figure 4.1 Implications of inseparability criteria 71 Figure 4.2 Gaussian one-way EPR steering visualized in phase space 72 Figure 4.3 Quadrature correlations of a two-mode squeezed state 77 Figure 5.1 Energy levels of
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