Average and maximal coherence based on the modified generalized Wigner–Yanase–Dyson skew information

Coherence is a fundamental feature of quantum mechanics and plays a crucial role in quantum information. In recent years, the quantification of coherence has aroused great interest, and various coherence quantifiers have appeared. We evaluate average coherence of a quantum state based on the modified generalized Wigner–Yanase–Dyson skew information with respect to all orthonormal bases as well as with respect to arbitrary mutually unbiased bases, and demonstrate their equivalence. We further evaluate average variance and average modified generalized variance, and establish several trade-off relations between them and linear entropy, the Brukner–Zeilinger invariant information and average coherence based on the Hilbert–Schmidt norm. In addition, we introduce two kinds of maximal coherence of a quantum state based on the modified generalized Wigner–Yanase–Dyson skew information and metric-adjusted skew information, respectively. Both of them extend the existing maximal coherence based on the Wigner–Yanase skew information and Wigner–Yanase–Dyson skew information. At the same time, it is verified numerically that the maximal coherence is almost equivalent to the average coherence in high-dimensional systems.