Average coherence with respect to generalized Wigner-Yanase-Dyson skew information and (N,M)-POVMs

Quantum coherence is a significant physical resource stemming from the principle of superposition of quantum states, and plays a pivotal role in elucidating the unique and intriguing properties of the quantum realm. At the same time, quantum measurement, serving as a critical tool for quantifying coherence, is intrinsically linked to the choice of measurement basis. Building on this foundation, we delve into the average coherence of arbitrary quantum states by leveraging generalized Wigner-Yanase-Dyson skew information and a broad class of informationally complete symmetric measurements, the latter can be viewed as a generalization encompassing of symmetric, informationally complete POVMs and mutually unbiased measurements. Furthermore, we define the maximal coherence of generalized Wigner-Yanase-Dyson skew information, and a series of trade-off relations between different maximal coherence and average coherence are derived. In order to verify the reliability of the obtained results, we have carried out detailed analysis with concrete examples.