Connection among entanglement, mixedness, and nonlocality in a dynamical context

We investigate the dynamical relations among entanglement, mixedness, and nonlocality, quantified by concurrence $C$, purity $P$, and maximum Bell function $B$, respectively, in a system of two qubits in a common structured reservoir. To this aim we introduce the $C$-$P$-$B$ parameter space and analyze the time evolution of the point representative of the system state in such a space. The dynamical interplay among entanglement, mixedness, and nonlocality strongly depends on the initial state of the system. For a two-excitation Bell state the representative point draws a multibranch curve in the $C$-$P$-$B$ space and we show that a closed relation among these quantifiers does not hold. By extending the known relation between $C$ and $B$ for pure states, we give an expression among the three quantifiers for mixed states. In this equation we introduce a quantity, vanishing for pure states, which in general does not have a closed form in terms of $C$, $P$ and $B$. Finally, we demonstrate that for an initial one-excitation Bell state, a closed $C$-$P$-$B$ relation instead exists and the system evolves, remaining always a maximally entangled mixed state.

Figure 1

$C$-$P$-$B$ space curve drawn by the system starting from the initial two-excitation Bell state $|\Psi \rangle$ for $\lambda ={10}^{-3}\Gamma$. The arrows indicate the time evolution and the numbering from 1 to 7 indicates the different branches (multibranch behavior) arising from the dynamics. A one-to-one correspondence among the three quantities is not possible here.

Figure 2

Projections of the $C$-$P$-$B$ space curve starting from the two-excitation Bell state $|\Psi \rangle$ on the planes (a) $B$-$C$, (b) $B$-$P$, and (c) $C$-$P$ for $\lambda ={10}^{-2}\Gamma$. Arrows indicate the time evolution, and multibranch behaviors are clearly shown. (a) Quantum states with inversion of entanglement ordering are crossed (for example, points ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ where $C_1>C_2$ but $B_1<B_2$). The plots in the insets are related to the case of a perfect cavity (single-mode reservoir). No one-to-one correspondence among couples of quantifiers exists in this case either.

Figure 3

$C$-$P$-$B$ space curve drawn by the system starting from the initial one-excitation Bell state $|+\rangle$ for $\lambda ={10}^{-3}\Gamma$. Arrows indicate the time evolution, and point A the maximum point reached after the first ascent. A one-to-one correspondence among the three quantities is clearly shown.