Bypassing State Initialisation in Perfect State Transfer Protocols on Spin-Chains

  • Carlo Di Franco
  • Mauro Paternostro
  • M. S. Kim
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6519)

Abstract

Although a complete picture of the full evolution of complex quantum systems would certainly be the most desirable goal, for particular Quantum Information Processing schemes such an analysis is not necessary. When quantum correlations between only specific elements of a many-body system are required for the performance of a protocol, a more distinguished and specialised investigation is helpful. Here, we provide a striking example with the achievement of perfect state transfer in a spin chain without state initialisation, whose realisation has been shown to be possible in virtue of the correlations set between the first and last spin of the transmission-chain.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Osterloh, A., et al: Nature 416, 608 (2002); Osborne, T.J., Nielsen, M.A.: Phys. Rev. A 66, 032110 (2002)Google Scholar
  2. 2.
    Schuch, N., et al.: Phys. Rev. Lett. 100, 030504 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Wilson, K.G.: Rev. Mod. Phys. 47, 773 (1975); White, S.R.: Phys. Rev. Lett. 69, 2863 (1992); Fannes, M., Nachtergaele, B., Werner, R.F.: Lett. Math. Phys. 25, 249 (1992); Vidal, G.: Phys. Rev. Lett. 91, 147902 (2003); Verstraete, F., Garcia-Ripoll, J.J., Cirac, J.I.: Phys. Rev. Lett. 93, 207204 (2004); Schollwöck, U.: Rev. Mod. Phys. 77, 259 (2005); Verstraete, F., Cirac, J.I. (2004), e-print arXiv:cond-mat/0407066; Vidal, G.: Phys. Rev. Lett. 99, 220405 (2007); Anders, S., et al.: Phys. Rev. Lett. 97, 107206 (2006); Anders, S., Briegel, H.-J., Dür, W.: New J. Phys. 9, 361 (2007)Google Scholar
  4. 4.
    Di Franco, C., Paternostro, M., Kim, M.S.: Phys. Rev. Lett. 101, 230502 (2008)CrossRefGoogle Scholar
  5. 5.
    Benjamin, S.C., Bose, S.: Phys. Rev. Lett. 90, 247901 (2003)CrossRefGoogle Scholar
  6. 6.
    DiVincenzo, D.P.: Mesoscopic Electron Transport. In: Kowenhoven, L., Schön, G., Sohn, L. (eds.) Kluwer, Dordrecht (1997)Google Scholar
  7. 7.
    Bose, S.: Phys. Rev. Lett. 91, 207901 (2003); Bose, S.: Contemp. Phys. 48, 13 (2007); Burgarth, D.: PhD thesis, University College London (2006)Google Scholar
  8. 8.
    Christandl, M., et al.: Phys. Rev. Lett. 92, 187902 (2004); Christandl, M., et al.: Phys. Rev. A 71, 032312 (2005); Nikolopoulos, G.M., Petrosyan, D., Lambropoulos, P.: Europhys. Lett. 65, 297 (2004); J. Phys.: Condens. Matter 16, 4991 (2004)Google Scholar
  9. 9.
    Di Franco, C., Paternostro, M., Palma, G.M.: Int. J. Quant. Inf. 6(Supp. 1), 659 (2008)CrossRefGoogle Scholar
  10. 10.
    The final state of spin N can be easily obtained from Eqs. (3)-(5). For instance, if N is even, \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_1\hat{X}_N(t^*)}\rangle=\langle{\hat{X}_1\hat{X}_N(0)}\rangle\) and \(\langle{\hat{X}_1\hat{Y}_N(t^*)}\rangle=\langle{\hat{Y}_1\hat{X}_N(0)}\rangle\). If qubit N has been projected onto ∣ ±N 〉 [for which \(\langle{\hat{X}_N(0)}\rangle=\pm(-1)^{\frac{N}{2}}\)], we have \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_1\hat{X}_N(t^*)}\rangle=\pm(-1)^{\frac{N}{2}}\langle{\hat{X}_1(0)}\rangle\) and \(\langle{\hat{X}_1\hat{Y}_N(t^*)}\rangle=\pm(-1)^{\frac{N}{2}}\langle{\hat{Y}_1(0)}\rangle\). The state of spin N, after the measurement performed on spin 1, will satisfy \(\langle{\hat{Z}_N(t^*)}\rangle=\langle{\hat{Z}_1(0)}\rangle\), \(\langle{\hat{X}_N(t^*)}\rangle=(-1)^{\frac{N}{2}}c\,\langle{\hat{X}_1(0)}\rangle\) and \(\langle{\hat{Y}_N(t^*)}\rangle=(-1)^{\frac{N}{2}}c\,\langle{\hat{Y}_1(0)}\rangle\), where c is the product of the measurement outcomes at 1 (after the evolution) and N (before the evolution). The state \((\hat{Z}^{\frac{N}{2}})\rho^{in}(\hat{Z}^{\frac{N}{2}})\) [\((\hat{Z}^{\frac{N}{2}+1})\rho^{in}(\hat{Z}^{\frac{N}{2}+1})\)] satisfies these conditions for c = 1 (c = − 1) Google Scholar
  11. 11.
    Di Franco, C., et al.: Phys. Rev. A 76, 042316 (2007)CrossRefGoogle Scholar
  12. 12.
    Markiewicz, M., Wiesniak, M.: Phys. Rev. A 79, 054304 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Carlo Di Franco
    • 1
  • Mauro Paternostro
    • 2
  • M. S. Kim
    • 3
    • 4
  1. 1.Department of PhysicsUniversity College CorkCorkRepublic of Ireland
  2. 2.School of Mathematics and PhysicsQueen’s UniversityBelfastUnited Kingdom
  3. 3.Institute for Mathematical SciencesImperial College LondonUnited Kingdom
  4. 4.QOLS, The Blackett LaboratoryImperial College LondonUnited Kingdom

Personalised recommendations