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Theoretical Study of Continuous-Variable Quantum Key Distribution Anthony Leverrier

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Theoretical Study of Continuous-Variable Quantum Key Distribution Anthony Leverrier Theoretical study of continuous-variable quantum key distribution Anthony Leverrier To cite this version: Anthony Leverrier. Theoretical study of continuous-variable quantum key distribution. Atomic Physics [physics.atom-ph]. Télécom ParisTech, 2009. English. tel-00451021 HAL Id: tel-00451021 https://pastel.archives-ouvertes.fr/tel-00451021 Submitted on 28 Jan 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. École Doctorale d’Informatique, Télécommunications et Électronique de Paris Thèse présentée pour obtenir le grade de docteur de l’Ecole Nationale Supérieure des Télécommunications Spécialité : Informatique et Réseaux Anthony LEVERRIER Etude théorique de la distribution quantique de clés à variables continues. Soutenue le 20 novembre 2009 devant le jury composé de Pr. Joseph Boutros Examinateur Pr. Nicolas Cerf Examinateur Pr. Jean Dalibard Examinateur Pr. Philippe Grangier Directeur de thèse Pr. Renato Renner Rapporteur Pr. Jean-Pierre Tillich Rapporteur Pr. Gilles Zémor Directeur de thèse École Doctorale d’Informatique, Télécommunications et Électronique de Paris PhD THESIS prepared and presented at Graduate School of Telecom ParisTech A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF SCIENCE Specialized in Network and Computer Science Anthony LEVERRIER Theoretical study of continuous-variable quantum key distribution. Prof. Joseph Boutros Examiner Prof. Nicolas Cerf Examiner Prof. Jean Dalibard Examiner Prof. Philippe Grangier Advisor Prof. Renato Renner Reviewer Prof. Jean-Pierre Tillich Reviewer Prof. Gilles Zémor Advisor i A ma famille. ii Abstract This thesis is concerned with quantum key distribution (QKD), a cryptographic prim- itive allowing two distant parties, Alice and Bob, to establish a secret key, in spite of the presence of a potential eavesdropper, Eve. Here, we focus on continuous-variable protocols, for which the information is coded in phase-space. The main advantage of these protocols is that their implementation only requires standard telecom components. The security of QKD lies on the laws of quantum physics: an eavesdropper will necessary induce some noise on the communication, therefore revealing her presence. A particularly difficult step of continuous-variable QKD protocols is the “reconcilia- tion” where Alice and Bob use their classical measurement results to agree on a common bit string. We first develop an optimal reconciliation algorithm for the initial protocol, then introduce a new protocol for which the reconciliation problem is automatically taken care of thanks to a discrete modulation. Proving the security of continuous-variable QKD protocols is a challenging problem because these protocols are formally described in an infinite dimensional Hilbert space. A solution is to use all available symmetries of the protocols. In particular, we introduce and study a class of symmetries in phase space, which is particularly relevant for continuous- variable QKD. Finally, we consider finite size effects for these protocols. We especially analyse the influence of parameter estimation on the performance of continuous-variable QDK protocols. Keywords: quantum cryptography, quantum key distribution, quantum optics, quan- tum communication, quantum bit commitment, phase-space representation, information theory, quantum information theory error correcting code. iii Résumé Cette thèse porte sur la distribution quantique de clés, qui est une primitive cryp- tographique qui permet à deux correspondants éloignés, Alice et Bob, d’établir une clé se- crète commune malgré la présence potentielle d’un espion. On s’intéresse notamment aux protocoles “à variables continues” où Alice et Bob encodent l’information dans l’espace des phases. L’intérêt majeur de ces protocoles est qu’ils sont faciles à mettre en œuvre car ils ne requièrent que des composants télécom standards. La sécurité de ces protocoles repose sur les lois de la physique quantique : acquérir de l’information sur les données échangées par Alice et Bob induit nécessairement un bruit qui révèle la présence de l’espion. Une étape particulièrement délicate pour les protocoles à variables continues est la “réconciliation” durant laquelle Alice et Bob utilisent leurs résultats de mesure classiques pour se mettre d’accord sur une chaîne de bits identiques. Nous proposons d’abord un algorithme de réconciliation optimal pour le protocole initial, puis introduisons un nouveau protocole qui résout automatiquement le problème de la réconciliation grâce à l’emploi d’une modulation discrète. Parce que les protocoles à variables continues sont formellement décrits dans un espace de Hilbert de dimension infinie, prouver leur sécurité pose des problèmes ma- thématiques originaux. Nous nous intéressons d’abord à des symétries spécifiques de ces protocoles dans l’espace des phases. Ces symétries permettent de simplifier considérable- ment l’analyse de sécurité. Enfin, nous étudions l’influence des effets de tailles finies, tels que l’estimation du canal quantique, sur les performances des protocoles. Mots-clés : cryptographie quantique, distribution quantique de clés, optique quan- tique, communications quantiques, mise en gage quantique, représentation dans l’espace des phases, théorie de l’information, théorie de l’information quantique, code correcteur d’erreurs. iv Contents Abstract i Résumé iii Remerciements ix List of Publications xii Résumé en français xv Introduction xxxv I From Quantum information to Quantum Key Distribution 1 1 Quantum information and communication 3 1.1 A rapid presentation of Quantum Mechanics . 3 1.1.1 Description of a quantum physical system. 4 1.1.2 Evolution of a physical system . 6 1.2 Information theory: the classical picture . 9 1.2.1 Shannon entropy . 10 1.2.2 Generalization of the Shannon entropy . 11 1.2.3 Operational interpretation: Shannon’s noisy-channel theorem . 12 1.3 Information theory in the quantum age . 17 1.3.1 Encoding quantum information . 17 1.3.2 Communication over a quantum channel . 23 1.3.3 Operational entropic quantities for quantum protocols . 23 v vi CONTENTS 2 Quantum information with continuous variables 27 2.1 Phase space representation . 27 2.1.1 Canonical quantization . 27 2.1.2 Measurements in phase space . 33 2.1.3 Wigner function . 34 2.2 Gaussian states and Gaussian operations . 36 2.2.1 Gaussian states . 36 2.2.2 Gaussian operations . 41 2.2.3 Partial measurements . 44 2.3 Quantum information with continuous variables . 44 2.3.1 von Neumann entropy . 44 2.3.2 Entropy of Gaussian states . 45 2.3.3 Extremality of Gaussian states . 46 2.3.4 Possible tasks and no-go theorems for Gaussian states with Gaus- sian operations . 47 2.3.5 Gaussian states: Hilbert space versus phase space representation . 48 3 Quantum Key Distribution 49 3.1 Quantum Key Distribution . 50 3.1.1 Security of a key . 50 3.1.2 QKD protocols . 51 3.2 Security analysis of QKD . 54 3.3 Continuous-variable QKD . 56 3.3.1 General presentation of continuous-variable protocols . 57 3.3.2 A brief history of CV QKD protocols: from EPR states to coherent states . 58 3.3.3 The GG02 protocol . 59 3.3.4 Security of CV QKD . 61 3.3.5 Estimation of the covariance matrix in the entanglement-based pro- tocol from data observed in the Prepare and Measure protocol . 64 3.3.6 Paranoid versus realistic mode . 67 II Increase the range of continuous-variable QKD 73 4 Reconciliation of correlated Gaussian random variables 75 4.1 Figures of merit for a QKD system. 76 4.2 The reconciliation problem for continuous-variable QKD . 79 4.3 Reconciliation and security . 81 4.4 Reconciliation of binary variables . 83 4.5 Reconciliation of Gaussian variables . 84 4.5.1 Gaussian modulation . 84 4.5.2 Rotations on 1, 3 and 7 ...................... 87 S S S 4.6 Application of the multi-dimensional reconciliation scheme to CVQKD . 88 CONTENTS vii 4.7 Conclusion and open questions . 90 5 Long distance CVQKD: protocols with a discrete modulation 93 5.1 Longer distances mean lower SNR . 94 5.2 Reconciliation at very low SNR . 96 5.3 Presentation of the new discrete-modulation protocols . 102 5.3.1 CV QKD protocols with a discrete modulation . 102 5.3.2 General outline of security proofs against collective attacks . 104 5.4 Security of the two-state protocol . 105 5.5 Security of the four-state protocol . 108 5.6 Performances of the protocols . 111 5.7 Remaining issues . 114 5.7.1 Potential issue with reverse reconciliation . 114 5.7.2 Other potential issues for long distance CV QKD . 115 5.8 Perspectives: convergence between DV and CV QKD at long distance . 117 III Security of continuous-variable quantum cryptography 119 6 Are collective attacks optimal? 121 6.1 Strategy for a proof . 122 6.1.1 Collective versus general attacks . 122 6.1.2 Symmetrization . 124 6.1.3 Symmetric states versus i.i.d. states . 126 6.2 Symmetries in phase space . 129 6.2.1 A symmetry group in phase space . 130 6.2.2 Single-party case: main properties of orthogonal invariance in phase space . 131 6.2.3 Bipartite case: application to continuous-variable QKD . 132 6.3 de Finetti theorem and postselection procedure in phase space . 134 6.3.1 de Finetti theorem in phase space representation . 135 6.3.2 Postselection technique in phase space . 139 6.4 Possible approaches to prove the unconditional security of CVQKD . 141 6.4.1 Characterization of isotropic states in phase space . 141 6.4.2 Further than symmetrization and postselection .
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