Quantum coherence in mutually unbiased bases

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.