Role of coherence during classical and quantum decoherence

The total correlation in a bipartite quantum system is measured by the quantum mutual information $\mathcal{I}$, which consists of quantum discord and classical correlation. However, recent results in quantum information show that coherence, which is a part of total correlation, is more general and more fundamental. The role of coherence in quantum resource theory is worthwhile to investigate. We first study the relationship between quantum discord and coherence by decreasing the difference between them. Then, we consider the dynamics of quantum discord, classical correlation, and quantum coherence under incoherent quantum channels. It is found that coherence indicates the behavior of quantum discord (classical correlation) for times $t<\overline{t}$, and indicates the behavior of classical correlation (quantum discord) for times $t>\overline{t}$. Moreover, the coherence freeze and decay characterize the quantum discord and classical correlation freeze and decay, respectively.

Figure 1

According to quantum discord and classical correlation, Bell-diagonal states in the tetrahedron can be divided into three regions. The sudden change of quantum discord and classical correlation appears in the contact area of those three areas. In the yellow region, $c=c_1$ and $\mathcal{D}\left({\rho }_{AB}\right)=\mathcal{C}\left({\rho }_{AB}^{{\sigma }_1}\right)$. In the red region, $c=c_2$ and $\mathcal{D}\left({\rho }_{AB}\right)=\mathcal{C}\left({\rho }_{AB}^{{\sigma }_2}\right)$. In the blue region, $c=c_3$ and $\mathcal{D}\left({\rho }_{AB}\right)=\mathcal{C}\left({\rho }_{AB}^{{\sigma }_3}\right)$. As for Werner states, ${\mathcal{C}}_r\left({\rho }_{AB}^{{\sigma }_1}\right)={\mathcal{C}}_r\left({\rho }_{AB}^{{\sigma }_2}\right)={\mathcal{C}}_r\left({\rho }_{AB}^{{\sigma }_3}\right)=\mathcal{D}\left({\rho }_{AB}\right)$.

Figure 2

The left is the trajectory of states with $c_3=0$, and the right is for the states with $c_3=0.2$. In the blue region, $c=c_3$ and $\mathcal{D}\left({\rho }_{AB}\right)=\mathcal{C}\left({\rho }_{AB}^{{\sigma }_3}\right)$, and outside the blue region, $\mathcal{CC}={\mathcal{C}}_{l_1}$. The trajectory of the time evolution of Bell-diagonal states under the phase-flip channel is close to the $c_3$ axis along a straight line. The trajectory of the time evolution for Bell-diagonal states will get to blue region over time $\overline{t}$, and the role of coherence suddenly changes at time $\overline{t}$.

Figure 3

For the initial states $c_1\left(0\right)=0.6$, $c_2\left(0\right)=-0.6$, $c_3\left(0\right)=1$, and $\gamma =0.1$, coherence (red dots) shows the frozen phenomenon of quantum discord (green line) and classical correlation (blue line) under the bit-flip channel.

Figure 4

For the initial states $c_1\left(0\right)=1$, $c_2\left(0\right)=-0.6$, $c_3\left(0\right)=0.6$, and $\gamma =0.1$, coherence (red dots) shows the decreasing phenomenon of quantum discord (green line) and classical correlation (blue line).