On multipartite symmetric states in quantum information theory [Elektronische Ressource] / von Tilo Eggeling

Tilo Eggeling

Sujets

• Physik
• Astronomie

Informations

• Publié par: technische_universitat_carolo-wilhelmina_zu_braunschweig
• Publié le: 01 janvier 2003
• Nombre de lectures: 78
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

On multipartite symmetric states in
Quantum Information Theory
Der Gemeinsamen Naturwissenschaftlichen Fakult˜at
der Technischen Universit˜at Carolo-Wilhelmina
zu Braunschweig
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
(Dr.rer.nat.)
genehmigte
D i s s e r t a t i o n
von Tilo Eggeling
aus Bad Harzburg1.Referent: Prof.Dr. Reinhard F. Werner
2.Referent: Prof.Dr. Martin Wilkens
Eingereicht am: 16. Januar 2003
Mundlic˜ he Prufung˜ (Disputation) am: 2. April 2003
Druck: 2003Teilergebnisse aus dieser Arbeit wurden mit Genehmigung der
Gemeinsamen Naturwissenschaftlichen Fakultat,¤ vertreten durch den Mentor
der Arbeit, in folgenden Beitragen¤ vorab veroffentlic¤ ht:
Publikationen
† T. Eggeling, R. F. Werner: Separability properties of tripartite states
withU›U›U symmetry, Phys.Rev.A 63 042111.
† T. Eggeling, R. F. Werner: Hiding classical data in multipartite
quantum states, Phys.Rev.Lett. 89 097905.
Eine vollstandige¤ Publikationsliste be ndet sich auf Seite 149.
Tagungsbeitrage¤
† T. Eggeling, R. F. Werner: Separabilitatseigensc¤ haften vonU›U›
U-invarianten Zustanden¤ dreigeteilter Systeme, (Vortrag), DPG
Fruhjahrstagung¤ 2000, Fachverband Theoretische und
Mathematische Grundlagen der Physik, Dresden (Germany), 20. 24.03.2000.
† T. Eggeling, R. F. Werner: Separability properties of tripartite states
withU›U›U symmetry, (Poster), Coherent Evolution in noisy
Environments, MPI fur¤ Physik komplexer Systeme, Dresden
(Germany), 20. 25.05.2001.
† T. Eggeling, R. F. Werner: Separability properties of tripartite states
ndwithU›U›U symmetry, (Poster),2 ESF QIT Conference, Gdansk
(Poland), 10. 18.07.2001.
† T. Eggeling, R. F. Werner: Hiding classical data in multipartite
quantum states, (Poster), QRandom II, MPI fur¤ Physik komplexer
Systeme, Dresden (Germany), 27.01. 01.02.2002.
† T. Eggeling, R. F. Werner: Hiding classical data in multipartite
quantum states, (Vortrag), DPG Fruhjahrstagung¤ 2002,
Fachver¤band Quantenoptik, Osnabruck (Germany), 04. 08.03.2002.
† T. Eggeling, R. F. Werner: Hiding classical data in multipartite
quantum states, (Poster), International Conference on Quantum
Information, Oviedo (Spain), 13. 18.07.2002.
† T. Eggeling, R. F. Werner, M. M. Wolf: Optimizing the residual
bipartite delity in multipartite systems, (Vortrag), ESF workshop
IQING, London (UK), 19. 22.09.2002.Introduction
More than 50 years after its publication, Shannon’s ?A mathematical
theory of communication? [Sha48] still in uences today’s physics. In
the late 50s Jaynes [Jay57a, Jay57b] succeeded in describing statistical
mechanics from an information theoretical point of view by using the
method of maximum-entropy inference (nowadays called Jaynes’
principle). The reconciliation of information theory and quantum theory,
however, took much longer. The starting point was given by Benioff
in 1980 ([Ben80], see also [Deu85, Fey86]) where he gave a
description of a Hamiltonian that can be interpreted as a Turing machine.
The vision of the quantum computer was born. In the following years
the capabilities of a like the exponentially growing
speed-up in factorizing large numbers (Shor’s algorithm [Sho94]) or in
database searches (Grover’s algorithm [Gro96]) and the phenomenon
+of quantum teleportation (see [BBC 93]) led to an almost exponential
interest in quantum computing from the military and the industrial
side. Recently, various proposals have been made on how to build such
a quantum computer. They involved ion traps, liquid NMR, quantum
dots and optical lattices. However, nowadays it seems as if for the next
ten years the quantum computer may remain a vision like the Holy
Grail due to the immense experimental demands.
Fortunately, in the light of this vision quantum information theory
evolved meanwhile to a large independent eld of research. It has
become a widely structured eld involving physicists, mathematicians,
computer scientists and electrical engineers. Its main pillars are
quantum computing, quantum communication and entanglement theory.
Quantum computation is concerned with the development of
quantum circuits and algorithms for future quantum computers. Latest
developments concern the hidden subgroup problem and search
algorithms. Methods of translating algorithms into quantum circuits have
already been developed. One of the major challenges is now to develop
good error correcting codes for arbitrary systems.
vQuantum communication comprises various communication
protocols that have been developed for or adapted to quantum information.
Among the best known protocols are quantum key distribution,
quantum teleportation and superdense coding. Quantum key distribution
was the rst application of quantum information to be realized in
experiments. In fact, rst implementations are already available
commercially (see www.idquantique.com).
Entanglement has been revalued by quantum information theory
from a fundamental property of quantum systems to a new resource
which may be used up or used as a catalyst in quantum information
processing. It is the glue that connects the various elements of
quantum information theory and at the same time the most important new
ingredient that enables us to perform new quantum protocols. In 1935
Schrodinger¤ [Sch35] coined the German word Verschranktheit ¤ for a
mysterious inseparability he encountered while investigating states of
compound systems. What he and Einstein, Podolsky and Rosen (see
[EPR35]) had found was the rst instance of quantum correlations
among particles/parties. In the last few years, a growing interest has
been devoted to the theory of entanglement. Most of the results were
obtained for low dimensional systems (e.g. qubits) and for systems that
could be simpli ed making use of symmetries.
This thesis is devoted to the study of entanglement in multipartite
systems. The characterisation of general multipartite states involves
ap2Nproximatelyd real parameters, whered is the dimension of the single
system andN is the number of systems. In order to be able to
investigate entanglement properties in multipartite systems, in chapter 2 we
therefore introduce families of states that can be described
with only few parameters. As a tool we use symmetry groups to reduce
the complexity of the problem. In chapter 3 we characterize the
separability/inseparability properties of this state family in the special case
ofN = 3. Chapters 4 and 5 are concerned with the operational aspects
of the entanglement contained in these states. In chapter 4 we use the
introduced states, amongst other things, to demonstrate the security of
multipartite quantum data hiding and to give a constructive scheme of
this protocol. The last chapter relates entanglement sharing to
quantum telecloning.
Note to the reader: The experienced reader may skip the rst
chapter which contains a very short overview of the concepts and
mathematical tools used in the following chapters. Chapters 3, 4 and 5 are
based on chapter 2 and can be read separately. A short summary of the
results presented in this thesis can be found on page 137.
viContents
Introduction v
1 Basic concepts 1
1.1 States and state transformations . . . . . . . . . . . . . . 1
1.1.1 States and measurements . . . . . . . . . . . . . . 2
1.1.2 State . . . . . . . . . . . . . . . . 4
1.1.3 Duality of states and state transformations . . . . 7
1.2 Classical and quantum correlations . . . . . . . . . . . . 7
1.2.1 Classical vs. quantum correlations . . . . . . . . . 8
1.2.2 Separability criteria . . . . . . . . . . . . . . . . . 10
1.2.3 Quantifying entanglement . . . . . . . . . . . . . . 14
1.3 Symmetries, groups and all that jazz . . . . . . . . . . . . 18
1.3.1 Symmetries, groups and representations . . . . . 18
1.3.2 C*-algebras . . . . . . . . . . . . . . . . . . . . . . 21
2 Multipartite symmetric states 25
2.1 Werner states . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Multipartite Werner states . . . . . . . . . . . . . . . . . . 29
2.2.1 Duality ofU(d) andS . . . . . . . . . . . . . . . . 29N
2.2.2 Commutative subfamilies of states . . . . . . . . . 32
2.3 Orthogonally symmetric states . . . . . . . . . . . . . . . 34
2.3.1 The chip representation . . . . . . . . . . . . . . 35
2.4 The power of reduced states . . . . . . . . . . . . . . . . . 38
2.4.1 Reducing symmetric states . . . . . . . . . . . . . 39
2.4.2 Extending . . . . . . . . . . . . . 41
3 Tripartite Werner states 45
3.1 Separability properties . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Fully separable states . . . . . . . . . . . . . . . . 52
3.1.2 Biseparable states . . . . . . . . . . . . . . . . . . . 57
3.1.3 Partially transposed permutations . . . . . . . . . 61
vii3.1.4 States having a positive partial transpose . . . . . 66
3.1.5 The realignment criteria . . . . . . . . . . . . . . . 75
3.2 Entanglement monotones and Bell violations . . . . . . . 76
3.2.1 Relative entropy and trace norm distance . . . . . 77
3.2.2 Bell inequalities for dichotomic observables