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

TL;DR

This paper introduces a unified formalism for constructing various operator bases on Hilbert spaces using d^2 x N matrices, encompassing known bases like SIC-POVMs and MUBs, and classifies star-product schemes based on matrix properties.

Contribution

It develops a general framework for generating operator bases from matrices, unifying known bases and classifying star-product schemes by matrix type.

Findings

Known bases are special cases of the general formulas.

Unitary matrices lead to self-dual star-product schemes.

Self-dual schemes are characterized by dequantizers that do not form POVMs.

Abstract

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.