# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 2 Entanglement Distillation with Fuzzy Relations for QST
In this view, the \(\mu(x)\) and \(\mu(y)\) are fuzzy sets that act as pseudo-likelihood functions obtained from measurement data (empirical). Since \(x\) and \(y\) are quantum states \(|\Phi_{d}\rangle\langle\Phi_{d}|\) for \(d\geq 0\), it can be stated that \(x=y\). The measurements of these quantum states in the form of fuzzy memberships may not be equal: \(\mu_{1}(x=y)\
eq\mu_{2}(x=y)\) and the normalization condition: \(\sum\limits_{A^{\prime}B^{\prime}}\rho_{A^{\prime}B^{\prime}}=1\) holds. Using the simplified notations for the pseudolikelihood measurements \(\mu_{1}(x)\sim\mu_{1}\) and \(\mu_{2}(x)\sim\mu_{2}\), Equation (4) then becomes: \(\rho(x,y)=1-|{(1-\mu_{1})}^{N}-{(1-\mu_{2})}^{N}|^{1/N}\). An interesting result is obtained from the symmetric semi-definite density matrix construction when the differential form of Equation (4) is obtained:  
\[\frac{\partial\rho}{\partial\mu_{2}}=-\frac{{(1-\mu_{2})}^{N-1}}{{(1-\mu_{1}) }^{N-1}}\frac{\partial\rho}{\partial\mu_{1}}\qquad\qquad\text{where}\qquad \qquad N=\frac{n+1}{log_{2}e} \tag{5}\]  
Equation (5) is then obtained by taking the partial differential for the density matrix in Equation (4) and assuming \(\mu\) and \(\rho\) to be continuous parameters. Similarly, the analysis is extended to seek for an analogue to approximate the factor in Equation (5) using the binomial formula and Maclaurin series:  
\[-\frac{{(1-\mu_{2})}^{N-1}}{{(1-\mu_{1})}^{N-1}}=-\frac{\sum_{ \lambda\geq 0}\binom{N-1}{k}{(-\mu_{2})}^{k}}{\sum_{\lambda\geq 0}\binom{N-1}{k}{(- \mu_{1})}^{k}}=-\frac{\sum_{\lambda>0}(-\mu_{2})^{k}/k!}{\sum_{\lambda\geq 0 }(-\mu_{1})^{k}/k!}\quad\approx-\frac{e^{-\mu_{2}}}{e^{-\mu_{1}}} \tag{6}\] \[= -e^{\ (\mu_{1}-\mu_{2})}\]  
Therefore, a generalized partial differential equation (PDE) independent of scale (i.e., cardinality or dimension) is obtained:  
\[\frac{\partial\rho}{\partial\mu_{2}}+e^{\ (\mu_{1}-\mu_{2})}\frac{\partial \rho}{\partial\mu_{1}}=0 \tag{7}\]  
The measurements \(\mu_{1}\) and \(\mu_{2}\) are treated as continuous parameters and the solution to the scale-free exact solution to the PDE in Equation (7) is obtained:  
\[\rho(\mu_{1},\mu_{2})=c_{0}e^{-\mu_{1}}-c_{0}e^{-\mu_{2}} \tag{8}\]  
where \(c_{0}\) is a constant. Considering \(\mu_{1}\) and \(\mu_{2}\) to be subsequent measurements, pseudo-likelihood measurements are conjectured to have the Markov property such that:  
\[\mu_{2}=\mathbf{P}\mu_{1} \tag{9}\]  
where \(\mathbf{P}\) is an \(n\times n\) transition or stochastic matrix. With this conjecture, the density matrix, \(\rho(x)\) is approximated using data from the current pseudo-likelihood measurement, \(\mu_{2}\) and the previous pseudo-likelihood function, \(\mu_{1}\). The current pseudo-likelihood measurement, \(\mu_{2}\) could be generated using the stochastic matrix, \(\mathbf{P}\) as in Equation (9). QST is highly efficient when estimating density matrices, \(\rho\) for small quantum systems. However, as the system scales-up and \(\rho\) increases in dimensions, effective QST becomes a challenging feat [28]. The fuzzy formulation for pseudo-likelihood relations presented in Equations (4)-(9) aim to enable effective estimation of density matrices, \(\rho\) at larger scales. Treating density matrices, \(\rho\) as symmetric and positive semi-definite are critical requirements for this analysis.