Quantum Information Processing

, Volume 12, Issue 12, pp 3725–3744 | Cite as

Dissipative and non-dissipative single-qubit channels: dynamics and geometry



Single-qubit dissipative and non-dissipative channels, set in the general scenario of a system’s interaction with a squeezed thermal bath, are compared in the Choi isomorphism framework, to bring out their contrasting rank and geometric properties. The equivalence of commutativity between the signal states and the Kraus operators to that between the system and interaction Hamiltonian, and thus to non-dissipativeness, is pointed out. Two distinct unitarily equivalent Kraus representations of the dissipative channel, one based on the Choi isomorphism, and the other based on an ansatz, are used to illustrate that the orthogonality of Kraus operators under the Hilbert–Schmidt inner product is not a unitary invariant. Unlike the non-dissipative (Pauli) channels, the dissipative (squeezed generalized amplitude damping) channels do not form a convex set. Further, whereas the rank of Pauli channels can be any positive integer up to 4, that of the amplitude damping ones is either 2 or 4. In the latter case, a noise range is identified where environmental squeezing counteracts the effect of thermal decoherence.


Dissipative Single-qubit 



Author S. Omkar acknowledges Manipal University, Manipal, India for accepting this work as a part of Ph.D. thesis.


  1. 1.
    Louisell, W.H.: Quantum Statistical Properties of Radiation. Wiley, Canada (1973)Google Scholar
  2. 2.
    Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica A 121, 587–616 (1983)MATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Zurek, W.H.: Decoherence and the transition from quantum to classical. Phys. Today 44, 36–44 (1991)CrossRefGoogle Scholar
  4. 4.
    Zurek, W.H.: Decoherence and the transition from quantum to classical. Prog. Theor. Phys. 87, 281 (1993)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Kraus, K.: States, Effects and Operations. Springer, Berlin (1983)MATHGoogle Scholar
  6. 6.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  7. 7.
    Sudarshan, E.C.G., Mathews, P., Rau, J.: Stochastic dynamics of quantum-mechanical systems. Phys. Rev. 121, 920–924 (1961)MATHMathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Cubitt, T.S., Eisert, J., Wolf, M.M.: The complexity of relating quantum channels to master equations. Commun. Math. Phys. 310, 383–426 (2012)MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Jamiolkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972)MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Horodecki, M., Horodecki, P., Horodecki, R.: General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 60, 1888–1898 (1999)MATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)CrossRefADSGoogle Scholar
  12. 12.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)MATHCrossRefGoogle Scholar
  13. 13.
    Leung, D.W.: Choi’s proof as a recipe for quantum process tomography. J. Math. Phys. 44, 528–533 (2003)MATHMathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Havel, T.F.: Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups. J. Math. Phys. 44, 534–557 (2003)MATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Rodrguez-Rosario, C.A., Modi, K.: Completely positive maps and classical correlations. J. Phys. A Math. Theor. 41, 205301 (2008)CrossRefADSGoogle Scholar
  16. 16.
    Devi, A.R.U., Rajagopal, A.K.: Open-system quantum dynamics with correlated initial states, not completely positive maps, and non-Markovianity. Phys. Rev. A 83, 022109 (2011)CrossRefADSGoogle Scholar
  17. 17.
    Srikanth, R., Banerjee, S.: Squeezed generalized amplitude damping channel. Phys. Rev. A 77, 012318 (2008)CrossRefADSGoogle Scholar
  18. 18.
    Uhlman, A.: On 1-qubit channels. J. Phys. A Math. Gen. 34, 7047–7055 (2001)CrossRefADSGoogle Scholar
  19. 19.
    Banerjee, S., Ghosh, R.: Dynamics of decoherence without dissipation in a squeezed thermal bath. J. Phys. A. Math. Theor. 40, 13735–13754 (2007)MATHMathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Shao, J., Ge, M.-L., Cheng, H.: Decoherence of quantum-nondemolition systems. Phys. Rev. E 53, 1243–1245 (1996)CrossRefADSGoogle Scholar
  21. 21.
    Mozyrsky, D., Privman, V.: Adiabatic decoherence. J. Stat. Phys. 91, 787–799 (1998)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gangopadhyay, G., Kumar, M.S., Dattagupta, S.: On dissipationless decoherence. J. Phys. A Math. Gen. 34, 5485–5495 (2001)MATHMathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Braginsky, V.B., Vorontsov, Y.I., Thorne, K.S.: Quantum nondemolition measurements. Science 209, 547–557 (1980)CrossRefADSGoogle Scholar
  24. 24.
    Caves, C.M., Thorne, K.D., Drever, R.W.P., Sandberg, V.D., Zimmerman, M.: On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle. Rev. Mod. Phys. 52, 341–392 (1980)CrossRefADSGoogle Scholar
  25. 25.
    Bocko, M.F., Onofrio, R.: On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress. Rev. Mod. Phys. 68, 755–799 (1996)CrossRefADSGoogle Scholar
  26. 26.
    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)MATHGoogle Scholar
  27. 27.
    Myatt, C.J., King, B.E., Turchette, Q.A., Sackett, C.A., et al.: Decoherence of quantum superpositions through coupling to engineered reservoirs. Nature 403, 269–273 (2000)CrossRefADSGoogle Scholar
  28. 28.
    Turchette, Q.A., Myatt, C.J., King, B.E., Sackett, C.A., et al.: Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs. Phys. Rev. A 62, 053807 (2000)CrossRefADSGoogle Scholar
  29. 29.
    Pryde, G.J., O’Brien, J.L., White, A.G., et al.: Measuring a photonic qubit without destroying it. Phys. Rev. Lett. 92, 190402 (2004)CrossRefADSGoogle Scholar
  30. 30.
    O’Brien, J.L., Pryde, G.J., White, A.G., et al.: Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003)CrossRefADSGoogle Scholar
  31. 31.
    Xu, J.-S., Xu, X.-Y., Li, C.-F., Zhang, C.-J., et al.: Discrete plasticity in sub-10-nm-sized gold crystals. Nat. Commun. 10, 1–8 (2010)Google Scholar
  32. 32.
    Krauter, H., Muschik, C.A., Jensen, K., Wasilewski, W., et al.: Entanglement generated by dissipation and steady state entanglement of two macroscopic objects. Phys. Rev. Lett. 107, 080503 (2011)CrossRefADSGoogle Scholar
  33. 33.
    Kennedy, T.A.B., Walls, D.F.: Squeezed quantum fluctuations and macroscopic quantum coherence. Phys. Rev. A 37, 152–157 (1988)CrossRefADSGoogle Scholar
  34. 34.
    Kim, M.S., Bužek, V.: Photon statistics of superposition states in phase-sensitive reservoirs. Phys. Rev. A 47, 610–619 (1993)CrossRefADSGoogle Scholar
  35. 35.
    Bužek, V., Knight, P.L., Kudryavtsev, I.K.: Three-level atoms in phase-sensitive broadband correlated reservoirs. Phys. Rev. A 44, 1931–1947 (1991)CrossRefADSGoogle Scholar
  36. 36.
    Georgiades, N.P., Polzik, E.S., Edamatsu, K., Kimble, H.J., Parkins, A.S.: Nonclassical excitation for atoms in a squeezed vacuum. Phys. Rev. Lett. 75, 3426–3429 (1995)CrossRefADSGoogle Scholar
  37. 37.
    Turchette, Q.A., Georgiades, N.P., Hood, C.J., Kimble, H.J., Parkins, A.S.: Squeezed excitation in cavity QED: experiment and theory. Phys. Rev. A 58, 4056–4077 (1998)CrossRefADSGoogle Scholar
  38. 38.
    Banerjee, S., Srikanth, R.: Geometric phase of a qubit interacting with a squeezed-thermal bath. Eur. Phys. J. D 46, 335–344 (2008)MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    Uhlman, A.: General Theory of Information Transfer and Combinatorics, pp. 413–424. Springer, Berlin (2006)CrossRefGoogle Scholar
  40. 40.
    Narang, G.: Simulating a single-qubit channel using a mixed-state environment. Phys. Rev. A 75, 032305 (2007)CrossRefADSGoogle Scholar
  41. 41.
    Bengtsson, I., Kyczkowski, K.: Geometry of Quantum States. Cambridge University Press, Cambridge (2006)MATHCrossRefGoogle Scholar
  42. 42.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)CrossRefADSGoogle Scholar
  43. 43.
    Maheshan, E.: Depolarizing behavior of quantum channels in higher dimensions. Quantum Inf. Comput. 11, 0466–484 (2011)Google Scholar
  44. 44.
    Bowdrey, M.D., Oi, D.K.L., Short, A.J., Banaszek, K., Jones, J.A.: Fidelity of single qubit maps. Phys. Lett. A 294, 258–260 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  45. 45.
    Nielsen, M.A.: A simple formula for the average gate fidelity of a quantum dynamical operation. Phys. Lett. A 303, 249–252 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  46. 46.
    Cortese, J.: Trends in Quantum Physics, pp. 125–172. Nova Science Publishers Inc., New York (2004)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Poornaprajna Institute of Scientific ResearchBangaloreIndia
  2. 2.Indian Institute of Technology RajasthanJodhpurIndia

Personalised recommendations