Imaginaring and deimaginaring power of quantum channels and the trade-off between imaginarity and entanglement

The resource theory of imaginarity provides a valuable framework for understanding the role of complex numbers. Quantum channels play a vital role in extraction and transmission of information, and they can both destroy and create imaginarity in quantum states. In this article, we introduce the concepts of imaginaring and deimaginaring power of quantum channels to describe quantum channels' ability to create and destroy imaginarity. Furthermore, the imaginaring and deimaginaring power for several typical single-qubit channels based on l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{1}$$\end{document} norm, robustness and relative entropy of imaginarity were computed. In addition, we define the non-imaginarity-generating channel as the completely positive trace-preserving map which does not generate quantum imaginarity from an real state. Several properties of non-imaginarity-generating channels are investigated. Finally, we explore the trade-off relationship between imaginarity and entanglement. The relationship between the deimaginaring power of quantum channels based on relative entropy of the imaginarity and entanglement is studied. Additionally, we explore the creation of imaginarity in a bipartite state. We demonstrate that it is dependent on both the mixedness of initial system and the minimum amount of entanglement that can be created. Our work further studies imaginarity, which will contribute to the development of quantum mechanics and quantum techniques.