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Communications in Mathematical Physics

, Volume 310, Issue 2, pp 419–454 | Cite as

Index Theory of One Dimensional Quantum Walks and Cellular Automata

  • D. Gross
  • V. Nesme
  • H. Vogts
  • R. F. Werner
Article

Abstract

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much “quantum information” as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems — namely quantum walks and cellular automata — we make this intuition precise by defining an index, a quantity that measures the “net flow of quantum information” through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S 1, S 2 can be “pieced together”, in the sense that there is a system S which acts like S 1 in one region and like S 2 in some other region, if and only if S 1 and S 2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S 1 into S 2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map \({S \mapsto {\rm ind} S}\) is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.

Keywords

Cellular Automaton Index Theory Quantum Walk Full Matrix Algebra Quantum Cellular Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ahlbrecht A., Vogts H., Werner A.H., Werner R.F.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    Arrighi, P., Nesme, V., Werner, R.F.: Unitarity plus causality implies locality. http://arxiv.org/abs/0711.3975v3 [quant-ph], 2009
  3. 3.
    Avron J., Seiler R., Simon B: The index of a pair of projections. J. Funct. Anal 120, 220–237 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beckman D., Gottesman D., Nielsen M.A., Preskill J.: Causal and localizable quantum operations. Phys. Rev. A 64, 052309 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. I+II. New York: Springer, 1979, 1997Google Scholar
  6. 6.
    Buchholz D.: The physical state space of quantum electrodynamics. Commun. Math. Phys 85, 49–71 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Gao X., Nguyen T., Strang G.: On factorization of m-channel paraunitary filterbanks. IEEE Trans. Signal Proc 49(7), 1433–1446 (2001)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Hedlund G.A.: Endomorphisms and automorphisms of the shift dynamical systems. Math. Syst. Th 3(4), 320–375 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hein M., Eisert J., Briegel H.: Multi-party entanglement in graph states. Phys. Rev. A 69, 062311 (2004)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Kari J.: Representation of reversible cellular automata with block permutations. Math. Syst, Th 29(1), 47–61 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kato T.: Perturbation theory of linear operators. Springer, Berlin-Heidelberg-Newyork (1995)zbMATHGoogle Scholar
  12. 12.
    Kitaev A.: Anyons in an exactly solved model and beyond. Ann. Phys 321, 2–111 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  14. 14.
    Roe, J.: Lectures on coarse geometry. Providence, RI: Amer. Math. Soc., 2003Google Scholar
  15. 15.
    Schlingemann D., Werner R.: Quantum error-correcting codes associated with graphs. Phys. Rev. A 65(1), 012308 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    Schumacher, B., Werner, R.: Reversible quantum cellular automata. http://arxiv.org/abs/quant-ph/0405174v1, 2004
  17. 17.
    Vogts, H.: Quanten-Zellularautomaten mit lokalen Erhaltungsgrößen. Diplomarbeit, Braunschweig 2005, http://www.imaph.tu-bs.de/ftp/vogts/dip_hv.pdf
  18. 18.
    Werner R.F.: Local preparability of states and the split property in quantum field theory. Lett. Math. Phys 13(4), 325–329 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Zanardi P.: Stabilizing quantum information. Phys. Rev. A 63, 12301 (2001)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Inst. f. Theoretical Physics, Univ. HannoverHannoverGermany
  2. 2.Inst. f. Theoretical Physics, ETH ZürichZürichSwitzerland
  3. 3.Univ. Potsdam, Inst. f. PhysikPotsdam-GolmGermany

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