Entropic uncertainty relations for SIC-POVMs and MUMs

TL;DR

This paper develops improved entropic uncertainty relations for SIC-POVMs and MUMs in finite-dimensional quantum systems, using inequalities involving Rényi entropy and coincidence indexes, with tight bounds for mixed states.

Contribution

It introduces new inequalities linking Rényi entropy and coincidence indexes to derive tighter state-dependent uncertainty relations for SIC-POVMs and MUMs.

Findings

Uncertainty relations are tight for sufficiently mixed states.

Relations are based on inequalities between Rényi entropy and coincidence indexes.

Comparisons with numerical results demonstrate improvements.

Abstract

We construct inequalities between R\'{e}nyi entropy and the indexes of coincidence of probability distributions, based on which we obtain improved state-dependent entropic uncertainty relations for general symmetric informationally complete positive operator-valued measures (SIC-POVM) and mutually unbiased measurements (MUM) on finite dimensional systems. We show that our uncertainty relations for general SIC-POVMs and MUMs can be tight for sufficiently mixed states, and moreover, comparisons to the numerically optimal results are made via information diagrams.