Capacity of a quantum memory channel correlated by matrix product states

  • Jaideep Mulherkar
  • V Sunitha


We study the capacity of a quantum channel where channel acts like controlled phase gate with the control being provided by a one-dimensional quantum spin chain environment. Due to the correlations in the spin chain, we get a quantum channel with memory. We derive formulas for the quantum capacity of this channel when the spin state is a matrix product state. Particularly, we derive exact formulas for the capacity of the quantum memory channel when the environment state is the ground state of the AKLT model and the Majumdar–Ghosh model. We find that the behavior of the capacity for the range of the parameters is analytic.


Quantum memory channels Quantum capacity Matrix product states 


  1. 1.
    Bayat, A., Burgarth, D., Mancini, S., Bose, S.: Memory effects in spin-chain channels for information transmission. Phys. Rev. A 77, 050306 (2008)ADSCrossRefGoogle Scholar
  2. 2.
    Benenti, G., D’Arrigo, A., Falci, G.: Enhancement of transmission rates in quantum memory channels with damping. Phys. Rev. Lett. 103, 020502 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Kretschmann, D., Werner, R.: Quantum channels with memory. Phys. Rev. A 72, 062323 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Caruso, F., Giovanetti, V., Lupo, C., Mancini, S.: Quantum channels with memory effects. Rev. Mod. Phys. 86, 1203 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    Plenio, M., Virmani, S.: Spin chains and channels with memory. Phys. Rev. Lett. 99, 120504 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Plenio, M., Virmani, S.: Many-body physics and the capacity of quantum channels with memory. New J. Phys. 10, 043032 (2008)CrossRefGoogle Scholar
  7. 7.
    Fannes, M., Nachtergaele, B., Werner, R.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144–3, 443–490 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Verstraete, F., Cirac, J.: Valence bond states for quantum computation. Phys. Rev. A 70, 060302(R) (2004)ADSCrossRefGoogle Scholar
  9. 9.
    Orus, R.: A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. 349, 117–121 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Majumdar, C., Ghosh, D.: On next nearest-neighbour interaction in linear chain. J. Math. Phys. 10, 1388–1402 (1969)ADSCrossRefGoogle Scholar
  11. 11.
    Affleck, A., Kennedy, T., Lieb, E., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477–528 (1988)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Deng, X., Porras, D., Cirac, J.: Effective spin quantum phases in systems of trapped ion. Phys. Rev. A 72, 063407 (2005)ADSCrossRefGoogle Scholar
  13. 13.
    Lewenstein, M., Sanpera, A., Ahufinger, V.: Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems. Oxford University Press, Oxford (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Korepin, V., Ying, X.: Entanglement in valence bond solid states. Int. J. Mod. Phys. B 24(11), 1361–1440 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Benenti, G., DArrigo, A., Falci, G.: Quantum capacity of dephasing channels with memory. New J. Phys. 9, 310 (2007)CrossRefGoogle Scholar
  16. 16.
    Lemos, G., Benenti, G.: Role of chaos in quantum communication through a dynamical dephasing channel. Phys. Rev. A 81, 062331 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Arshed, N., Toor, A., Lidar, D.: Channel capacities of an exactly solvable spin-star system. Phys. Rev. A 81, 062353 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Cover, T., Thomas, J.: Elements of Information Theory. Wiley, Hoboken (2006)zbMATHGoogle Scholar
  19. 19.
    Marchand, K., Mulherkar, J., Nachtergaele, B.: Entropy rate calculations using algebraic measures. In: Proc. IEEE Int. Symp. on Inf. Theory (2012)Google Scholar
  20. 20.
    Accardi, L., Frigerio, A.: Markovian cocycles. Proc. R. Irish Acad. 83, 251–263 (1983)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kennedy, T., Tasaki, H.: Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains. Commun. Math Phys. 147, 431–484 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stephen, D., Wang, D., Prakash, A., Wei, T., Raussendorf, R.: Computational power of symmetry-protected topological phases. Phys. Rev. Lett. 119, 010504 (2017)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dhirubhai Ambani Institute of Information and Communication TechnologyGandhinagarIndia

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