Advertisement

Foundations of Physics

, Volume 35, Issue 12, pp 1985–2006 | Cite as

Minimal Informationally Complete Measurements for Pure States

  • Steven T. FlammiaEmail author
  • Andrew Silberfarb
  • Carlton M. Caves
Article

Abstract

We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PS I-complete) POVM. We show that a measurement with 2D−1 outcomes cannot be PS I-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PS I-complete POVM has 2D outcomes. We also consider PS I-complete POVMs that have only rank-one POVM elements and construct an example with 3D−2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PS I-complete POVM is left open.

Keywords

Quantum state tomography informationally complete measurement Positive Operator-Valued Measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Prugovečki, E. 1977Int J Theor Phys.16321Google Scholar
  2. 2.
    Busch, P., Lahti, P.J. 1989Found Phys.19633MathSciNetGoogle Scholar
  3. 3.
    A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, The Netherlands, 1993). POVMs are discussed in Secs. 9-5 and 9-6, and PS I-complete measurements in Sec. 3–5.Google Scholar
  4. 4.
    Caves, C.M., Fuchs, C.A., Schack, R. 2002J Math Phys.434537ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M. 2004J Math Phys.452171ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Renes, J.M. 2004Phys Rev A70052314ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fuchs C.A., “On the quantumness of a Hilbert space,” arXiv.org e-print quant-ph/0404122.Google Scholar
  8. 8.
    Fuchs, C.A., Sasaki, M. 2003Quant Inf Comp.3377MathSciNetGoogle Scholar
  9. 9.
    W. Pauli, in Handbuch der Physik, Vol. XXIV, Pt. 1, edited by H. Geiger and K.~Scheel (Springer, Berlin, 1933), p. 98; reprinted in Encyclopedia of Physics, Vol. V, Part 1 (Springer, Berlin, 1958), p. 17.Google Scholar
  10. 10.
    Weigert, S. 1992Phys Rev A457688ADSCrossRefGoogle Scholar
  11. 11.
    Weigert, S. 1996Phys Rev A532078ADSMathSciNetGoogle Scholar
  12. 12.
    Corbett, J.V., Hurst, C.A. 1978J Austral Math Soc B20182MathSciNetGoogle Scholar
  13. 13.
    Amiet, J.-P., Weigert, S. 1999J Phys A322777ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Amiet, J.-P., Weigert, S.J 1999J Opt B1L5MathSciNetGoogle Scholar
  15. 15.
    Peres, A., Wootters, W.K. 1992Phys Rev A661119Google Scholar
  16. 16.
    Iyanaga, S.Kawada, Y. eds. 1980Encyclopedic Dictionary of MathematicsMIT PressCambridge, MA682Google Scholar
  17. 17.
    Wyler, J.A. 1974Gen Rel Grav.5175CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Steven T. Flammia
    • 1
    Email author
  • Andrew Silberfarb
    • 1
  • Carlton M. Caves
    • 1
  1. 1.Department of Physics and AstronomyUniversity of New MexicoNew MexicoUSA

Personalised recommendations