Article

Journal of Russian Laser Research

, 32:56

First online:

Unitary and non-unitary matrices as a source of different bases of operators acting on hilbert spaces

  • Sergey N. FilippovAffiliated withMoscow Institute of Physics and Technology (State University)P. N. Lebedev Physical Institute, Russian Academy of Sciences Email author 
  • , Vladimir I. Man’koAffiliated withMoscow Institute of Physics and Technology (State University)P. N. Lebedev Physical Institute, Russian Academy of Sciences

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Abstract

Columns of d 2 ×N matrices are shown to create different sets of N operators acting on d-dimensional Hilbert space. This construction corresponds to a formalism of the star-product of operator symbols. The known bases are shown to be partial cases of generic formulas derived using d 2 ×N matrices as a source for constructing arbitrary bases. The known examples of the SIC-POVM, MUBs, and the phase-space description of qubit states are considered from the viewpoint of the unified approach developed. Star-product schemes are classified with respect to associated d 2 ×N matrices. In particular, unitary matrices correspond to self-dual schemes. Such self-dual star-product schemes are shown to be determined by dequantizers which do not form the POVM.

Keywords

finite-dimensional Hilbert space basis of operators star-product scheme unitary matrix self-dual scheme