Average quantum coherence of pure-state decomposition

We study the average quantum coherence over the pure state decompositions of a mixed quantum state. An upper bound of the average quantum coherence is provided, and sufficient conditions for the saturation of the upper bound are shown. These sufficient conditions always hold for two- and three-dimensional systems. This provides a tool to estimate the average coherence experimentally by measuring only the diagonal elements, which remarkably requires less measurements compared with state tomography. We then describe the pure state decompositions of the qubit state in the Bloch sphere geometrically. For any given qubit state, the optimal pure state decomposition achieving the maximal average quantum coherence as well as three other pure state decompositions are shown in the Bloch sphere. The order relations among their average quantum coherence are invariant for any coherence measure. The results presented in this paper are universal and suitable for all coherence measures.

Figure 1

The average coherence (solid lines) and MSD (dashed lines) with respect to the pure state decomposition $\mathfrak{D}\left(x\right)$ for $\rho \left(x\right)$ with $x=0.1,0.2,0.3,0.4$. The plots are symmetric at $\alpha =\frac{\pi }{4}$ in the interval $\left[0,\frac{\pi }{2}\right]$. In the interval $\left[0,\frac{\pi }{4}\right]$, the maximum average coherence of $\rho \left(x\right)$ is 1, which is arrived at $\alpha =0$. The minimum average coherence of $\rho \left(x\right)$ is $2x$ for $\alpha \in \left[\frac{1}{2}arccos2x,\frac{\pi }{4}\right]$. The average coherence of $\rho \left(x_1\right)$ and $\rho \left(x_2\right)$ overlap on the interval $\left[0,\frac{1}{2}arccos2x_2\right]$ for $0\le x_1\le x_2\le \frac{\pi }{4}$. The MSD reaches its maximum at $\alpha =\frac{1}{2}arccos2x$ and minimum at $\alpha =0$ and $\alpha =\frac{\pi }{4}$.

Figure 2

The four pure state decompositions ${\mathfrak{D}}^{\left(M\right)},{\mathfrak{D}}^{\left(s\right)},{\mathfrak{D}}^{\left(m_1\right)}$, and ${\mathfrak{D}}^{\left(m_2\right)}$ in the Bloch sphere. (a) The pure state decomposition ${\mathfrak{D}}^{\left(M\right)}$, (b) the spectral decomposition ${\mathfrak{D}}^{\left(s\right)}$, (c) the pure state decomposition ${\mathfrak{D}}^{\left(m_1\right)}$, and (d) the pure state decomposition ${\mathfrak{D}}^{\left(m_2\right)}$. The central (yellow) disk in (a) is the set of mixed states with spectral decompositions reaching the maximal average coherence.