On Kirkwood--Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

Abstract

An issue which has attracted increasing attention in contemporary researches are Kirkwood--Dirac quasiprobabilities. List of their use includes many questions of quantum physics. Applications of complex tight frames in quantum information science were recently demonstrated. It is shown in this paper that quasiprobabilities naturally appear in the context of unravelings of a quantum channel. Using vectors of the given tight frame to build principal Kraus operators generates quasiprobabilities with interesting properties. For an equiangular tight frame, we characterize the Hilbert--Schmidt and spectral norms of the matrix consisted of quasiprobabilities. Hence, novel uncertainty relations in terms of R\'{e}nyi and Tsallis entropies are obtained. New inequalities for characterizing the location of eigenvalues are derived. They give an alternative to estimating on the base of Ger\v{s}gorin's theorem. A utility of the presented inequalities is exemplified with symmetric informationally complete measurement in dimension two.