Localized endomorphisms in Kitaev's toric code on the plane

Reviews in Mathematical Physics

Volumn 23, Issue 4, 2011, Pages 347-373

  Naaijkens, Pieter ^a  

^a RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

Author keywords

anyons; Localized endomorphisms; superselection sectors; toric code

Indexed keywords

EID: 79957935723     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X1100431X     Document Type: Article

References (39)

1. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303 (2003) 2-30. (Pubitemid 36177417)
2. C. Nayak, S. H. Simon, A. Stern,M. Freedman and S. Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083-1159.
3. Z. Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, Vol. 112, Published for the Conference Board of the Mathematical Sciences (Amer. Math. Soc., 2010).
4. R. Alicki, M. Fannes and M. Horodecki, A statistical mechanics view on Kitaev's proposal for quantum memories, J. Phys. A 40 (2007) 6451-6467.
5. E. Dennis, A. Kitaev and J. Preskill, Topological quantum memory, J. Math. Phys. 43 (2002) 4452-4505.
6. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 1, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, New York, 1987).
7. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, Berlin, 1997).
8. R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, Berlin, 1996).
9. D. Buchholz and K. Fredenhagen, Locality and the structure of particle states, Comm. Math. Phys. 84 (1982) 1-54.
10. S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. I, Comm. Math. Phys. 23 (1971) 199-230.
11. S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. II, Comm. Math. Phys. 35 (1974) 49-85.
12. F. Nill and K. Szlachányi, Quantum chains of Hopf algebras with quantum double cosymmetry, Comm. Math. Phys. 187 (1997) 159-200.
13. K. Szlachányi and P. Vecsernyés, Quantum symmetry and braid group statistics in G-spin models, Comm. Math. Phys. 156 (1993) 127-168.
14. 2 gauge theories, Comm. Math. Phys. 92 (1983) 81-119.
15. F. A. Bais, P. van Driel and M. de Wild Propitius, Quantum symmetries in discrete gauge theories, Phys. Lett. B 280 (1992) 63-70.
16. 2 gauge theories, Comm. Math. Phys. 113 (1987) 403-417.
17. N-Higgs gauge model, Comm. Math. Phys. 171 (1995) 27-86.
18. N-Higgs models, Comm. Math. Phys. 191 (1998) 409-466.
19. B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem, Comm. Math. Phys. 265 (2006) 119-130. (Pubitemid 43747216)
20. D. E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Mathematical Monographs (The Clarendon Press Oxford University Press, New York, 1998).
21. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II, Graduate Studies in Mathematics, Vol. 16 (American Mathematical Society, Providence, RI, 1997).
22. D. Buchholz and K. Fredenhagen, Locality and the structure of particle states in gauge field theories, in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, Vol. 153 (Springer, 1982), pp. 368-371.
23. M. Keyl, T. Matsui, D. Schlingemann and R. F. Werner, Entanglement Haag-duality and type properties of infinite quantum spin chains, Rev. Math. Phys. 18 (2006) 935-970. (Pubitemid 44847163)
24. T. Matsui, Spectral gap, and split property in quantum spin chains, J. Math. Phys. 51 (2010) 015216, 8 pp.
25. P. Naaijkens, On the extension of stringlike localised sectors in 2+1 dimensions, Comm. Math. Phys. 303 (2011) 385-420.
26. H. Halvorson, Algebraic quantum field theory, in Philosophy of Physics, eds. J. Butterfield and J. Earman (Elsevier, 2006), pp. 731-922.
27. K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance, Rev. Math. Phys. 4 (1992) 113-157.
28. R. Longo and J. E. Roberts, A theory of dimension, K-Theory 11 (1997) 103-159. (Pubitemid 127499217)
29. T. Matsui, The split property and the symmetry breaking of the quantum spin chain, Comm. Math. Phys. 218 (2001) 393-416.
30. M. Takesaki, Theory of Operator Algebras. I, Encyclopaedia of Mathematical Sciences, Vol. 124 (Springer-Verlag, Berlin, 2002).
31. C. D'Antoni and R. Longo, Interpolation by type I factors and the flip automorphism, J. Funct. Anal. 51 (1983) 361-371.
32. M. Takesaki, On the direct product of W*-factors, Tôhoku Math. J. (2) 10 (1958) 116-119.
33. C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Vol. 155 (Springer- Verlag, New York, 1995).
34. B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series, Vol. 21 (American Mathematical Society, Providence, RI, 2001).
35. M. Müger, Abstract duality for symmetric tensor *-categories, appendix to [26].
36. E. Rowell, R. Stong and Z. Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 (2009) 343-389.
37. P. Etingof and S. Gelaki, Isocategorical groups, Int. Math. Res. Notices 2001 (2001) 59-76.
38. S. Doplicher and J. E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157-218.
39. R. Dijkgraaf, V. Pasquier and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models. Recent advances in field theory (Annecy-le-Vieux, 1990), Nuclear Phys. B Proc. Suppl. 18B (1991) 60-72.