Maximum and minimum relative entropies of imaginarity

In this work, we investigate two measures of imaginarity: the maximum and minimum relative entropies of imaginarity, and provide their corresponding operational interpretations. For the maximum relative entropy of imaginarity, we demonstrate that it not only characterizes the maximum overlap between a given state and the maximally imaginary state through real operations, but also provides a lower bound for the efficiency of imaginarity distillation. For the minimum relative entropy of imaginarity, we show that it is related to the maximum probability of transformation between pure states using real operations, as well as the minimum time required for a unitary evolution to convert a given pure state into a real state. Furthermore, by introducing the concepts of smooth maximum and minimum relative entropies of imaginarity, as well as the one-shot imaginarity cost and one-shot distillable imaginarity, we establish that the smooth maximum relative entropy of imaginarity provides a lower bound for the one-shot imaginarity cost, while the smooth minimum relative entropy of imaginarity offers an upper bound for the one-shot distillable imaginarity. Finally, we prove that any nontrivial imaginarity measure is not additive under the tensor product of quantum states. Based on this, we prove that the regularized maximum and minimum relative entropies of imaginarity, as well as the regularized relative entropy of imaginarity are all equal to zero for any states, which highlights the distinction between the resource theory of imaginarity and those of entanglement and coherence.