# Uncertainty relations for quantum coherence with respect to mutually unbiased bases

## TL;DR

This paper develops new uncertainty relations for quantum coherence across mutually unbiased bases, providing bounds on coherence measures like relative entropy and geometric coherence, and introduces state-independent relations using min-entropy.

## Contribution

It introduces novel uncertainty relations for quantum coherence with respect to MUBs, including bounds for multiple coherence measures and state-independent relations.

## Key findings

- Derived upper bounds on coherence sums in MUBs
- Established lower bounds as uncertainty relations for coherence
- Presented state-independent uncertainty relations using min-entropy

## Abstract

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.