# Quantum coherence in mutually unbiased bases

## TL;DR

This paper explores the $l_{1}$ norm of quantum coherence in mutually unbiased bases, deriving new results for specific quantum states and introducing the concept of autotensor of mutually unbiased bases to analyze coherence.

## Contribution

It introduces the autotensor of mutually unbiased bases (AMUB) and provides analytical results for the $l_{1}$ norm of coherence in various quantum states within this framework.

## Key findings

- Sum of squared $l_{1}$ coherence for single qubit is less than two.
- The $l_{1}$ norm of coherence for three classes of $X$ states in 4D is equal.
- Coherence levels of Bell-diagonal, Werner, and isotropic states are characterized in AMUB.

## Abstract

We investigate the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit is less than two. We derive the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space is equal. We proposed "autotensor of mutually unbiased basis(AMUB)" by the tensor of mutually unbiased bases, and depict the level surface of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We find the $l_{1}$ norm of coherence of Werner states and isotropic states in AMUB is equal respectively.