  It has been a long-standing debate that why quantum mechanics uses complex
numbers but not only real numbers. To address this topic, in recent years, the
imaginarity theory has been developed in the way of quantum resource theory.
However, the existing imaginarity theory mainly focuses on the quantum systems
with finite dimensions. Gaussian states are widely used in many fields of
quantum physics, but they are in the quantum systems with infinite dimensions.
In this paper we establish a resource theory of imaginarity for bosonic
Gaussian states. To do so, under the Fock basis, we determine the real Gaussian
states and real Gaussian channels in terms of the means and covariance matrices
of Gaussian states. Also, we provide two imaginary measures for Gaussian states
based on the fidelity.
