144 Pages
English

Entanglement distribution in quantum networks [Elektronische Ressource] / Sébastien Perseguers

-

Gain access to the library to view online
Learn more

Informations

Published by
Published 01 January 2010
Reads 48
Language English
Document size 2 MB

¨ ¨Technische Universitat Munchen
Max-Planck–Institut fu¨r Quantenoptik
Entanglement Distribution
in Quantum Networks
S´ebastien Perseguers
Vollst¨andiger Abdruck der von der Fakult¨at fu¨r Physik
der Technischen Universit¨at Mu¨nchen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender : Univ.-Prof. J. J. Finley, Ph.D.
Pru¨fer der Dissertation : 1. Hon.-Prof. I. Cirac, Ph.D.
2. Univ.-Prof. Dr. P. Vogl
Die Dissertation wurde am 25.02.2010 bei der
Technischen Universit¨at Mu¨nchen eingereicht und
durch die Fakult¨at fu¨r Physik am 15.04.2010 angenommen.Every great and deep difficulty bears in itself its own“ solution. Itforcesustochangeourthinkinginorder
to find it.

– Niels BohrAbstract
This Thesis contributes to the theory of entanglement distribution in
quantum networks, analyzing the generation of long-distance entangle-
ment in particular. We consider that neighboring stations share one par-
tially entangled pair of qubits, which emphasizes the difficulty of creating
remote entanglement in realistic settings. The task is then to design local
quantum operations at the stations, such that the entanglement present
in the links of the whole network gets concentrated between few parties
only, regardless of their spatial arrangement.
First, we study quantum networks with a two-dimensional lattice struc-
ture, where quantum connections between the stations (nodes) are de-
scribed by non-maximally entangled pure states (links). We show that
the generation of a perfectly entangled pair of qubits over an arbitrarily
long distance is possible if the initial entanglement of the links is larger
than a threshold. This critical value highly depends on the geometry of
the lattice, in particular on the connectivity of the nodes, and is related to
a classicalpercolation problem. We then develop a genuine quantum strat-
egy based onmultipartite entanglement, improving both the threshold and
the success probability of the generation of long-distance entanglement.
Second, we consider a mixed-state definition of the connections of the
quantum networks. This formalism is well-adapted for a more realistic
description of systems in which noise (random errors) inevitably occurs.
New techniques are required to create remote entanglement in this set-
ting, and we show how to locally extract and globally process some error
syndromes in order to create useful long-distance quantum correlations.
Finally, we turn to networks that have a complex topology, which is
the case for most real-world communication networks such as the Inter-
net for instance. Besides many other characteristics, these systems have in
common thesmall-world feature, stating that any two nodes are separated
by a few links only. Based on the theory of random graphs, we propose
a model of quantum complex networks, which exhibit some totally unex-
pected properties compared to their classical counterparts.Contents
Introduction 1
I Pure states 7
1 Entanglement manipulation in basic networks 9
1.1 Entanglement swapping . . . . . . . . . . . . . . . . . . . 10
1.1.1 Joint measurement at the middle station . . . . . . 10
1.1.2 Figures of merit . . . . . . . . . . . . . . . . . . . . 13
1.1.3 Optimal measurements . . . . . . . . . . . . . . . . 15
1.2 Maximally-entangled states are not always optimum . . . . 18
1.2.1 Two consecutive entanglement swappings . . . . . . 19
1.2.2 A single square network . . . . . . . . . . . . . . . 22
1.3 An infinite chain of quantum relays . . . . . . . . . . . . . 25
1.3.1 Exponential decay of the entanglement . . . . . . . 26
1.3.2 SCP under ZZ measurements . . . . . . . . . . . . 27
2 Long-distance entanglement in planar graphs 31
2.1 Deterministic methods . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Hierarchical graphs . . . . . . . . . . . . . . . . . . 33
2.1.2 Regular lattices . . . . . . . . . . . . . . . . . . . . 36
2.2 Strategies based on bond percolation . . . . . . . . . . . . 39
2.2.1 Classical entanglement percolation . . . . . . . . . 40
2.2.2 Quantum entanglement percolation . . . . . . . . . 42
2.3 Multipartite entanglement percolation . . . . . . . . . . . 46
2.3.1 Generalized entanglement swapping . . . . . . . . . 47
2.3.2 An illustrative example . . . . . . . . . . . . . . . . 49
2.3.3 The superiority of multipartite strategies . . . . . . 53
2.4 On optimal protocols . . . . . . . . . . . . . . . . . . . . . 57
3 Quantum complex networks 61
3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.1 Random graphs . . . . . . . . . . . . . . . . . . . . 62vi Contents
3.1.2 Erd˝os-R´enyi networks in the quantum world . . . . 64
3.2 Joint measurements help . . . . . . . . . . . . . . . . . . . 66
3.2.1 Creation of W states . . . . . . . . . . . . . . . . . 67
3.2.2 Creation of GHZ states . . . . . . . . . . . . . . . . 68
3.3 A complete collapse of the critical exponents . . . . . . . . 71
3.3.1 The Λ subgraph . . . . . . . . . . . . . . . . . . . . 71
3.3.2 General subgraphs . . . . . . . . . . . . . . . . . . 73
II Mixed states 77
4 Towards noisy quantum networks 79
4.1 Rank-two mixed states . . . . . . . . . . . . . . . . . . . . 81
4.1.1 From pure to mixed states and vice versa . . . . . . 81
4.1.2 Quantum complex networks . . . . . . . . . . . . . 83
4.2 Full-rank mixed states . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Elementary operations on Werner states . . . . . . 85
4.2.2 Quantum repeaters . . . . . . . . . . . . . . . . . . 86
4.2.3 Lower bound for long-range entanglement . . . . . 88
5 One-shot protocol in square lattices 89
5.1 Network with bit-flip errors only . . . . . . . . . . . . . . . 90
5.1.1 Propagating a large GHZ state . . . . . . . . . . . 91
5.1.2 Network-based bit-flip error correction . . . . . . . 92
5.2 A fault-tolerant protocol via encoding . . . . . . . . . . . . 97
5.2.1 Required physical and temporal resources . . . . . 98
5.2.2 Towards a realistic scenario . . . . . . . . . . . . . 102
6 Fidelity threshold in cubic quantum networks 105
6.1 Quantum networks and cluster states . . . . . . . . . . . . 105
6.1.1 A mapping to noisy cluster states . . . . . . . . . . 107
6.2 Long-distance entanglement generation . . . . . . . . . . . 108
6.2.1 Measurement pattern and quantum correlations . . 109
6.2.2 Error correction and fidelity of the final state . . . 111
6.2.3 Numerical estimation of the fidelity threshold . . . 116
Bibliography 121
Acknowledgment 133List of Figures
1 Examples of quantum networks . . . . . . . . . . . . . . . 3
1.1 Entanglement swapping . . . . . . . . . . . . . . . . . . . 11
1.2 SCP after two consecutive entanglement swappings . . . . 20
1.3 A single square network . . . . . . . . . . . . . . . . . . . 23
1.4 SCP of the single square network . . . . . . . . . . . . . . 26
1.5 Entanglement swappings in a one-dimensional network . . 27
2.1 “Diamond” graph . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Double binary tree . . . . . . . . . . . . . . . . . . . . . . 35
2.3 SCP of the double binary tree . . . . . . . . . . . . . . . . 36
2.4 “Centipede” in the square lattice . . . . . . . . . . . . . . 37
2.5 Two-dimensional centipedes . . . . . . . . . . . . . . . . . 38
2.6 Bond percolation . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Honeycomb lattice with double bonds . . . . . . . . . . . . 42
2.8 Asymmetric triangular lattice . . . . . . . . . . . . . . . . 44
2.9 Square lattices of double size . . . . . . . . . . . . . . . . . 45
2.10 Generalized entanglement swapping . . . . . . . . . . . . . 48
2.11 Multipartite entanglement percolation . . . . . . . . . . . 50
2.12 Lattice transformations under GHZ measurements . . . . . 54
2.13 Percolation probabilities for multipartite strategies . . . . 57
3.1 Classical random graphs . . . . . . . . . . . . . . . . . . . 62
3.2 Quantum random graphs . . . . . . . . . . . . . . . . . . . 65
3.3 Construction of W states . . . . . . . . . . . . . . . . . . . 67
3.4 Graphical representation of|Ki . . . . . . . . . . . . . . 69c
3.5 Construction of GHZ states . . . . . . . . . . . . . . . . . 70
3.6 Construction of the Λ state . . . . . . . . . . . . . . . . . 72
4.1 Squeezed-light entanglement . . . . . . . . . . . . . . . . . 80
4.2 Quantum repeaters . . . . . . . . . . . . . . . . . . . . . . 87viii List of Figures
5.1 Propagation of a large GHZ state through the lattice . . . 91
5.2 Bit-flip error correction . . . . . . . . . . . . . . . . . . . . 92
5.3 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . 96
5.4 Physical resources required at each station . . . . . . . . . 98
5.5 Universal computation on a line . . . . . . . . . . . . . . . 101
6.1 Non-local control phase . . . . . . . . . . . . . . . . . . . . 107
6.2 Measurement pattern . . . . . . . . . . . . . . . . . . . . . 109
6.3 Missing syndromes in the sublattices . . . . . . . . . . . . 113
6.4 Harmful paths of errors . . . . . . . . . . . . . . . . . . . . 114
6.5 Inference of the missing syndromes . . . . . . . . . . . . . 118
6.6 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . 119