Unitary and non-unitary matrices as a source of different bases of operators acting on hilbert spaces
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- Filippov, S.N. & Man’ko, V.I. J Russ Laser Res (2011) 32: 56. doi:10.1007/s10946-011-9189-z
Columns of d2×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 d2×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 d2×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.