Coherence and imaginarity of quantum states

Abstract Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states [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) = 12[C(ρ) +C(ρ∗)] such that C′(ρ) = C′(ρ∗). We also establish some similar results for bosonic Gaussian states.