1. Entanglement and Topological order in 1D & 2D cluster states
Wonmin Son
Centre for Quantum Technology,
National University of Singapore

2. Ground state degeneracy insensitive to local perturbation.
Topological order
A phase which cannot be described by the Landau framework of symmetry breaking.
Three characterization of quantum topological order.
Ground state degeneracy to the boundary condition.
Insensitivity to local perturbation.
Topological entropy. (Kitaev, Preskill vs Wen)
Conceptual relationship between topological order and entanglement.
Global property ; Non-local Order Parameter - Nonlocality (Bell’s inequality)
Degeneracy by Symmetry-breaking -Mixedness
Insensitivity to Local Perturbation- Invariance under Local Unitary Operation
Z. Nussinov, G. Ortiz, Annals of Physics 324 (2009), 977
Relationship between the topological order and fault tolerance.
Ground state degeneracy insensitive to local perturbation.
What is topological order.
Point 1, It is a phase that cannot be characterized by conventional way.
Point 2, Quantum case is not well defined.
Point 3, Useful for fault tolerant quantum computation (Anyonic operation)
Point 4, Properties of the phase are same as entanglement.
Point 5, It is related with geometric configuration.

3. Order tree Long range order (e.g. 2D Ising) Short range order
Different 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, finite
T order ?
Symmetry breaking
This diagram tell us the type of phases which can be characterized by the different quantities.
Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)

4. Questions
Can the topological quantum phase be in arbitrary system (e.g.1D, thermal) ?
How does the TO can be related with the surface topology.

5. Contents XX model & quantum phase transition.
1D Cluster states & topological order.
Dual mapping & boundary effect.
2D systems, mappings and geometry.
W. Son, V. Vedral, arXiv:
OSID volume 2-3:16 (2009)
W. Son, L. Amico, S. Saverio, R. Fazio, A. Hamma, V. Vedral, arXiv:

6. QPT in XX model
What is quantum phase (transition) in many-body system? (XX model) 1 2 3
Shell we start short speculation on the quantum phase transition.
Definition of Quantum phase transition…
Question 1) Criticality,,, (continuous phase transition)
Question 2) Relation with Entanglement.
Question 3) High temperature case… 1 2 3

7. Thermal state and purity (XX model)
Purity simply can detect such a criticality
Purity is important for (1) QPT & (2) mixdeness (degeneracy)

8. Cluster states (1D) Construction of the cluster state.
Hamiltonian for cluster state.
Usefulness of cluster states for measurement based quantum computation Recent review; H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf,
M. Van den Nest, Nature Physics 5 1, (2009) CP
Quick review on cluster state.

9. 1D Cluster state – Topologically ordered or not?
Cyclic & Non-cyclic boundary condition.
No degeneracy, Zero topological entropy
Open boundary condition.
Degeneracy can be occurred if there is missing stabilizers. (cf, AKLT - HALDANE phase )
Is the degenerated mani-fold robust against ANY local perturbation? Not really…

10. Symmetry protected TOarXiv:1103.0251
The degeneracy is possible to be protect under local perturbation if it is controlled under Z2 * Z2 symmetry.
Impossible with Z2 symmetry only.

11. Dual transformation (Fradkin-Susskind).
Definition.
Duality
Emergence of qusi-particles (discuss XX).
Identification of critical point.
Global transformation only with two-body unitary operation
(Controlled flip gate.)
Sensitivity to the boundary condition in the dual transformation.
Duality mapping is a way to map many spins into different spins.
And typical example of unitary operation achiving global transformation only by two body unitary operation -
-The ising model with external perturbation is self-dual under the duality mapping.
-And it identifies critical points…
-Sensitive to bounary.

12. Mapping of 1D Cluster into Ising
1D Cluster Hamiltonian.
State transformation.
The transformed Hamiltonian of cluster state without boundary term is exact Ising state.
We applied the mapping to cluster state and realized that the state is mapped into ising model.
But what one should careful about the mapping is that the boundary term is not trivially transformed.
The transformed boundary operator has string operator form (nonlocal shape) and act as an perturbation resulting energy level splitting.

13. Self-dual Cluster Hamiltonian
Model
Solution
Geometric
entanglement
and criticality

14. Mapping in 2D models
(3)
(1)
(2)
Skew dual mapping.
Row dual mapping.
Even site local unitary transformation.
All the mappings are sensitive to boundary conditions.

15. Skew diagonal dual mapping Cluster Wen

16. Diagonalizing Wen model through Fermionization (JW)
Reinventing the approach by Chen & Hu (07) with boundary terms.

17. Wen’s Model to Kitaev model

18. Topological effects from topology
Imbedding the lattices into the surfaces of different topology.
Euler-Poincare Characteristics.
number of sites, links, plaqutte, and number of handles (genus)

19. Number of degeneracies in 2D
To imbed lattice to surface with different topology, consistently at the thermodynamic limit, the lattice should have defects. Which means…
Number of missing spins = 2 * number of genus = 2 * number of degeneracy.
Topology protects the ground state degeneracy

20. Summary & Discussions
We studied QPT through complete characterization of XX model. (State & identification KT by entanglement)
TO in 1D cluster state with symmetry.
Mapping between different 2D models.
Developed new skills for exactly solvable model.
TO & Entanglement.
Applied standard methods of statistical physics and solid state to computing.
Can all topological phases support computing?
Could we map between circuits, clusters, 2D models?
Measurement based quantum computation & Topological quantum computation?

21. Conclusion & natural conjecture…
Our definition of TO is consistent with all the existing notion of topological quantum order… And
In the definition of quantum topological entropy, the degree of mixedness, “purity of state”, should be included.
Where p is purity

22. References
L. Amico, R. Fazio, A. Osterloh, V. V, Rev. Mod.Phys (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:
A. Kitaev, J. Preskill, Phys. Rev. Lett. 96 (2006)
R. Raussendorf, D.E. Browne, H.J. Briegel, Phys. Rev. A 68 (2003)
W. Son, L. Amico, S. Saverio, R. Fazio, A. Hamma, V. Vedral, arXiv:

23. Thanks to Collaborators. And funding.
Luigi Amico (Catania), Rosario Fazio (Pisa),
Alioscia Hamma (Perimeter), Saverio Pascazio (Bari),
Benjamin J. Brown (Imperial Collage),
Christina V. Kraus (MPS),
And special thanks to ;
Vlatko Vedral (Oxford & Singapore)
And funding.