  Complex numbers are widely used in both classical and quantum physics, and
are indispensable components for describing quantum systems and their dynamical
behavior. Recently, the resource theory of imaginarity has been introduced,
allowing for a systematic study of complex numbers in quantum mechanics and
quantum information theory. In this work we develop theoretical methods for the
resource theory of imaginarity, motivated by recent progress within theories of
entanglement and coherence. We investigate imaginarity quantification, focusing
on the geometric imaginarity and the robustness of imaginarity, and apply these
tools to the state conversion problem in imaginarity theory. Moreover, we
analyze the complexity of real and general operations in optical experiments,
focusing on the number of unfixed wave plates for their implementation. We also
discuss the role of imaginarity for local state discrimination, proving that
any pair of real orthogonal pure states can be discriminated via local real
operations and classical communication. Our study reveals the significance of
complex numbers in quantum physics, and proves that imaginarity is a resource
in optical experiments.
