Quantum Information Processing

, Volume 13, Issue 7, pp 1607–1637 | Cite as

Quantum networks: anti-core of spin chains

  • E. Jonckheere
  • F. C. Langbein
  • S. Schirmer


The purpose of this paper is to exhibit a quantum network phenomenon—the anti-core—that goes against the classical network concept of congestion core. Classical networks idealized as infinite, Gromov hyperbolic spaces with least-cost path routing (and subject to a technical condition on the Gromov boundary) have a congestion core, defined as a subnetwork that routing paths have a high probability of visiting. Here, we consider quantum networks, more specifically spin chains, define the so-called maximum excitation transfer probability \(p_{\max }(i,j)\) between spin \(i\) and spin \(j\) and show that the central spin has among all other spins the lowest probability of being excited or transmitting its excitation. The anti-core is singled out by analytical formulas for \(p_{\mathrm{max}}(i,j)\), revealing the number theoretic properties of quantum chains. By engineering the chain, we further show that this probability can be made vanishingly small.


Quantum networks Spin chains Transfer probability Distance Gromov negatively curved networks  Anti-core 



This research was supported by ARO MURI Contract W911NF-11-1-0268 and by NSF Grant CNS NetSE 1017881.


  1. 1.
    Ariaei, F., Lou, M., Jonckheere, E., Krishnamachari, B., Zuniga, M.: Curvature of indoor sensor network: clustering coefficient. EURASIP J. Wirel. Commun. Netw. 20, Article ID 213185. doi: 10.1155/2008/2131185 (2008)
  2. 2.
    Baryshnikov, Y., Tucci, G.: Asymptotic traffic flow in a hyperbolic network. In: International Symposium on Communications, Control, and Signal Processing (ISCCSP), Rome, Italy, May 2–4 (2012)Google Scholar
  3. 3.
    Bridson, Martin R., Haefliger, André: Metric Spaces of Non-positive Curvature, volume 319 of A Series of Comprehensive Surveys in Mathematics. Springer, New York (1999)Google Scholar
  4. 4.
    Jonckheere, E., Ariaei, F., Lohsoonthorn, P.: Scaled Gromov four-point condition for network graph curvature computation. Internet Math. 7(3), 137–177 (2011). doi: 10.1080/15427951.2011.601233 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jonckheere, E., Langbein, F.C., Schirmer, S.G.: Curvature of quantum rings. In: Proceedings of the 5th International Symposium on Communications, Control and Signal Processing (ISCCSP 2012), Rome, Italy, May 2–4 (2012)Google Scholar
  6. 6.
    Jonckheere, E., Lohsoonthorn, P., Bonahon, F.: Scaled Gromov hyperbolic graphs. J. Graph Theory 57, 157–180 (2008). doi: 10.1002/jgt.20275 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jonckheere, E., Schirmer, S., Langbein, F.: Geometry and curvature of spin networks. In: IEEE Multi-conference on Systems and Control, pp. 786–791, Denver, CO (2011). Available at arXiv:1102.3208v1 [quant-ph]
  8. 8.
    Jonckheere, Edmond, Lou, Mingji, Bonahon, Francis, Baryshnikov, Yuliy: Euclidean versus hyperbolic congestion in idealized versus experimental networks. Internet Math. 7(1), 1–27 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Lectures in Mathematics. Birkhauser, Basel (1997)CrossRefGoogle Scholar
  10. 10.
    Lou Mingji: Traffic Pattern Analysis in Negatively Curved Network. PhD thesis, Department of Electrical Engineering-Systems, University of Southern California (2008)Google Scholar
  11. 11.
    Narayan, O., Saniee, I.: Large-scale curvature of networks. Phys. Rev. E, 84, 066108-1–066108-8 (2011)Google Scholar
  12. 12.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  13. 13.
    Trifonov, D.A.: On the ‘Polarized Distances Between Quantum States and Observables’. arXiv: quant-ph/0410045v1 (2004)
  14. 14.
    Wang, X., Pemberton-Ross, P., Schirmer, S.G.: Symmetry and Controllability for Spin Networks with a Single-Node Control. arXiv:1012.3695v2 [quant-ph] (2011)
  15. 15.
    Wootters, W.K.: Statistical distance and Hilbert space. Phys. Rev. D 23, 357–362 (1981)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Electrical Engineering & Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Computer Science & InformaticsCardiff UniversityCardiffUK
  3. 3.College of Science (Physics)Swansea UniversitySwanseaUK

Personalised recommendations