  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.
