Polar codes in quantum information theory

0.59 MB·English

by  Christoph Hirche

LEIBNIZ UNIVERSITÄT HANNOVER
FAKULTÄT FÜR MATHEMATIK UND PHYSIK
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Polar codes in quantum information theory
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Master’s thesis
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v CHRISTOPH HIRCHE
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r January 6, 2015
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st supervisor : Prof. Dr. Reinhard F. Werner
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nd supervisor : Prof. Dr. Tobias J. Osborne
co-supervisor : Dr. Ciara Morgan
Christoph Hirche: Polar codes in quantum information theory, Master’s thesis,
6 2015
© January ,
ABSTRACT
Polar codes are the first capacity achieving and efficiently implementable
codesforclassicalcommunication.Recentlytheyhavealsobeengeneralized
to communication over classical-quantum and quantum channels. In this
workwepresentourrecentresultsforpolarcodinginquantuminformation
theory,includingapplicationstoclassical-quantummultipleaccesschannels,
interference channels and compound communication settings, including
the first proof of channel coding achieving the Han-Kobayashi rate region
of the interference channel without the need of a simultaneous decoder.
Moreover we add to the existing framework by extending polar codes to
achieve the asymmetric capacity and improving the block error probability
for classical-quantum channels. In addition we use polar codes to prove a
new achievable rate region for the classical-quantum broadcast channel.
We also discuss polar codes for quantum communication over quantum
channels and state results towards codes for compound quantum channels
in this setting.
Weconcludebystatingalistofinterestingopenquestionstoinvitefurther
research on the topic.
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ACKNOWLEDGMENTS
Firstly I would like to thank Reinhard F. Werner and Tobias J. Osborne for
taking over the supervision of my thesis.
I am especially grateful to Ciara Morgan for the enormous amount of
help, requiring a lot of time and work, in the process of writing this thesis.
I am also very grateful to Mark M. Wilde for the enjoyable and fruitful
collaboration, including introducing me to the topic.
Last but not least I would like to thank all members of the Quantum
Information Group in Hannover for making my time here so pleasant.
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CONTENTS
List of Figures ix
1 introduction 1
2 preliminaries 5
21 5
. Notation and definitions . . . . . . . . . . . . . . . . . . . . . .
22 8
. Multi-user channels . . . . . . . . . . . . . . . . . . . . . . . . .
3 classical quantum channel 11
-
31 11
. Polar codes for classical-quantum channels . . . . . . . . . . .
311 11
. . Achieving the symmetric capacity . . . . . . . . . . . .
312 14
. . Achieving the asymmetric capacity . . . . . . . . . . .
313 16
. . Improved block error probability . . . . . . . . . . . . .
32 19
. Universal polar codes for cq-channel . . . . . . . . . . . . . . .
33 22
. Polar codes for the Multiple Access Channel . . . . . . . . . .
331 22
. . Two-user MAC . . . . . . . . . . . . . . . . . . . . . . .
332 28
. . k-user MAC . . . . . . . . . . . . . . . . . . . . . . . . .
34 30
. Universal polar codes for the compound MAC . . . . . . . . .
35 34
. Interference channel . . . . . . . . . . . . . . . . . . . . . . . .
36 36
. Broadcast channel . . . . . . . . . . . . . . . . . . . . . . . . . .
361 37
. . Marton-Gelfand-Pinsker region for private messages .
362 41
. . Marton-Gelfand-Pinsker region with common messages
4 quantum channel 43
41 43
. Polar Codes for quantum channels . . . . . . . . . . . . . . . .
411 43
. . Simultaneous scheme . . . . . . . . . . . . . . . . . . .
412 45
. . Concatenated scheme . . . . . . . . . . . . . . . . . . .
42 47
. Towards universal polar codes for quantum channel. . . . . .
421
. . Universal polar codes for degraded channels in the
47
simultaneous scheme. . . . . . . . . . . . . . . . . . . .
5 conclusions and outlook 49
51 49
. Further directions . . . . . . . . . . . . . . . . . . . . . . . . . .
52 51
. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Bibliography
vii

LIST OF FIGURES
31 19
Figure . Polar codes for different channels. . . . . . . . . . . . .
32 20
Figure . Coding for the compound channel. . . . . . . . . . . .
33 2 23
Figure . Order for decoding the -user MAC . . . . . . . . . . .
34 3 29
Figure . Order for decoding the -user MAC. . . . . . . . . . .
35 32
Figure . Coding for the compound MAC. . . . . . . . . . . . . .
36 2 35
Figure . -user interference channel. . . . . . . . . . . . . . . . .
37 35
Figure . Rate region of the interference channel. . . . . . . . . .
38 40
Figure . Coding for the broadcast channel. . . . . . . . . . . . .
ix