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Quantum Communication in Noisy Environments Quantum communication in noisy environments Hans Aschauer M¨unchen2004 Quantum communication in noisy environments Hans Aschauer Dissertation an der Sektion Physik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Hans Aschauer aus Bad Reichenhall M¨unchen, den 30. Januar 2004 Erstgutachter: Hans J. Briegel Zweitgutachter: Ignacio Cirac Tag der m¨undlichen Pr¨ufung:27. April 2005 Contents Zusammenfassung xi 1 Introduction 1 2 Noisy quantum operations and channels 7 2.1 Quantum states, operations, and measurements ........ 7 2.1.1 Quantum states ...................... 7 2.1.2 Quantum operations ................... 8 2.1.3 Quantum state measurements .............. 9 2.2 Entanglement and quantum channels .............. 10 2.2.1 Composite quantum systems ............... 10 2.2.2 Separable and entangled states ............. 12 2.2.3 Quantum teleportation .................. 13 2.3 Noise in quantum mechanics ................... 14 2.3.1 Noise channels ....................... 14 2.3.2 Teleportation with imperfect EPR pairs ........ 16 2.4 Entanglement purification .................... 17 2.4.1 2-Way Entanglement Purification Protocols ...... 18 2.4.2 Purification with imperfect apparatus .......... 21 2.4.3 The quantum repeater .................. 24 2.5 Quantum error correcting codes ................. 27 2.5.1 Classical codes ...................... 27 2.5.2 The Shor code ....................... 28 2.5.3 CSS codes and stabilizer codes .............. 31 2.5.4 Errors and quantum error correcting codes ....... 33 2.6 Quantum cryptography ...................... 33 2.6.1 The BB84 Protocol .................... 35 2.6.2 The Ekert protocol .................... 36 vi CONTENTS 2.6.3 Security Proofs ...................... 37 3 Factorization of Eve 41 3.1 The security proof ........................ 41 3.1.1 The effect of noise .................... 42 3.1.2 Binary pairs ........................ 46 3.1.3 Bell-diagonal initial states ................ 51 3.1.4 Numerical results ..................... 55 3.1.5 Non-Bell-diagonal pairs .................. 58 3.2 How to calculate the flag update function ............ 64 3.2.1 Unitary transformations and errors ........... 64 3.2.2 Measurements and measurement errors ......... 66 3.2.3 The reset rule ....................... 67 3.3 Discussion ............................. 69 4 Cluster state purification 71 4.1 The cluster purification protocol ................. 72 4.1.1 Cluster states ....................... 72 4.1.2 Description and analytical treatment of the protocol .. 73 4.1.3 Numerical analysis of the protocol ............ 75 4.1.4 Results ........................... 76 4.2 Noisy operations ......................... 77 4.2.1 One-qubit white noise .................. 78 4.2.2 Results ........................... 80 4.3 On the security of the protocol ................. 80 4.3.1 The flag update function ................. 82 4.3.2 The conditional fidelity .................. 84 4.4 Generalized cluster states .................... 86 5 Entanglement purification protocols from quantum codes 89 5.1 Creating purification protocols from coding circuits ...... 90 5.1.1 Encoding and decoding .................. 90 5.1.2 Error detection vs. error correction ........... 93 5.2 The hashing protocol and quantum codes ............ 94 5.3 Numerical results ......................... 98 5.3.1 Purification curves .................... 98 5.3.2 Efficiency of the protocols ................ 102 Contents vii 6 The tensorspace software library 109 6.1 Introduction ............................ 109 6.2 Basic concepts ........................... 110 6.2.1 Parties ........................... 110 6.2.2 States ........................... 111 6.2.3 Operations ......................... 112 6.3 Diagonal density operators .................... 113 6.4 qtensorspace by examples ................... 114 6.4.1 CNOT operation ..................... 114 6.4.2 Teleportation ....................... 115 6.4.3 Teleportation using noisy EPR pairs .......... 116 6.4.4 Cluster states ....................... 117 6.5 Mathematics of qtensors ..................... 119 7 Local invariants for multi-partite quantum states 121 7.1 State tomography ......................... 122 7.2 Invariant decomposition of the state space ........... 123 7.3 Other invariants .......................... 128 viii Contents List of Figures 2.1 The entanglement purification protocol and process. ...... 19 2.2 The purification curve for the IBM protocol .......... 22 2.3 The nested entanglement purification protocol ......... 26 2.4 Quantum logic network of the Shor code ............ 30 2.5 The Ekert protocol ........................ 37 3.1 The lab demon .......................... 44 3.2 Entanglement purification with binary pairs .......... 49 3.3 Fixpoint map for binary entanglement purification ....... 50 3.4 Illustration of the purification curve for variouse noise levels . 52 3.5 The actual purification curve ................... 53 3.6 Evolution of the Bell-diagonal elements in the subensembles . 56 3.7 Evolution of the fidelity and conditional fidelity ........ 57 3.8 Three purification regimes .................... 59 3.9 Fidelity ”phase-transition” at the purification threshold .... 60 3.10 Efficiency of the purification protocol .............. 61 4.1 The cluster purification protocol ................. 73 4.2 Combinations of the sub-protocols ................ 75 4.3 Minimum fidelity in the cluster purification protocol ...... 77 4.4 Fidelities F and F cond in the cluster purification protocol ... 86 4.5 Graph states ............................ 87 5.1 Equivalence between quantum coding/decoding and EPP ... 91 5.2 An example of an encoding circuit ................ 92 5.3 A map of quantum error correcting codes. ........... 97 5.4 Purification curves for the [[5, 1, 3]] and the [[11, 1, 5]] EDM-EPP.101 5.5 Purification curves of the noisy[[5, 1, 3]] EDM and ECM EPP . 103 x List of Figures 5.6 Purification curves of the noisy [[11, 1, 5]] EDM and ECM EPP 104 5.7 Efficiency of the protocols without errors ............ 106 Abstract In this thesis, we investigate how protocols in quantum communication theory are influenced by noise. Specifically, we take into account noise during the transmis- sion of quantum information and noise during the processing of quantum infor- mation. We describe three novel quantum communication protocols which can be accomplished efficiently in a noisy environment: (1) Factorization of Eve: We show that it is possible to disentangle transmitted qubits a posteriori from the quantum channel’s degrees of freedom. (2) Cluster state purification: We give multi-partite entanglement purification protocols for a large class of entangled quantum states. (3) Entanglement purification protocols from quantum codes: We describe a constructive method to create bipartite entanglement purification protocols form quantum error correcting codes, and investigate the properties of these protocols, which can be operated in two different modes, which are related to quantum communication and quantum computation protocols, respectively. In dieser Arbeit wird untersucht, wie Quantenkommunikationsprotokolle durch Rauschen beeinflusst werden. Insbesondere berucksichtigen¨ wir Rauschen w¨ahrend der Ubertragung¨ der Quanteninformation und Rauschen w¨ahrend ihrer Verar- beitung. Wir beschreiben drei neue Quantenkommunikationsprotokolle, die in ei- ner verrauschten Umgebung effizient umgesetzt werden k¨onnen: (1) Abfaktori- sierung von Eve: Wir zeigen, dass es m¨oglich ist, bereits ubertragene¨ Qubits nachtr¨aglich von Freiheitsgraden des Kommunikationskanals zu entschr¨anken. (2) Cluster-Zustands-Reinigung: Wir geben viel-parteien Verschr¨ankungsreinigungs- protokolle fur¨ eine große Klasse von verschr¨ankten Quantenzust¨anden an. (3) Verschr¨ankungsreinigungsprotokolle von Quantencodes: Wir beschreiben eine kon- struktive Methode, um bipartite Verschr¨ankungsreinigungsprotokolle aus Fehler korrigierenden Codes zu erzeugen, und untersuchen die Eigenschaften dieser Pro- tokolle, die in zwei Betriebsarten existieren, die mit Quantenkommunikationspro- tokollen bzw. mit Quantenrechenprotokollen in Verbindung stehen. xii Abstract Chapter 1 Introduction ‘I can’t believe that!’ said Alice. ‘Can’t you?’ the Queen said in a pitying tone. ‘Try again: draw a long breath, and shut your eyes.’ Alice laughed. ‘There’s no use trying,’ she said; ‘one cannot believe impossible things.’ ‘I daresay you haven’t had much practice,’ said the Queen. ‘When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.’ Lewis Carroll, Through the Looking Glass Quantum mechanics and its interpretation During the last century, quantum theory has proved to be a very successful theory, which accurately describes the physical reality of the microscopic and mesoscopic world. To- day, no physical experiment is known which contradicts the predictions made by quantum theory. This is even more remarkable, since measurement ac- curacy has increased, and the size of the systems under consideration has decreased at a fast pace. The fact that quantum theory allows for an accurate description of real- ity is obvious from many physical experiments, and has probably never been seriously disputed. On the other hand, for the interpretation of quantum mechanics, things could not be more different: ever since the theory of quan- tum mechanics has been developed, the question How can the mathematical formulation of quantum mechanics be interpreted? lead to a discussion, in which people with different
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