Remark: Exactness and exclusions

The physical tensor factorization remains that of $N$ sites. The $N$-level matrix $S$ is a computational reduction, not a new factorization for evaluating entanglement.

Uniform relaxation is a hypothesis of this scalar-survival theorem. Nonuniform relaxation still preserves the vacuum-plus-one-excitation block class, but generally replaces $-{\gamma }_{\mathrm{r}}R$ by $-\{D,R\}/2$, with $D$ the diagonal rate matrix; the universal scalar factorization above then need not hold. Bath raising, higher-excitation preparations, and vacuum–excitation coherences are not covered by the theorem.