Entanglement production in nonideal cavities and optimal opacity

We compute analytically the distributions of concurrence $\text{C}$ and squared norm $\text{N}$ for the production of electronic entanglement in a chaotic quantum dot. The dot is connected to the external world via one ideal and one partially transparent lead, characterized by the opacity $\gamma$. The average concurrence increases with $\gamma$ while the average squared norm of the entangled state decreases, making it less likely to be detected. When a minimal detectable norm ${\text{N}}_0$ is required, the average concurrence is maximal for an optimal value of the opacity ${\gamma }^{☆}\left({\text{N}}_0\right)$ which is explicitly computed as a function of ${\text{N}}_0$. If ${\text{N}}_0$ is larger than the critical value ${\text{N}}_0^{☆}\simeq 0.3693\cdots$, the average entanglement production is maximal for the completely ideal case, a direct consequence of an interesting bifurcation effect.

Figure 1

Sketch of the orbital entangler.

Figure 2

Probability density of the concurrence (6) compared with numerical simulations (black triangles) for different values of $\gamma$.

Figure 3

Analytical prediction for the average concurrence $\overline{\text{C}}$ as a function of $\gamma$ (straight lines) together with numerical simulations (black circles). Inset: Comparison between the average square norm $\overline{\text{N}}$ and numerical simulations.

Figure 4

Behavior of the optimal ${\gamma }^{☆}$ as a function of the minimal required squared norm ${\text{N}}_0$. When ${\text{N}}_0$ is greater than ${\text{N}}_0^{☆}\simeq 0.3693\cdots$, the maximal average entanglement production is reached for the case of ideal leads. Inset: Distribution of $\overline{\left(\text{C}\right|{\text{N}}_0)}$ for different values of ${\text{N}}_0$, where the bifurcation effect around $\gamma =0$ is now explicit.