Time-invariant entanglement and sudden death of nonlocality

We investigate both theoretically and experimentally the dynamics of entanglement and nonlocality for two qubits immersed in a global pure dephasing environment. We demonstrate the existence of a class of states for which entanglement is forever frozen during the dynamics, even if the state of the system does evolve. At the same time nonlocal correlations, quantified by the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, either undergo sudden death or are trapped during the dynamics.

Figure 1

The blue octahedron in the middle, marked with dashed lines, represents separable Bell-diagonal states. The states filling the red and orange regions, connected to the Bell states $|{\psi }^{+}\rangle$ and $|{\psi }^{-}\rangle$, respectively, give rise to time-invariant entanglement.

Figure 2

Dynamics of (a) entanglement $E\left(\rho \right)$ and (b) CHSH violation $B\left(\rho \right)$ as a function of the dimensionless time $t\mathrm{\Gamma }$. The initial Bell-diagonal states are chosen as $c_1=1, c2=0.4$, and $c_3=-0.4$ (red solid line) and $c_1=c_3=-0.5$ and $c2=-1$ (orange dashed line). The black dotted line shows the nonlocality threshold.

Figure 3

Our experimental setup. PBS, polarizing beam splitter; QWP, quarter-wave plate; HWP, half-wave plate; QP, quartz plate; TBS, tunable beam splitter; and ES, entangled photon source. The experiment is performed in three stages that are separated in the figure with two dotted vertical lines. The first stage is the preparation of the Bell-diagonal states. As the second stage deals with the introduction of the global pure dephasing dynamics, the last stage consists of the CHSH measurements and tomography.

Figure 4

(a) Entanglement and (b) CHSH inequality violation versus effective path difference. The experimentally prepared initial Bell-diagonal states are $c_1=1, c2=0.4$, and $c_3=-0.4$ (red squares) and $c_1=c_3=-0.5$ and $c2=-1$ (orange circles). In panel (b), we show the violation of the CHSH inequality for many different measurement settings. The optimal violation is given by the highest points in the $y$ axis for each state, which are marked with black data points.

Figure 5

Degree of CHSH inequality violation for different measurement configurations around the numerically calculated optimal angles for the Bell-diagonal state with $c_1=1, c_2=0.4$, and $c_3=-0.4$. While H1 (H2, H3, H4) forms the half-wave plate of basis $\vec{a}$ (${\vec{a}}^{\prime }, \vec{b}, {\vec{b}}^{\prime }$), Q1 (Q2, Q3, Q4) forms the quarter-wave plate of basis $\vec{a}$ (${\vec{a}}^{\prime }, \vec{b}, {\vec{b}}^{\prime }$).