Uncertainty relations for quantum measurements from generalized equiangular tight frames

Abstract

The current study aims to examine uncertainty relations for measurements from generalized equiangular tight frames. Informationally overcomplete measurements are a valuable tool in quantum information processing, including tomography and state estimation. The maximal sets of mutually unbiased bases are the most common case of such measurements. The existence of $d+1$ mutually unbiased bases is proved for $d$ being a prime power. More general classes of informationally overcomplete measurements have been proposed for various purposes. Measurements of interest are typically characterized by some inner structure maintaining the required properties. It leads to restrictions imposed on generated probabilities. To apply the considered measurements, these restrictions should be converted into information-theoretic terms. It is interesting that certain restrictions hold irrespectively to overcompleteness. To describe the amount of uncertainty quantitatively, we use the Tsallis and R\'{e}nyi entropies as well as probabilities of separate outcomes. The obtained results are based on estimation of the index of coincidence. The derived relations are briefly exemplified.