# Entanglement Distillation Optimization Using Fuzzy Relations for Quantum State Tomography
## 1 Introduction  
Entanglement distillation is a critical aspect of quantum information systems. The key idea of entanglement distillation is to preserve or restore the quality of diluted entanglement states of quantum information transmitted over large distances. Decoherence effects during transmission cause the dilution of entanglement states. Past theoretical and experimental research works have focused on the study of quantum distillation frameworks [1; 2; 3]. Recently, pairs of single photons (entangled in multiple degrees of freedom) were used to experimentally determine the domain of distillable states and their relative fidelity [4]. In [PERSON] et al., 2021 [4], comparative studies were also carried out on various distillation schemes to gain a deeper understanding in terms of resilient quantum network design. A proof-of-concept experiment was recently conducted to study the application of filtering protocols (in atomic ensembles) for constructing quantum repeater nodes [5]. The experiment was conducted in a crystal (rare-earth-ion-doped). In that setting, the entanglement states were prepared. The relationship between bit thread, entanglement distillation, and entanglement purification (in the holographic framework) was recently studied [6]. In [PERSON] et al., 2021 [6], a bit thread interpretation of the one-shot entanglement distillation tensor network was provided. It was shown that the holographic entanglement purification process could be viewed as a special case of a type of surface growth scheme. The objective of the study in [PERSON] et al., 2021 [6] was to develop an accurate framework for describing physical entanglement structures. Another interesting work is seen in the theoretical investigation of entanglement distillability presented in [7]. In that work, the authors studied the mentioned subject with regards to the undistillability conjecture of specific Werner states and gained deeper understanding of the distillability problem. Recent developments of entanglement distillation are seen in [8], [PERSON] et al., 2021 [9], [PERSON] and [PERSON], 2021 [10], [PERSON] et al., 2022 [11], [PERSON] and [PERSON], 2022 [12], and [PERSON] et al., 2021 [13]. In the past few years, much research in quantum information systems has been directed towards optimization. This can be seen in the work of [PERSON] and [PERSON] [14]. In that work, the optimization of the read-out procedure of local unitaries of a high-retrieval efficiency quantum memory was performed. The authors also studied the retrieval efficiency of quantum memory and the signal-to-noise ratio. A similar work is seen in [PERSON] et al., 2020 [15], where an approach for obtaining the optimal quantum state and computational path evaluation for gate-model quantum computing devices was proposed. Efficient algorithms for attaining generation time and transmission fidelity of entangled pairs between the end nodes of quantum chains is seen in [PERSON] et al., [16]. Quantum state tomography (QST) is an effective method to reconstruct or estimate quantum states from measurements of identical quantum states. This directly impacts the development of reliable quantum resources, as well as quantum devices for quantum information processing. A recent interesting study on information gain in QST is seen in the work of [PERSON] et al., [17]. In that work, the authors discuss the effects of chaotic dynamics on the information transfer during QST. Their investigations also uncovered operational insights into the mechanisms of fidelity gain during actual quantum information tomography protocols. In [PERSON] et al., [18], the authors improve the accuracy of observable measurements from tomographic data. This was done by using a QST scheme that approximates a probability distribution over a measurement (informationally complete) within a variational manifold characterized by a convolutional neural network.