### 5.3.3 Persistence Probability

The aim of this subsection is to study another kind of reachability of quantum Markov chains, namely persistence. Intuitively, persistence means that a desired condition is always satisfied from a certain point of time. As pointed out in the last subsection, we can focus our attention on ${\mathcal{E}}_{\infty }\left(\mathcal{H}\right)$ because ${\mathcal{E}}_{\infty }{\left(\mathcal{H}\right)}^{\perp }$ is a transient subspace.

Definition 5.3.3

Let $\mathcal{C}=〈\mathcal{H},\mathcal{E}〉$ be a quantum Markov chain and X a subspace of ${\mathcal{E}}_{\infty }\left(\mathcal{H}\right)$. Then ...

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