Entanglement and Topological order in self-dual cluster states
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Transcript of Entanglement and Topological order in self-dual cluster states
Entanglement and Topological order in self-dual cluster states
Vlatko Vedral
University of Oxford, UK &National University of Singapore
ContentsTopological order and Entanglement.
XX model.
Cluster states.
Dual transformation.
Boundary effects, Phase transition and criticality.
Entanglement as an order parameter.W. Son, L. Amico, S. Saverio, R. Fazio, V. V., arXiv:1001.2565
Topological order A phase which cannot be described by the Landau
framework of symmetry breaking. Three different characterization of the topological order.
◦ Insensitivity to local perturbation. ◦ Ground state degeneracy to the boundary condition.◦ Topological entropy.
Relationship between the topological order and fault tolerance.
Conceptual relationship between topological order and entanglement.◦ Entanglement is global properties in the system.◦ Entanglement is sensitive to degeneracy (Pure vs Mixed )
Criticality indicatorLong range order
Off-diagonal LRO
Even more creative : Two dimensional phase transitions.
Entanglement order? (c.f. Wen) Fractional Quantum Hall effects.
| |† ( ) ( ) x yx y const
| |† †( ) ( ) ( ) ( ) x yx x y y const
Order treeDifferent Orders
Long range order(e.g. 2D Ising)
Short range order(e.g. KT)
Off-diagonal LRO(e.g. BCS)
Quantum – ground state –Topological(e.g. FQHE)
Topological, finiteT order ?
Symmetry breaking
Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)
Entanglement (Block ent. & Geometric Ent.)Separability
Block entanglement (Entropy)
Geometric entanglement
)]([)( AALogTrS
2||min)( sep
QPT in XX model
What is quantum phase (transition) in many-body system? (XX model)
1
23
12
3
Thermal state and purity (XX model)
Cluster statesConstruction of the cluster state.
Hamiltonian for cluster state.
Usefulness of cluster states for measurement based quantum computation.
CP CP CP CP CP
)(iNk
kii XZSi
iSH
)(iNk
kii XZS
CZCi
nin
i
N
i
CPiUC
Full Spectrums of Cluster HamitonianFull Spectrums
For the case of N=4
CZCi
nin
i
C
CZi
CZZ ji
CZZZ kji
CZZZZ 4321
40 E
21 E
02 E
23 E
44 E
}1,0{in
Geometric entanglementPhysical meaning;
Mean field correspondence.
Numerical evaluation.
Symmetries can be applied for closest separable state. (XX model with perturbation.) Can entanglement b
e a
topological order parameter?
Entanglement as Energy
Think of phase transition as tradeoff between energy and entropy:
TdSdUdF
Quantum phase transitions: tradeoff between entanglement and entropy:
)ln(~ GpSE
Clusters:
12ln
2
1
1ln
2 /2
JT
e
NC
N
kTJ
Diagonalising ClusterJordan Wigner transformation leads
to free fermions (“hopping” between next to nearest neighbours)
Probability looks like N independent fermions
Then do the FT and Bogoliubov…
nnnnn
l
l
k
zkl
ccccH
c
22
1
1
Dual transformation (Fradkin-Susskind).
Definition.
Duality ◦ Emergence of qusi-particles (discuss XX).◦ Identification of critical point. ◦ Change of state and entanglement.
Sensitivity to the boundary condition in the dual transformation.
Mapping of Cluster into Ising 1D Cluster Hamiltonian.
State transformation.
Hamiltonian without boundary term.
Ising state.
i
iiiC XZXH 11
Cluster
Self-dual Cluster Hamiltonian
Model
Solution
Geometric entanglement and criticality
Topological order in Cluster stateInsensitivity to local perturbation.
No degeneracy in the ground state.
String order
Highly entangled state (E~N/2).
CNCH c
1)1(1
21
N
k
yN
zk
yNO
DiscussionApplied standard methods of statistical
physics and solid state to computing; Can think of entanglement as
equivalent to energy (free energy)Should do the same analysis in 2D (JW
ambiguity)Can all topological phases support
computing?Could we map between circuits and
clusters?
ReferencesL. Amico, R. Fazio, A. Osterloh, V. V, Rev.
Mod.Phys. 80 (2008)
Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)
W. Son, L. Amico, F. Plastina, V. V Phys. Rev. A 79(2009)
W. Son, V. V., OSID volume 2-3:16 (2009) Michal Hajdušek and V. V. New J. Phys. 12 (2010) A. Kitaev, Chris Laumann, arXiv:0904.2771 A. Kitaev, J. Preskill, Phys. Rev. Lett. 96 (2006) R. Raussendorf, D.E. Browne, H.J. Briegel, Phys. Rev.
A 68 (2003)