# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 2 Entanglement Distillation with Fuzzy Relations for QST  
In this work, a bipartite practical entanglement distillation model is considered, where the central idea is to convert a state, \(\rho_{AB}\) (density matrix form), into a state which is close to a maximally entangled state, utilizing only local operations and classical communication. In contrast to theoretical entanglement distillation, practical entanglement distillation frameworks allow for the possibility of failure [26]. The mentioned communication takes place between two nodes of a communication network \(A\) and \(B\). This can be represented mathematically as follows:  
\[F=\langle\Phi_{d}|\eta_{\hat{A}\hat{B}}|\Phi_{d}\rangle\ \ \text{such that}\ \ |\Phi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle_{\hat{A}}|i \rangle_{\hat{B}} \tag{1}\]  
where \(F\in(0,1)\) is the fidelity--i.e., closeness of the converted state to the maximally entangled state. \(A\) and \(B\) are the input registers while \(\hat{A}\) and \(\hat{B}\) are the output registers. \(d\) is the dimension of the quantum state and \(\eta_{\hat{A}\hat{B}}\) is the converted state (\(\rho_{AB}\rightarrow\eta_{\hat{A}\hat{B}}\)). \(|\Phi_{d}\rangle\) is the maximally entangled state across output registers \(\hat{A}\) and \(\hat{B}\). If the dimension, \(d=|A|=|B|=2\), then one possible example of \(\rho_{AB}\) would be as follows:  
\[\rho_{AB}=(1-p)|01\rangle\langle 01|+p|\Phi_{2}\rangle\langle\Phi_{2}| \tag{2}\]  
In this view, a technically simplified subcategory of local operations and classical communication (between network nodes, \(A\) and \(B\)) called measure and exchange operations scheme is employed as presented in [PERSON] et al., [27]. In this scheme, the positive operator-valued measurement performed on the input register, \(A\) is defined as \(\{M_{A}^{0},\ M_{A}^{1}\}\) with \(M_{A}^{1}=\left(A_{A}^{1}\right)^{\dagger}A_{A}^{1}\) and \(M_{A}^{0}=\left(A_{A}^{0}\right)^{\dagger}A_{A}^{0}=\mathbb{I}-M_{A}^{1}\) where \(A_{A}^{1}=\sqrt{\omega}|0\rangle\langle 0|+|1\rangle\langle 1|\). The parameter, \(\omega\) defines the trade-off between fidelity and the probability of success. This measurement follows the map: \(\Lambda_{A\to A,\ F_{A}}(\rho)=\sum_{f_{A}\in\{0,1\}}A_{A}^{f_{A}} \rho\left(A_{A}^{f_{A}}\right)^{\dagger}\otimes|f_{A}\rangle\langle f_{A}|_{F _{A}}\), where the symbol, \(\otimes\) represents the Kronecker product. A similar formulation would apply for a positive operator-valued measurement performed on the input register, \(B\). Both measurements are declared a success if \(f_{A}=f_{B}=1\) for option \(P=|11\rangle\langle 11|_{F_{A}F_{B}}\). Since practical entanglement distillation frameworks allow for the possibility of failure, the fidelity parameter would only be relevant to the analysis if the entanglement distillation is a success [26]. This then creates a multilevel scenario with the probability of distillation success, \(P(\delta)\in(0,1)\) cascaded by the fidelity parameter, \(F\). Since the local quantum memory utilized to store the quantum entanglement is imperfect/non-ideal, this entanglement cannot be preserved for an arbitrary amount of time. Hence, the probability of distillation success, \(P(\delta)\) would control the rate at which high-fidelity entanglement between the different nodes in the network is aimed. Practical entanglement distillation uses schemes involving the application of local operation and measurement on \(A\) and \(B\) registers. This is then followed by a measurement outcome (single exchange) using classical communication to determine distillation success/failure. In this work, the practical quantum distillation scheme presented in [PERSON] et al., [27] is considered. The central idea is to search for the following optimal parameters: (i) output dimension, \(d\); (ii) input state, \(\rho_{AB}\); and (iii) quantum channels (i.e., quantum operations).