We study the average quantum coherence over the pure state decompositions of a mixed quantum state. An upper bound of the average quantum coherence is provided, and sufficient conditions for the saturation of the upper bound are shown. These sufficient conditions always hold for two- and three-dimensional systems. This provides a tool to estimate the average coherence experimentally by measuring only the diagonal elements, which remarkably requires less measurements compared with state tomography. We then describe the pure state decompositions of the qubit state in the Bloch sphere geometrically. For any given qubit state, the optimal pure state decomposition achieving the maximal average quantum coherence as well as three other pure state decompositions are shown in the Bloch sphere. The order relations among their average quantum coherence are invariant for any coherence measure. The results presented in this paper are universal and suitable for all coherence measures.

中文翻译：

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纯态分解的平均量子相干

我们研究了混合量子态的纯态分解的平均量子相干性。提供了平均量子相干的上限，并且示出了上限饱和的充分条件。这些足够的条件对于二维和三维系统始终成立。这提供了一种仅通过测量对角线元素来通过实验来估计平均相干性的工具，与状态层析成像相比，这显然需要更少的测量。然后，我们用几何方法描述Bloch球体中qubit态的纯态分解。对于任何给定的量子位态，在布洛赫球中显示了实现最大平均量子相干性的最佳纯态分解以及其他三个纯态分解。它们的平均量子相干性之间的顺序关系对于任何相干性度量都是不变的。本文介绍的结果具有普遍性，适用于所有相干性度量。