# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 1 Introduction
In that work, the authors regenerated high classical-fidelity states, which performed more efficiently as compared to standard techniques (e.g., maximum likelihood estimation). Another recent approach in QST is observed in the work of [PERSON] et al., [19]. In that work, the authors developed a formulation for pure quantum state reconstruction via eigenvalue decomposition. In [PERSON] et al., [19], the authors demonstrated that the proposed approach is robust against depolarizing noise (high-strength white noise) where the quantum states were reconstructed accurately similar to the noiseless case. A computational-focused strategy for QST is seen in the work of [PERSON] et al., 2022 [20]. In that research, the authors proposed a simultaneous perturbation stochastic approximation algorithm with high-speed convergence. The central idea in that work was to develop a computational technique for estimating quantum states under limited computational conditions using a Barzilai-Borwein two-point step size gradient method. In [PERSON], 2022 [21], a low-complexity effective QST scheme was proposed. The proposed scheme requires the measurement of only three observables for systems of any scale. The key concept was the coupling of the system and the 'pointer' of a single qubit. Complex nonlinear dynamical systems have been successfully modeled using chaotic maps such as coupled map lattices (CML). For instance, in [PERSON] et al., [22] an urban rail transit system under cascading failure was modeled with respect to network vulnerability using CMLs. In that work, the authors quantified the relationship between passenger flow and the perturbation index. In addition to investigations of the anti-risk resistance capability of stations, the dependency of cascading failures on location and type of stations were identified. In [PERSON] and [PERSON], 2022 [23], CMLs were for a biomedical-focused application. In that work, the authors utilized spatiotemporal chaos in the CML to simulate cardiac reentry. Among their key research findings were clinical manifestations of certain types of tachycardia in the electrocardiogram, as well as a novel type of dynamical pattern (with wavefronts comprised of harmonized bursts and cardiac plateaus). In [PERSON] and [PERSON], 2021 [24], a one-dimensional, two-parameter with a wide-range system mixed coupled map lattice model was proposed for image encryption. The authors of that work performed simulations to establish the effectiveness of the proposed encryption algorithm for grayscale and color images. In addition, they also carried out security tests to ensure that the proposed method could hold against conventional security attacks. CMLs have also been recently employed in the field of wireless communication. For instance, in [PERSON] et al., [25], a CML model was developed to assess the vulnerability of an unmanned aerial vehicle (UAV) network. The authors in that work showed that precision interference on critical UAV nodes could inflict significant damage to the entire UAV network. In addition, the authors of [PERSON] et al., 2022 [25] also discovered that network vulnerability increases with the intensity of external interference. This work reformulates the practical entanglement distillation problem as a bilevel optimization model. This work extends the practical entanglement distillation problem to account for approximating the quantum state (density matrix) using QST using fuzzy relations. The bilevel practical entanglement distillation optimization problem is then solved using the combined CML and the dual annealing approach. The central idea is to leverage on the complex behavior of the CML to enhance the optimization capabilities of the dual annealing algorithm. This paper is organized as follows: the Section 2 describes the model formulation for the bilevel practical entanglement distillation problem. In this section, the novel QST formulation using fuzzy relations is presented. The Section 3 discusses the CML and its hybridization with the dual annealing algorithm.