1. Entanglement and Topological order in self-dual cluster states Vlatko Vedral University of Oxford, UK & National University of Singapore

2. Contents Topological 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

3. 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 )

4. Criticality indicator Long range order Off-diagonal LRO Even more creative : Two dimensional phase transitions. Entanglement order? (c.f. Wen) Fractional Quantum Hall effects.

5. Order tree 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 Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)

6. Entanglement (Block ent. & Geometric Ent.) Separability Block entanglement (Entropy) Geometric entanglement

7. QPT in XX model What is quantum phase (transition) in many-body system? (XX model) 1 2 3 1 2 3

8. Thermal state and purity (XX model)

9. Cluster states Construction of the cluster state. Hamiltonian for cluster state. Usefulness of cluster states for measurement based quantum computation. CP

10. Full Spectrums of Cluster Hamitonian Full Spectrums For the case of N=4

11. Geometric entanglement Physical meaning; Mean field correspondence. Numerical evaluation. Symmetries can be applied for closest separable state. (XX model with perturbation.) Can entanglement be a topological order parameter?

12. Entanglement as Energy Think of phase transition as tradeoff between energy and entropy: Quantum phase transitions: tradeoff between entanglement and entropy: Clusters:

13. Diagonalising Cluster Jordan Wigner transformation leads to free fermions (“hopping” between next to nearest neighbours) Probability looks like N independent fermions Then do the FT and Bogoliubov…

14. 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.

15. Mapping of Cluster into Ising 1D Cluster Hamiltonian. State transformation. Hamiltonian without boundary term. Ising state.

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

17. Topological order in Cluster state Insensitivity to local perturbation. No degeneracy in the ground state. String order Highly entangled state (E~N/2).

18. Discussion Applied 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?

19. References L. 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)