# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 4 Computational Analysis  
Similar to many research efforts in QST, the ideas presented in this work have been built on previous instrumental efforts in quantum parameter estimation--e.g., estimating one-parameter unitary gates for qubit systems and the use of entanglement for quantum parameter estimation to improve overall measurement stability [36; 37; 38; 39; 40]. The Stackelberg game-theoretic framework was utilized for the entanglement distillation problem, with a QST using fuzzy relations to estimate the quantum state. The pseudolikelihood measurements were simulated and the optimization problem was then solved using: the combined CML and dual annealing technique (CML-DA); and the conventional dual annealing approach with pseudo-random number generators (PRNG-DA). The numerical experiments were performed using the Python programming language on Google Collaboratory platform on a cloud with Python 3 Google Compute Engine (RAM 12.68 GB and Disk space:107.72 GB). Each technique was executed a total of 40 times, where each time the technique was run 3 times and the best solution taken for each execution. Therefore, both techniques were individually executed a total of 120 times. The computational results obtained using both techniques were measured using the weighted hypervolume indicator for the bilevel problem:  
\[wHVI=w_{1}(x^{*}-x)+w_{2}(x^{*}_{o}-x_{o}) \tag{15}\]  
where the optimal solution candidate is \((x^{*},x^{*}_{0})\) and the nadir point is \((x,x_{o})\). The weights \(w_{1}\) and \(w_{2}\) enable the relative importance of the contribution of the upper-level problem (1) and lower-level problem (2). In these experiments, the weights \(w_{1}=0.7\) and \(w_{1}=0.3\). The nadir point is for the upper-level problem (or fidelity objective) (1), and the lower-level subproblem (or probability of success) (2) is \((x=1\) and \(x_{o}=1\)). The larger the value of the wHVI metric, the better the optimization performance. Due to the complexity of the optimization problem given in Equation (3), the existence of a global optima cannot be confirmed. Thus, the computational techniques present a framework to find a local optimal for the given entanglement distillation protocol. The rated individual solutions obtained using the PRNG-DA approach are given in Table 3, and the optimal quantum state of the best solution (pure state density matrix) is shown in Figure 1: \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Logistic map parameter, \(\theta\) & 3.7 \\ Maximum iteration & 100 \\ Lattice vertex count (\(N\)) & 100 \\ Coupling parameter (\(\epsilon\)) & 0.5 \\ Parametric interval (\(\Delta r\)) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings for the CML. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Best** & **Median** & **Worst** \\ \hline \(d\) & 4 & 4 & 4 \\ \(P(\delta)\) & 0.9897 & 0.9878 & 0.9839 \\ \(F\) & 0.8873 & 0.5201 & 0.1218 \\ Iterations & 341 & 397 & 557 \\  & [0.10023951, & [0.1290744, & [0.07538315, \\ \(\rho^{T}_{A^{\prime}B^{\prime}}\) & 0.18421193, & 0.46015852, & 0.81589332, \\  & 0.20252628, & 0.36462488, & 0.07227743, \\  & 0.69502228] & 0.0461422] & 0.0364461] \\ wHVI & 0.918 & 0.6604 & 0.3805 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rated individual solutions obtained using the PRNG-DA approach. The individual solutions obtained using the CML-DA technique are ranked and shown in Table 4. The generated optimal quantum state (pure density matrix) for the individual best solution is given in Figure 2:  
The best individual solutions reached by the PRNG-DA and CML-DA techniques have a quantum state, \(\rho_{A^{\prime}B^{\prime}}^{T}\), with the dimensions: \(d=4\) and \(d=8\), respectively. The difference in terms of the level of optimality reached by the individual best solutions, measured using the wHVI, was about 7.722%. The scatter plot in Figure 3 shows the spread of the individual solutions generated using the PRNG-DA and CML-DA: