Fidelity and trace-norm distances for quantifying coherence

We investigate the coherence measures induced by fidelity and trace norm, based on the coherence quantification recently proposed by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We show that the fidelity of coherence does not in general satisfy the monotonicity requirement as a measure of coherence under the subselection of the measurement condition. We find that the trace norm of coherence can act as a measure of coherence for qubits and some special class of qutrits with some restrictions on the incoherent operators, while the general case needs to be explored further.

Figure 1

Comparison between $C_f\left(\rho \right)$ and $p_1{C}_f\left({\rho }_1\right)$. The black solid line is obtained by taking ${\left|a\right|}^2=1,{\left|b\right|}^2=\frac{1}{4},{\left|c\right|}^2=\frac{3}{4},r_x^2+r_y^2=\frac{1}{2}$, and $-\frac{\sqrt{2}}{2}\le r_z\le \frac{\sqrt{2}}{2}$ in Eqs. (12) and (14). The red dashed line is obtained by taking the same values in Eq. (8).