Quantum Information Processing

, Volume 1, Issue 5, pp 409–424 | Cite as

On a Parametrization of Purifications of a Qubit

  • Tiberiu Constantinescu
  • Viswanath Ramakrishna


This work provides, constructively, explicit one–one parametrizations of all purifications of a mixed state in dimension 2 and all joint purifications, if any, of two mixed states in the same dimension. The former is parametrized by SO(3, R), while the latter is parametrized by SO(2, R), except when the state being purified is already pure. These parametrizations are derived without any passage to the spectral decompositions of the state(s) being purified. Using this, we show how to calculate certain measures of quantum information. The appendix considers an alternative one–one parametrization of mixed states in C2 ⊗ C2, which provides a different explicit construction of joint purifications.

PACS: 03.67-a; 03.67-Hk; 03.67-Lx

Purifications parametrizations maximal entangled fraction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 1999).Google Scholar
  2. 2.
    R. Horn and C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1986).Google Scholar
  3. 3.
    W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C 2nd ed (Cambridge University Press, Cambridge, 1992).Google Scholar
  4. 4.
    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. P. Wooters, Phys. Rev. A 54, 3824 (1996).Google Scholar
  5. 5.
    G. Vidal, D. Jonathan, and M. A. Nielsen, Phys. Rev. A 62, 012304 (2000).Google Scholar
  6. 6.
    P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 62, 012311 (2000).Google Scholar
  7. 7.
    S. Bandyopadhyay, Phys. Rev. A 65 022302 (2002).Google Scholar
  8. 8.
    J. Grondalski, D. M. Etlinger, and D. F. V James, Phys Lett A 300, 573 (2002).Google Scholar
  9. 9.
    F. Verstraete and H. Verschelde, Phys. Rev. A 66, 022307 (2002).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Center for Signals, Systems and CommunicationsUniversity of Texas at DallasRichardson

Personalised recommendations