Letters in Mathematical Physics

, Volume 109, Issue 1, pp 1–9 | Cite as

All unital qubit channels are 4-noisy operations

  • Alexander Müller-Hermes
  • Christopher PerryEmail author


We show that any unital qubit channel can be implemented by letting the input system interact unitarily with a four-dimensional environment in the maximally mixed state and then tracing out the environment. We also provide an example where the dimension of such an environment has to be at least 3.


Quantum channels Open quantum systems Noisy operations Factorizable maps Operator Schmidt rank 

Mathematics Subject Classification

81P45 46L07 46L60 


  1. 1.
    Anantharaman-Delaroche, C.: On ergodic theorems for free group actions on noncommutative spaces. Probab. Theory Relat. Fields 135(4), 520–546 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chefles, A.: Quantum operations, state transformations and probabilities. Phys. Rev. A 65(5), 052314 (2002)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choi, M.D., Wu, P.Y.: Convex combinations of projections. Linear Algebra Appl. 136(Supplement C), 25–42 (1990).
  6. 6.
    Dür, W., Vidal, G., Cirac, J.I.: Optimal conversion of nonlocal unitary operations. Phys. Rev. Lett. 89(5), 057,901 (2002)CrossRefGoogle Scholar
  7. 7.
    Haagerup, U., Musat, M.: Factorization and dilation problems for completely positive maps on von Neumann algebras. Commun. Math. Phys. 303(2), 555–594 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haagerup, U., Musat, M.: An asymptotic property of factorizable completely positive maps and the Connes embedding problem. Commun. Math. Phys. 338(2), 721–752 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Horodecki, M., Horodecki, P., Oppenheim, J.: Reversible transformations from pure to mixed states and the unique measure of information. Phys. Rev. A 67(6), 062,104 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kadison, R.V., Pedersen, G.K.: Means and convex combinations of unitary operators. Math. Scand. 57, 249–266 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Landau, L., Streater, R.: On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras. Linear Algebra Appl. 193, 107–127 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications. Springer, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Müller-Hermes, A., Nechita, I.: Restrictions on the Schmidt rank of bipartite unitary operators beyond dimension two. arXiv preprint arXiv:1612.07616 (2016)
  14. 14.
    Musat, M.: In preparationGoogle Scholar
  15. 15.
    Musz, M., Kuś, M., Życzkowski, K.: Unitary quantum gates, perfect entanglers, and unistochastic maps. Phys. Rev. A 87(2), 022,111 (2013)CrossRefGoogle Scholar
  16. 16.
    Nielsen, M.A.: Quantum information theory. arXiv preprint arXiv:quant-ph/0011036 (2000)
  17. 17.
    Ruskai, M.B., Szarek, S., Werner, E.: An analysis of completely-positive trace-preserving maps on M2. Linear Algebra Appl. 347(1–3), 159–187 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Scharlau, J., Mueller, M.P.: Quantum Horn’s lemma, finite heat baths, and the third law of thermodynamics. Quantum 2, 54 (2018). CrossRefGoogle Scholar
  19. 19.
    Życzkowski, K., Bengtsson, I.: On duality between quantum maps and quantum states. Open Syst. Inf. Dyn. 11(01), 3–42 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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