# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 4 Computational Analysis
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Best** & **Median** & **Worst** \\ \hline \(d\) & 9 & 9 & 4 \\ \(P(\delta)\) & 0.9745 & 0.9813 & 0.9699 \\ \(F\) & 0.9991 & 0.5105 & 0.0435 \\ Iterations & 315 & 495 & 443 \\  & [5.91809983 \(\times\) 10\({}^{-3}\), & \\  & 2.26849719 \(\times\) 10\({}^{-4}\), & \\  & 2.75340776 \(\times\) 10\({}^{-2}\), & \\  & 1.5435405 \(\times\) 10\({}^{-1}\), & 0.03569338, 0.2363382, & [0.33783083, \\ \(\rho_{A^{\prime}B^{\prime}}^{T}\) & 5.56746335 \(\times\) 10\({}^{-2}\), & 0.34003655, 0.20481441, & 0.09176041, \\  & 4.33851377 \(\times\) 10\({}^{-1}\), & 0.00174142, 0.08249384, & 0.19728488, \\  & 4.06979776e \(\times\) 10\({}^{-3}\), & 0.04225155] & \\  & 3.17812492 \(\times\) 10\({}^{-1}\), & \\  & 5.59267748 \(\times\) 10\({}^{-4}\)] & \\ wHVI & 0.9917 & 0.6517 & 0.3214 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rated individual solutions obtained using the CML-DA approach. Figure 1: The quantum state, \(\rho_{A^{\prime}B^{\prime}}^{T}\) for the best individual solution generated using the PRNG-DA approach. Figure 2: The quantum state, \(\rho_{A^{\prime}B^{\prime}}^{T}\) for the best individual solution generated using the CML-DA approach. In Figure 3, it can be seen that the shaded area highlights the optimal region in the objective space, where the optimality of the fidelity objective is given more importance as compared to the cascaded objective function: probability of success. Thus, more solutions generated by the CML-DA technique fall into the optimal region as compared to the PRNG-DA approach. This is reflected in the value of the overall optimality of the solutions measured using the weighted HVI. The overall optimality of the CML-DA and PRNG-DA approaches are 27,109 and 25,718, respectively. The CML-DA outperforms the PRNG-DA in terms of overall optimization performance by approximately 5.267%. The key factor influencing the techniques in this study is the CML component. The chaotic dynamics exhibited by the CML enables wide-range dynamical behavior as compared the standard PRNG. This boosts the performance of the CML-DA approach, enabling it to perform a more thorough search of the objective landscape as compared to standard techniques (PRNG-based approaches). The CML is thus able to adapt better to the multilevel structure of the optimization problem and avoid scenarios that may cause stagnation. In other words, the CML manages to escape local optima traps in the objective space. The programs developed using the mentioned computational techniques did not encounter convergent issues and performed smoothly during execution. The execution time taken for the overall solution generation using the CML-DA technique was 156,804 s while the PRNG-DA approach took about 27 s. Thus, the trade-off experienced by the CML-DA in terms of optimization efficiency is the execution time as compared to the PRNG-DA approach. The additional algorithmic complexity embedded in the CML chaotic simulator comes with an additional computational cost. Nevertheless, both techniques showed stable performance when solving the entanglement distillation problem with the fuzzy relations for QST. In that work, the authors also developed a similarly scalable efficient maximum likelihood computational framework for approximating states from incomplete statistical data.