# Quantifying Quantum Coherence Using Machine Learning Methods
## 2 Materials and Methods
### The Robustness of Coherence  
Let us analyze the quantifier of quantum coherence within the framework of quantum resource theory (QRT). A QRT consists of three main components: free states, resource states, and free or restricted operations. Mathematically, a quantum state is represented by a normalized Hermitian operator \(\rho\) that satisfies \(\rho^{\dagger}=\rho\) and \(\text{tr}\{\rho\}=1\). In a specific basis, it is simply a Hermitian matrix with a trace of one. A quantum operation is defined as a linear, completely positive map from the set of density operators to itself [36]. A consistent QRT ensures that resource states cannot be generated from free states using free operations. We adhere to the framework of quantum coherence QRT as described by [PERSON] et al. [2]. In this framework, the free states are those with diagonal density matrices in a specific basis, represented as  
\[\delta=\sum_{i}\delta_{i}|i\rangle\langle i|, \tag{1}\]  
where \(\{|i\rangle\}\) is a fixed reference basis in a finite-dimensional Hilbert space and \(\delta_{i}\) forms a probability distribution. In the context of quantum coherence QRT, these free states are referred to as incoherent states, and the set of incoherent states is denoted by \(\mathcal{I}\). The resource states, known as coherent states, are those that cannot be expressed in this form. In order to characterize the set of free operations, we recall that quantum operations are specified by a set of Kraus operators \(\{K_{n}\}\) satisfying \(\sum_{n}K_{n}^{\dagger}K_{n}=I\). For a quantum operation, the corresponding Kraus representation is not unique. The free operations, called incoherent operations, are thus defined as those operations for which there exist a Kraus representation \(\{K_{n}\}\) such that \(K_{n}\rho K_{n}^{\dagger}/\operatorname{tr}\left(K_{n}\rho K_{n}^{\dagger} \right)\in\mathcal{I}\) for all \(n\) and all \(\rho\in\mathcal{I}\). The restriction guarantees that in an overall quantum operation \(\rho\mapsto\sum_{n}K_{n}\rho K_{n}^{\dagger}\), quantum coherence cannot be generated from incoherent input states, even if someone has access to individual measurement outcomes \(n\), not even probabilistically. Now, let us look at the robustness quantifier for quantum coherence, i.e., the robustness of coherence (ROC). Note that quantum coherence can be treated as a useful resource in certain quantum tasks, and the robustness of a resource can be defined within a general resource theory [37]. Let \(\mathcal{D}(\mathbb{C}^{d})\) be the convex set of density operators acting on a \(d\)-dimensional Hilbert space. The ROC of a quantum state \(\rho\in\mathcal{D}(\mathbb{C}^{d})\) is defined as  
\[\mathcal{C}_{\mathcal{R}}(\rho)=\min_{\tau\in\mathcal{D}(\mathbb{C}^{d})}\bigg{\{} s\geq 0\bigg{|}\frac{\rho+s\tau}{1+s}=:\delta\in\mathcal{I}\bigg{\}}. \tag{2}\]  
Intuitively, given a quantum state \(\rho\), one can mix it with another state \(\tau\) according to the weight \(s\). The resulting normalized state may be either coherent or not. Hence, the ROC of \(\rho\) can be viewed as the minimum weight of another state \(\tau\), when their convex mixture yields an incoherent state \(\delta\). For an incoherent state \(\rho\) which already belongs to \(\mathcal{I}\), it is no need to be mixed with another state to achieve incoherence. Hence, the ROC for an incoherent state is zero. Moreover, in the Equation (2), \(\tau\) can be a coherent state as well as an incoherent state, otherwise, the minimum \(s\) would diverge for any state \(\rho\) that possesses nonzero coherence, for the non-diagonal elements of the state matrix cannot be eliminated. In order to show the property of computability for the ROC, one must first realize that any state \(\rho\) can be reduced to an incoherent one through a dephasing operation \(\Delta(\rho)=\sum_{i}\lvert i\rangle\langle i\rvert\rho\lvert i\rangle\langle i\rvert\) in the reference basis \(\{\lvert i\rangle\}\). Then, one must introduce the notion of coherence witnesses.