Theorem: Generation and dissipative decay

For the default localized preparation, finite rates, and finite detunings, $C_{01}>0$ on a sufficiently short positive-time interval. Every cut separating sites $0,1$ is entangled there.

If ${\gamma }_{\mathrm{r}}>0$, every pairwise concurrence and bipartite logarithmic negativity tends to zero. The concurrence $C_{01}$ attains a strictly positive global maximum at finite positive time.

Proof. The leading coefficient in the physical-time expansion is $2J>0$. A cut separating the two sites contains the nonzero coherence in $R_{AB}$, so Theorem thm:negativity applies. For ${\gamma }_{\mathrm{r}}>0$, $q\to 0$, and (eq:Cbound), (eq:ENbound) prove decay. A continuous function vanishing at zero and infinity but positive somewhere attains a positive maximum on a sufficiently large compact time interval, which is also its global maximum.