Average coherence and entropy
Yajing Fan, Nan Li, Shunlong Luo · Physical Review A · 2023
Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. To eliminate the influence of the reference basis on the coherence of a quantum state and uncover its intrinsic properties, it is common to study coherence by averaging over different reference bases. Using the metric-adjusted skew information, we explore three natural approaches to average coherence of a state: average over all orthonormal bases, average over all elements of an operator orthonormal base, and average over a complete family of mutually unbiased bases. We establish the equivalence among these three types of average coherence and interpret the unified average coherence as the coherence of a quantum state relative to a depolarizing channel. Additionally, we employ the unified average coherence to introduce a notion of quantum $f$ entropy (where $f$ is an operator monotone function associated with the metric-adjusted skew information) and demonstrate that quantum $f$ entropy possesses properties analogous to the ubiquitous von Neumann entropy. Furthermore, we illuminate some connections between quantum $f$ entropy and quasientropy, and compare $f$ entropy with von Neumann entropy, R\'enyi entropy, and Tsallis entropy for some typical states.