Coherence and imaginarity of quantum states

Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states [\href{http://dx.doi.org/10.1103/PhysRevLett.113.140401}{Phys. Rev. Lett. 113, 140401 (2014)}]. In BCP framework, a quantum state is called incoherent if it is diagonal in the fixed orthonormal basis, and a coherence measure should satisfy some conditions. For a fixed orthonormal basis, if a quantum state $\rho$ has nonzero imaginary part, then $\rho$ must be coherent. How to quantitatively characterize this fact? In this work, we show that any coherence measure $C$ in BCP framework has the property $C(\rho )-C($Re$\rho )\geq 0$ if $C$ is invariant under state complex conjugation, i.e., $C(\rho )=C(\rho ^{\ast })$, here $\rho ^{\ast }$ is the conjugate of $\rho ,$ Re$\rho$ is the real part of $\rho .$ If $C$ does not satisfy $C(\rho )=C(\rho ^{\ast }),$ we can define a new coherence measure $C^{\prime }(\rho )=\frac{1}{2}[C(\rho )+C(\rho ^{\ast })]$ such that $C^{\prime }(\rho )=C^{\prime }(\rho ^{\ast }).$ We also establish some similar results for bosonic Gaussian states.