Entanglement production in non-ideal cavities and optimal opacity

We compute analytically the distributions of concurrence $\mathcal{C}$ and squared norm $\mathcal{N}$ 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 ${\mathcal{N}}_0$ is required, the average concurrence is maximal for an optimal value of the opacity ${\gamma }^{\star }({\mathcal{N}}_0)$ which is explicitly computed as a function of ${\mathcal{N}}_0$. If ${\mathcal{N}}_0$ is larger than the critical value ${\mathcal{N}}_0^{\star }\simeq 0.3693\dots$, the average entanglement production is maximal for the completely ideal case, a direct consequence of an interesting bifurcation effect.