Maximum coherence in the optimal basis

The resource theoretic measure of quantum coherence is basis dependent, and the amount of coherence contained in a state is different in different bases. We obtained analytical solutions for the maximum coherence by optimizing the reference basis and highlighted the essential role of the mutually unbiased bases (MUBs) on attaining the maximum coherence. Apart from the relative entropy of coherence, we showed that the MUBs are optimal for the robustness of coherence, the coherence weight, and the modified skew information measure of coherence for any state. Moreover, the MUBs are optimal for all the faithful coherence measures if the state is pure or is of the single qubit. We also highlighted an upper bound for the $l_1$ norm of coherence and compared it with the other bounds as well as the maximum one attainable by optimizing the reference basis.

Figure 1

The absolute value of the right-hand side of Eq. (20) versus $a$, with the sign of $sin{\theta }_1$ and $sin{\theta }_3$ being the same (the black solid line) and different (the red solid line).

Figure 2

Maximum $l_1$ norm of coherence $C_{l_1}^{max}\left(\mathrm{\Lambda }\right)$ for $d=4$, 5, and 6 (from top to bottom), where the black solid lines (red dashed lines) correspond to $O_d$ (${\mathcal{B}}_d$), and every blue solid circle was the numerical result obtained by ${10}^9$ equally distributed unitaries generated according to the Haar measure. Moreover, the insets at the top-right corner are the relative deviations of $C_{l_1}^{max}\left(\mathrm{\Lambda }\right)$ from ${\mathcal{B}}_d$, and the insets at the bottom-left corner are the distribution of $C_{l_1}\left(U\mathrm{\Lambda }{U}^{†}\right)$ for $p=0.4$.