Imaginarity measures induced by real part states and the complementarity relations

Abstract: Complex numbers are indispensable in quantum mechanics and the resource theory of imaginarity has been developed recently. In this paper, we propose a method to construct imaginary measures by real part states. Specifically, we propose an imaginarity measure in terms of fidelity and explore its properties. The analytical expression of the imaginarity measure is presented in qubit systems. The relations between the proposed imaginarity measure and some other imaginarity measures (such as geometric imaginarity, Tsallis relative entropy imaginarity and trace norm imaginarity) are derived. The complementarity relations of the imaginarity measure under a complete set of mutually unbiased bases are provided in low-dimensional systems. This work not only highlights the prominent role of the real part state in the imaginarity resource theory, but also reveals the constraint of imaginarity on a complete set of mutually unbiased bases physically.