Coherence and imaginarity of quantum states

Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states [{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 ρ has nonzero imaginary part, then ρ 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(ρ )-C(Reρ )≥ 0 if C is invariant under state complex conjugation, i.e., C(ρ )=C(ρ ), here ρ is the conjugate of ρ , Reρ is the real part of ρ . If C does not satisfy C(ρ )=C(ρ ), we can define a new coherence measure C(ρ )=1/2[C(ρ )+C(ρ )] such that C(ρ )=C(ρ ). We also establish some similar results for bosonic Gaussian states.