Fidelity is crucial for characterizing transformations of quantum states under various quantum channels, serving as a fundamental tool in resource theories. An interesting work is to unify different concepts of fidelity proposed in the literature under a single framework. Firstly, we define an \(\alpha \) - \(z\) -fidelity as a significant quantity in quantum information theory and analyze its properties for different orders \(\alpha \) and \(z\) . Secondly, by investigating the \(\alpha \) -z-fidelity under the evolution of different types of quantum channels, including unitary orbits, quantum channels, unital channels, and mixed unitary channels, we derive explicit formulas for their maximum and minimum values. In addition, we have extended the \(\alpha \) -z-Rényi relative entropy, providing new insights into its relevance for resource quantification. Finally, we offer a geometric interpretation for measuring the distance between quantum states, advancing the understanding of the operational and transformative power of dynamical quantum resources in various physical settings.
