Entanglement and Topological order in 1 D 2

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

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. Ground state degeneracy ◦ Global property ; Non-local Order Parameter - Nonlocality (Bell’s inequality) ◦ Degeneracy by Symmetry-breaking -Mixedness ◦ Insensitivity to Local Perturbation- Invariance under Local Unitary Operation insensitive to local perturbation. Z. Nussinov, G. Ortiz, Annals of Physics 324 (2009), 977 � Relationship between the topological order and fault tolerance.

Order tree Different Orders Long range order (e. g. 2 D 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)

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

Contents � XX model & quantum phase transition. � 1 D Cluster states & topological order. � Dual � 2 D mapping & boundary effect. systems, mappings and geometry. W. Son, V. Vedral, ar. Xiv: 0905. 3065 OSID volume 2 -3: 16 (2009) W. Son, L. Amico, S. Saverio, R. Fazio, A. Hamma, V. Vedral, ar. Xiv: 1103. 0251

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

Thermal state and purity (XX model) Purity is important for (1) QPT & (2) mixdeness (degeneracy)

Cluster states (1 D) Construction CP of the cluster state. CP Hamiltonian CP CP CP 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, 19 -26 (2009)

1 D 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…

Symmetry protected TOar. Xiv: 1103. 0251 The degeneracy is possible to be protect under local perturbation if it is controlled under Z 2 * Z 2 symmetry. Impossible with Z 2 symmetry only.

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.

Mapping of 1 D Cluster into Ising 1 D Cluster Hamiltonian. State The transformation. transformed Hamiltonian of cluster state without boundary term is exact Ising state.

Self-dual Cluster Hamiltonian Model Solution Geometric entanglement and criticality

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

Skew diagonal dual mapping Cluster Wen

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

Wen’s Model to Kitaev model

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)

Number of degeneracies in 2 D To imbed lattice to surface with different topology, consistently at 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

Summary & Discussions � We studied QPT through complete characterization of XX model. (State & identification KT by entanglement) � TO in 1 D cluster state with symmetry. � Mapping between different 2 D 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, 2 D models? � Measurement based quantum computation & Topological quantum computation?

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

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, ar. Xiv: 0904. 2771 � 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, ar. Xiv: 1103. 0251

Thanks to Collaborators. ◦ ◦ ◦ 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.