Remark: The localized hypothesis matters

A general mixed $S_0$ need not generate entanglement: $S_0=I_N/N$ is already stationary conditionally. The generation theorem is therefore not asserted for all one-excitation density matrices. With no dissipation, decay also need not occur; the disorder-free two-site chain has $C_{01}(\tau )=|\sin (2\tau )|$.