# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 3 Chaotic Dual Annealing Optimization  
In recent times, optimization problems have grown in terms of complexity--i.e., nonlinearity, nonconvexity, multilevel, and uncertainty in parameters. This directly results in stagnation in the solution method upon implementation, where the algorithmic technique gets trapped in a local optimum. This causes the task of optimizing such problems to become exceedingly difficult. An effective approach to tackle such issues is the utilization of stochastic optimization approaches. One such approach is simulated annealing (SA) [31]. SA is based on the idea of the physical concept of annealing, where a solid is repeatedly heated and cooled until it reaches a configurational state of minimal energy. In this work,we use the dual annealing optimization approach, which is a coupled technique inspired by the classical simulated annealing as well as the fast simulated annealing approaches [32]. The dual annealing optimization approach used in this work implements a distorted Cauchy-Lorentz visiting distribution, \(g(\Delta x(t))\) with \(q\) (the distribution's shape controlling parameter):  
\[g(\Delta x(t))=k\left[\frac{T(t)^{-A_{1}}}{(1+(q-1)A_{2})^{A_{3}}}\right],\ A_{1}= \left[\frac{D}{(3-q)}\right],\ A_{2}=\left[\frac{\Delta x(t)^{2}}{\left(T(t)^{ \frac{A_{2}}{2}}\right)}\right],\ \ \ \ A_{3}=\frac{1}{q-1}+\frac{D-1}{2} \tag{10}\]  
where \(k\) and \(D\) are constants, \(t\) is artificial time, \(\Delta x(t)\) is the trial jump distance and \(T(t)\) is the artificial temperature. The candidate solution's acceptance probability is set by defining an acceptance parameter, \(q^{\prime}<1\) is determined as follows:  
\[P=\min\{1,\ f\big{(}q^{\prime}\big{)}\}\ \ \ \text{where}\ \ \ f(q)=\big{|}1-\big{(}1-q^{\prime}\big{)}\beta\Delta E\big{|}^{\frac{1}{1-q ^{\prime}}} \tag{11}\]  
The decrease of the artificial temperature, \(T(t)\) is given by the following relation:  
\[T(t)=T(1)\left[\frac{2^{q-1}-1}{(1+t)^{q-1}-1}\right] \tag{12}\]  
In this work, the dual annealing technique is coupled with coupled map lattices (CML) to enhance its optimization capability by simulating chaotic non-linear dynamics. CMLs have applications in diverse fields of research and implementation: system identification, process modeling, design optimization, internet of things (IoT), and cryptography [24; 33; 34; 35]. For a specific discreet-time dynamical map, CMLs consist of discreet interacting elements. These elements make up a multidimensional dynamical system. Hence, the CML model could be designed to simulate a dynamical system, where its macroscopic field variables are defined on a lattice. Independent processes then decompose the dynamical system. In this sense, each process represents a procedure in the CML. Nonlinear transformations are generated using each state variable (i.e., lattice points) and coupling terms, which are executed iteratively. Since CMLs commonly generate chaotic behavior, a logistic map is introduced in this work to induce such dynamics by setting the constant, \(\theta>3.57\). The logistic map is given as follows:  
\[x_{i+1}=\theta x_{i}(1-x_{i}) \tag{13}\]  
where \(i\in[1,m]\) is the iteration count. The recursive map for CML is as follows:  
\[x_{i+1}=\epsilon|rx_{i}(1-x_{i})|_{j}+(1-\epsilon)|rx_{i}(1-x_{i})|_{j-1} \tag{14}\]  
where \(s\) is the index for the vertices on the lattice and the coupling parameter, \(\epsilon\in[0,1]\). The second term in the CML recursive map is for the neighboring lattice points. In this work a unidirectional coupling approach is taken with the convex map. The proposed techniques that combine the CML and the dual annealing optimization approaches is given in Algorithm 1:  
```
1. Initialize dual annealing parameters: maximum iterations, \(T(1),q,q^{\prime}\). 2. Initialize CML parameters: Logistic map parameter, \(\theta\) number of lattice vertices (\(N\)), coupling parameter (\(\epsilon\)). 3.