§IV, p. 4 — for χ ⊥ |0ⁿ⟩ some monomial P of the σ_-^{(k)} has ⟨0ⁿ|P|χ⟩ ≠ 0

Let $d_k = \sigma_-^{(k)}$, $k = 1,\dots,n$, be the qubit lowering operators. For every nonzero vector $\chi$ orthogonal to $|0^n\rangle$ there is a finite word $k_1,\dots,k_r$ (a monomial $P(\chi) = d_{k_1}\cdots d_{k_r}$ of the jump operators) such that

This is the property that rules out a nonzero subspace invariant under all $d_k$ and orthogonal to $|0^n\rangle$, and hence (via Theorem 2) shows that $|0^n\rangle$ is the unique stationary state of $\mathcal L_{\{d_k\}}$.

Formalization Note "$|\chi\rangle$" is a state, so $\chi\neq 0$ is assumed (the zero vector is orthogonal to everything). The empty word is the identity.