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

الأصل · EN

Columns of d² 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 d² 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 d² 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.