## 2 Quantum Markov Chains  
In this section, we recall from literature the required notions and notations about quantum systems and quantum Markov chains. For a more thorough discussion, we refer the interested reader to, e.g., [44, 50, 56]. Given a finite dimensional Hilbert space \(\mathcal{H}\), let \(\mathcal{L}(\mathcal{H})\) be the set of linear operators on it. Let \(\mathcal{S}(\mathcal{H})\) be the set of _super-operators_, where a super-operator is a completely positive linear operator on \(\mathcal{L}(\mathcal{H})\). We identify two specific super-operators in \(\mathcal{S}(\mathcal{H})\), namely \(\mathcal{I}_{\mathcal{H}}\) and \(0_{\mathcal{H}}\), that are the identity and the null super-operator, respectively. Given a finite set \(J\) of indices, for simplicity of presentation, we exchange freely a super-operator and its Kraus representation, that is, for a given super-operator \(\mathcal{E}\), we write \(\mathcal{E}=\{\,E_{j}\mid j\in J\,\}\) where \(\{\,E_{j}\mid j\in J\,\}\) is a set of Kraus operators of \(\mathcal{E}\); that is, \(\mathcal{E}(A)=\sum_{j\in J}E_{j}AE_{j}^{\dagger}\) for all \(A\in\mathcal{L}(\mathcal{H})\), where \(E_{j}^{\dagger}\) is the complex conjugate and transpose of \(E_{j}\). For any \(\mathcal{E},\mathcal{F}\in\mathcal{S}(\mathcal{H})\), we denote the composition of \(\mathcal{E}\) and \(\mathcal{F}\) by \((\mathcal{E}\circ\mathcal{F})(A)=\mathcal{E}(\mathcal{F}(A))\). To simplify the notation, we may drop the symbol \(\circ\) and just write \(\mathcal{E}\mathcal{F}\) instead of \(\mathcal{E}\circ\mathcal{F}\). Super-operators can be (pre-)ordered according to their preservation of the trace. The _trace_ of a linear operator \(A\in\mathcal{L}(\mathcal{H})\) is defined as \(\sum_{i=1}^{d}\left\langle\eta_{i}\right|A\left|\eta_{i}\right\rangle\) where \(\{\,\left|\eta_{i}\right\rangle\mid 1\leq i\leq d\,\}\) is an orthonormal basis of the Hilbert space \(\mathcal{H}\) with dimension \(d\). Let \(\mathcal{D}(\mathcal{H})\) denote the set of partial density operators in \(\mathcal{L}(\mathcal{H})\), where we say that a linear operator \(A\in\mathcal{L}(\mathcal{H})\) is a partial density operator if \(A\) is positive (i.e., \(\left\langle\eta\right|A\left|\eta\right\rangle\geq 0\) for all \(\left|\eta\right\rangle\in\mathcal{H}\)) and its trace \(\mathsf{tr}(A)\) satisfies \(\mathsf{tr}(A)\leq 1\); we denote by \(\mathcal{D}^{1}(\mathcal{H})\) the set of density operators, that is, the partial density operators whose trace is \(1\). Recall that the trace of a partial density operator denotes the probability that the corresponding (normalized) quantum state is reached [57]. Given two super-operators \(\mathcal{E},\mathcal{F}\in\mathcal{S}(\mathcal{H})\), we write \(\mathcal{E}\lesssim\mathcal{F}\) if for any \(\rho\in\mathcal{D}(\mathcal{H})\), we have \(\mathsf{tr}(\mathcal{E}(\rho))\leq\mathsf{tr}(\mathcal{F}(\rho))\). Intuitively, \(\mathcal{E}\lesssim\mathcal{F}\) means that the success probability of performing \(\mathcal{E}\) is always at most the probability of performing \(\mathcal{F}\), whatever the initial state is. We define \(\widetilde{\,\sim\,}\) to be \(\lesssim\cap\,\gtrsim\). For any \(\rho\in\mathcal{D}(\mathcal{H})\), the _support_\(\text{supp}(\rho)\) is defined to be the space spanned by the eigenvectors of \(\rho\) with non-zero eigenvalues. Let \(\{X_{k}\}\) be a family of subspaces of \(\mathcal{H}\). The _join_ of \(\{X_{k}\}\) is defined to be \(\bigvee_{k}X_{k}=\text{span}(\bigcup_{k}X_{k})\). We denote by \(\mathcal{S}^{\mathcal{I}}(\mathcal{H})=\{\,\mathcal{E}\in\mathcal{S}(\mathcal{ H})\mid 0_{\mathcal{H}}\lesssim\mathcal{E}\lesssim\mathcal{I}_{\mathcal{H}}\,\}\) the set of trace-nonincreasing super-operators over \(\mathcal{H}\). In particular, this means that \(\mathcal{E}\in\mathcal{S}^{\mathcal{I}}(\mathcal{H})\) if, and only if, for any \(\rho\in\mathcal{D}(\mathcal{H})\), we have that \(\mathsf{tr}(\mathcal{E}(\rho))\in[0,1]\).