A new fidelity of quantum channel evolution and its geometric interpretation

TL;DR

This paper introduces a new $ ext{α-}z$-fidelity measure for quantum channels, explores its properties under various evolutions, extends related entropy concepts, and provides a geometric interpretation for quantum state transformations.

Contribution

It defines the $ ext{α-}z$-fidelity, analyzes its extremal values under different quantum channels, extends $ ext{α-}z$-Rényi entropy, and offers a geometric perspective on quantum state distances.

Findings

Derived limit formulas for $ ext{α-}z$-fidelity extremal values.

Extended $ ext{α-}z$-Rényi relative entropy for resource quantification.

Provided a geometric interpretation of quantum state distances.

Abstract

Fidelity is crucial for characterizing transformations of quantum states under various quantum channels, which can be served as a fundamental tool in resource theories. Firstly, we define an $\alpha$-$z$-fidelity as a significant quantity in quantum information theory and give the properties of the fidelity with orders $\alpha$ and $z$. Secondly, by analyzing the $\alpha$-$z$-fidelity under the evolution of different types of quantum channels (single orbit, all quantum channels, unitary quantum channels, and mixed unitary quantum channels), we propose a limit formula for the maximum and the minimum of the $\alpha$-$z$-fidelity. In addition, we have extended the $\alpha$-$z$-R\'enyi relative entropy, providing new insights into its relevance for resource quantification. Finally, we offer a geometric interpretation for measuring the distance between quantum states, contributing to the broader understanding of the operational and transformative power of dynamical quantum resources across various physical settings.