Dynamics of geometric and entropic quantifiers of correlations in open quantum systems

We extend the Hilbert-Schmidt (square norm) distance, previously used to define the geometric quantum discord, to define also geometric quantifiers of total and classical correlations. We then compare the dynamics of geometric and entropic quantifiers of the different kinds of correlations in a non-Markovian open two-qubit system under local dephasing. We find that qualitative differences occur only for quantum discords. This is taken to imply that geometric and entropic discords are not, in general, equivalent in describing the dynamics of quantum correlations. We then show that geometric and entropic quantifiers of total correlations also present qualitative disagreements in the state space. This aspect indicates that the differences found for quantum discord are not attributable to a different separation, introduced by each measure, between the quantum and classical parts of correlations. Finally, we find that the Hilbert-Schmidt distance formally coincides with a symmetrized form of linear relative entropy.

Figure 1

Panel (a) REB quantum discord $D$ (blue solid line) and normalized geometric quantum discord $2D_{\mathrm{g}}$ (purple dashed line) vs $\nu =t/2\tau$, with $\tau =5$s and $\left|\alpha \right|=1$s${}^{-1}$, for an initial Bell-diagonal state. Panel (b) REB total correlations $T$ (orange solid line) and normalized geometric total correlations $2T_{\mathrm{g}}$ (green dashed line) vs $\nu$. The values of initial coefficients are $c_1\left(0\right)=1$, $c_2\left(0\right)=-0.6,$ and $c_3\left(0\right)=0.6$ [${\lambda }_1^{+}\left(0\right)=0.2$, ${\lambda }_1^{-}\left(0\right)={\lambda }_2^{-}\left(0\right)=0$, ${\lambda }_2^{+}\left(0\right)=0.8$].

Figure 2

Panel (a) REB quantum discord $D$ (blue solid line) and normalized geometric quantum discord $2D_{\mathrm{g}}$ (purple dashed line) vs $\nu$, for an initial Bell-diagonal state with $c_1\left(0\right)=0.6$, $c_2\left(0\right)=0,$ and $c_3\left(0\right)=0.4$ [${\lambda }_1^{+}\left(0\right)=0.3$, ${\lambda }_1^{-}\left(0\right)=0$, ${\lambda }_2^{+}\left(0\right)=0.5$, ${\lambda }_2^{-}\left(0\right)=0.2]$. Panel (b) REB total correlations $T$ (orange solid line) and normalized geometric total correlations $2T_{\mathrm{g}}$ (green dashed line) vs $\nu$. Values of other parameters for both panels: $\tau =5$s, $\left|\alpha \right|=1$s${}^{-1}$.

Figure 3

REB total correlations $T$ (orange solid line) and normalized geometric total correlations $2T_g$ (green dashed line), in the space state, as functions of $c_1$, with $c_2=\sqrt{0.25-c_1^2}$ and $c_3=0.2$.

Figure 4

Panel (a) REB discord $D$ (blue solid line) and normalized geometric discord $2D_{\mathrm{g}}$ (purple dashed line), in the space state, as functions of $c_1$. Panel (b) REB classical correlations $C$ (red solid line) and normalized geometric classical correlations $2C_{\mathrm{g}}$ (black dashed line) as functions of $c_1$. The other coefficients are $c_2=\sqrt{0.25-c_1^2}$ and $c_3=0.2$.