# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 2 Entanglement Distillation with Fuzzy Relations for QST
To optimize quantum operations, Choi isomorphism can be utilized, where a one-to-one correspondence between quantum channels and quantum states is established (with certain properties) ([PERSON] et al., 2013). A unique Choi state, \(C_{RT^{\prime}}=\Gamma_{T\to R}\otimes 1_{T^{\prime}}(\Phi_{TT^{\prime}})\), exists for any quantum channel, \(\Gamma_{T\to R}\) from a system, \(T\) to a system, \(R\) with \(\Phi_{TT^{\prime}}\) as the density matrix of the normalized maximally entangled state, as presented in Equation (1) with dimensions, \(D=|T|\). This unique Choi state satisfies \(C_{RT^{\prime}}\geq 0\) and \(C_{T^{\prime}}=\mathbb{I}_{T^{\prime}}/|T^{\prime}|\). In the measurement and exchange operations, the Choi states take a product representation achieved using the Kronecker product operations. After a series of simplifications presented in [PERSON] et al., the Choi states for the case of register, \(A\) are obtained: \(\widehat{C}_{AFA^{\prime}A^{\prime}}=\sum_{f_{A}\in\{0,1\}}\widehat{C}_{f_{A} AA^{\prime}}\otimes|f_{A}\rangle\langle f_{A}|_{F_{A}}\). A similar formulation could be achieved for register, B. This way, the isomorphism carries over all the information from the original channel to the Choi state. Following the problem formulation in [PERSON] et al., [27], the bilevel optimization formulation of entangle distillation is as follows:  
\[Maximize\to\ F=\frac{|A||B|}{P(\delta)}\;\mathrm{Tr}\;[|\Phi_{d}\rangle\langle \Phi_{d}|_{A\hat{B}}\otimes\,\rho^{T}_{A^{\prime}B^{\prime}}(\hat{C}_{1,AA^{ \prime}}\otimes\hat{C}_{1,BB^{\prime}})]\]  
subject to,  
\[ \begin{array}{l}Maximize\to\,P(\delta)=\,|A||B|\;\mathrm{Tr}\;[\rho^{T}_{A^ {\prime}B^{\prime}}(\hat{C}_{1,A^{\prime}}\otimes\hat{C}_{1,B^{\prime}})]\ \ \ \text{such that,}\\ \rho^{T}_{A^{\prime}B^{\prime}}\ \geq 0\\ C_{1,AA^{\prime}}\geq 0\,,\ \hat{C}_{1,B^{\prime}}\geq 0\\ \hat{C}_{1,A^{\prime}}\leq\frac{1_{A^{\prime}}}{|A|}\,,\ \hat{C}_{1,B^{\prime}}\leq\frac{ \mathbb{I}_{B^{\prime}}}{|B|},\ \ |A|=|B|=d\geq 0\ :d\in\mathbb{N}\end{array}  \tag{3}\]  
where \(A^{\prime}\) and \(B^{\prime}\) are Choi state equivalent for output registers \(A\) and \(B\). Similarly, \(\hat{C}_{1,AA^{\prime}}\), \(\hat{C}_{1,BB^{\prime}},\hat{C}_{1,A^{\prime}}\) and \(\hat{C}_{1,B^{\prime}}\) are matrices depicting Choi states which correspond to quantum channels. The symbol \(\otimes\) represents the Kronecker product and the dimensions of the identity matrices, \(\mathbb{I}_{A^{\prime}}\) and \(\mathbb{I}_{B^{\prime}}\) depend on the dimensions of the registers \(A^{\prime}\) and \(B^{\prime}\). In designing practical quantum information processing systems, quantum state tomography (QST) is employed to approximate a certain state of the density matrix. This is done by using measurement results of repeated state preparations. Unfortunately, QST becomes very challenging as the system scales up in terms of size--i.e., dimensions, \(d\). Current methods for QST formulation include linear inversion, maximum likelihood, Bayesian, hybrid Bayesian-Monte Carlo, and neural network approaches [28; 29]. In this work, a fuzzy approach is developed and employed for QST. In Recasens, [30], it was demonstrated that symmetric positive semi-definite matrices could be constructed from the application of fuzzy relations on certain sets (fuzzy subsets). For example, if sets \(X\) and \(Y\), with both having cardinality of \(n\) and \(\mu\), are fuzzy subsets of \(X\) and \(Y\) with the additive generator of the \(t\)-norm being: \(t_{N}(x)=(1-x)^{N}\), then the following matrix is constructed:  
\[\rho(x,y)=1-|(1-\mu(x))^{N}-(1-\mu(y))^{N}|^{1/N}\;\text{where}\;\;\;\;\;N=\frac{ n+1}{log_{2}e} \tag{4}\]  
In the case of estimating the density matrix, \(\rho_{A^{\prime}B^{\prime}}\) in Equations (2) and (3), \(\rho_{A^{\prime}B^{\prime}}\sim\rho\) is considered as symmetric and positive semi-definite with the additional normalization condition: \(\mathrm{Tr}\;(\rho_{A^{\prime}B^{\prime}})=1\).