Skip to main content
SpringerLink
Log in

All unital qubit channels are 4-noisy operations

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Note that \(p\succ q\) is equivalent to \(\left( q_1,\dots ,q_k\right) \in \text {Conv}\left\{ \left( p_{\sigma (1)},\dots ,p_{\sigma (k)}\right) :\sigma \in S_k\right\} \) where we append zeros to p or q as necessary to make them the same size (see [12, Corollary B.3]).

  2. With respect to the Hilbert-Schmidt inner product.

  3. \(\sigma _1=\begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix}\), \(\sigma _2=\begin{pmatrix} 0 &{} -i \\ i &{} 0 \end{pmatrix}\), \(\sigma _3=\begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}\).

References

  1. Anantharaman-Delaroche, C.: On ergodic theorems for free group actions on noncommutative spaces. Probab. Theory Relat. Fields 135(4), 520–546 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2017)

    Book  MATH  Google Scholar 

  3. Chefles, A.: Quantum operations, state transformations and probabilities. Phys. Rev. A 65(5), 052314 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  4. Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Choi, M.D., Wu, P.Y.: Convex combinations of projections. Linear Algebra Appl. 136(Supplement C), 25–42 (1990). https://doi.org/10.1016/0024-3795(90)90019-9. http://www.sciencedirect.com/science/article/pii/0024379590900199

  6. Dür, W., Vidal, G., Cirac, J.I.: Optimal conversion of nonlocal unitary operations. Phys. Rev. Lett. 89(5), 057,901 (2002)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Kadison, R.V., Pedersen, G.K.: Means and convex combinations of unitary operators. Math. Scand. 57, 249–266 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications. Springer, New York (1979)

    MATH  Google Scholar 

  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. Musat, M.: In preparation

  15. Musz, M., Kuś, M., Życzkowski, K.: Unitary quantum gates, perfect entanglers, and unistochastic maps. Phys. Rev. A 87(2), 022,111 (2013)

    Article  Google Scholar 

  16. Nielsen, M.A.: Quantum information theory. arXiv preprint arXiv:quant-ph/0011036 (2000)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Scharlau, J., Mueller, M.P.: Quantum Horn’s lemma, finite heat baths, and the third law of thermodynamics. Quantum 2, 54 (2018). https://doi.org/10.22331/q-2018-02-22-54

    Article  Google Scholar 

  19. Życzkowski, K., Bengtsson, I.: On duality between quantum maps and quantum states. Open Syst. Inf. Dyn. 11(01), 3–42 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Alexander Müller-Hermes & Christopher Perry

Authors
  1. Alexander Müller-Hermes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher Perry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher Perry.

Additional information

We thank Magdalena Musat for interesting discussions and comments that helped us to improve this manuscript. We also thank Karol Życzkowski for bringing up a question regarding the relationship between unital qubit channels and n-noisy operations at Institut Henri Poincaré during the trimester on “Analysis in Quantum Information Theory”. We acknowledge financial support from the European Research Council (ERC Grant Agreement No 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Müller-Hermes, A., Perry, C. All unital qubit channels are 4-noisy operations. Lett Math Phys 109, 1–9 (2019). https://doi.org/10.1007/s11005-018-1104-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-018-1104-x

Keywords

Mathematics Subject Classification

Search

Navigation