Imaginarity of quantum channels: Refinement and Alternative

At present, quantum channels have been widely concerned, and many ways to quantify quantum channels have been proposed, which has led to the generation of many resource theories for quantum channels. We add strong monotonicity and convexity to the requirement of imaginarity measure of quantum channels to make the measure proper. We also introduce an alternative framework to simplify the process of verifying whether a quantifier is a proper measure. We present three imaginarity measures of quantum channels via on the robustness, the trace norm and entropy, respectively. Some properties are also given.