Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces

Details Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces

2010-12-29

arxiv.org/abs/1012.6045v2

Journal of Russian Laser Research 32 (2011) 56-67

Physics

Quantum Physics

12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser Research

Scientific paper

10.1007/s10946-011-9189-z

Columns of d^2 x 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 by using d^2 x 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 developed unified approach. Star-product schemes are classified with respect to associated d^2 x 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 POVM.

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