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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. Contents. Topological order and Entanglement. XX model. Cluster states. Dual transformation. Boundary effects, Phase transition and criticality.

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Entanglement and Topological order in self-dual cluster states

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  1. Entanglement and Topological order in self-dual cluster states VlatkoVedral 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. Topological, finite T order ? Symmetry breaking Order tree Different Orders Short range order (e.g. KT) Quantum – ground state – Topological (e.g. FQHE) Long range order (e.g. 2D Ising) Off-diagonal LRO (e.g. BCS) 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 CP CP CP 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)

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