Journal of Russian Laser Research

, Volume 31, Issue 3, pp 211–231

Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

Article

Abstract

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments on the Lie algebraic structure of SIC-POVMs are presented.

Keywords

SIC-POVM probability representation of quantum mechanics star-product-quantization scheme quantum tomography Lie algebraic structure 

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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (State University)Moscow RegionRussia
  2. 2.P. N. Lebedev Physic, Russian Academy of SciencesMoscowRussia

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