Imaginarity of Gaussian states

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.