在談論量子理論時，劃定量子世界與經典世界邊界的問題引人入勝，而開放系統中的量子關聯引發了我們對動力學分析的興趣。真正捕捉本質的特性被稱為經典性或非經典性。與我們熟悉的性質相比，它有一種不同的定義，該定義描述了動態過程，其中的策略是根據HE（Hamiltonian Ensemble）進行制定。此外，根據潛在的李代數結構，它可以被重塑成群上的傅立葉變換（Fourier-transform-on-group, FToG）形式，稱為CHER（Canonical Hamiltonian Ensemble Representation），我們可以在頻域中分析動力學。最終，整個過程將由聯合概率分布決定。通過計算分佈的負值，我們可以清晰地觀察到溫度上升時量子特性的消散。然而，對聯合概率分布的預測在數值方法中顯得非常艱巨，所以有必要提出另一種方法來彌補這個缺陷。
隨著計算機科學的發展和硬件技術的提升，深度學習也取得了巨大的進展。深度學習是基於神經網絡的概念。它是一類使用多層次逐步從原始輸入中提取更高級別特性的算法。經典的例子是卷積神經網絡（Convolution Neuron Networks, CNN）。CNN主要處理圖像處理，圖像識別和計算機視覺。另一項非凡的創新是生成對抗網絡（Generative adversarial Networks, GAN），這是一種生成模型，可以生成圖像或重構破壞性圖像，甚至提高圖像的分辨率。它也有許多衍生概念，如DGAN，CGAN等。
在我們的工作中，我們將基於深度學習生成一個解碼器(decoder)模型，通過CHER管理的三個邊緣輸入來預測聯合概率分布。主要架構將是一個身份塊和反卷積塊的集合，每個塊都包含反卷積層，並且兩者都連接為殘差結構以增加層次的深度。然後，我們可以通過解碼器模型重建聯合概率分布並量化其負值（非經典性）。
除了前面提到的例子，我們也能夠運用此模型來預測不同的維格納函數，例如諧振狀態、凝聚態以及貓態。在量子力學的領域中，維格納函數佔據了一個非常重要的地位，特別是其能夠在相空間中表現出量子態的能力。這是一個寶貴的工具，為我們提供了解量子世界的奇特性質的寶貴見解。
透過維格納函數對這些狀態進行數學表示，雖然嚴謹，但要實驗上驗證它們卻是繁重且耗時的。在量子力學的複雜世界中，實驗需要精確的控制與測量，這通常需要大量的時間、資源和技術專業知識。此外，隨著量子系統的維度增加，與維格納函數的數學評估相關的計算成本也會呈指數型增長。
這就是解碼器模型潛在的作用所在。這裡的想法是利用人工智慧和機器學習的力量，創建一個可以預測給定量子態的維格納函數的模型。這樣的模型可以幫助簡化計算和理解維格納函數的過程。通過從眾多的量子態及其對應的維格納函數中學習，該模型可以被訓練以精確預測新的、未見過的態的維格納函數。這將可能節省大量時間和資源，從而提高研究過程的效率。

Speaking of the quantum theory, it is an intriguing issue to classify the boundary between quantum and classical worlds, and the quantum correlation within the open system has arise our interests of analysing the dynamics. The properties which really capture the essence is called the classicality or the nonclassicality. Compared to the ones we familiar with, it has an alternative definition which characterize the dynamical process, wherein the strategy is formulated in terms of a Hamiltonian ensemble(HE). Additionally, according to the underlying Lie-algebraic structure, it can be recast into the Fourier-transform-on-group (FToG) formalism, as referred to canonical Hamiltonian ensemble representation (CHER), which we can analysis the dynamics in frequency domain. Finally, the whole process will be determined by the joint quasi distribution. By calculating the negativity of the distribution, we can clearly observe the dissipation of the quantum traits when the temperature goes up. However, the prediction of the joint quasi distribution turns out to be the formidable task with the numerical methods, so it is necessary to come up with an alternative method to compensate this deficiency.
With the development of the computer science and the enhancement of the hardware technology, deep learning has also ushered a great process. Deep learning is the concepted which based on the neural networks. It is a class of the algorithms that use multiple layers to progressively extract higher-level features from the raw input. The classical example is the Convolution Neuron Networks (CNN). CNN is mainly deal with the image processing, image recognition and computer vision. Another extraordinary invention is Generative adversarial Networks (GAN), which is a generating model that can generate images or reconstruct the destructive images, or even enhance the resolution of the images. It has also many derivate concepts like DGAN, CGAN, and so on.
In our work, we will generate a decoder model based on the deep learning to predict the joint quasi distribution by the input of the three marginals governed by the CHER. The main architecture will be a collection of the identity blocks and devolution blocks which each of them is consist of the deconvolution layers, and both are connected as a residual structure to increase the depth of the layers. After that we can rebuild the joint quasi distribution by the decoder model and quantify the negativity (nonclassicality) of it.
Despite the case aforementioned, it can also applied to predict different Wigner function like harmonic state, coherent state and cat state.In the realm of quantum mechanics, the Wigner function holds an integral position, particularly with its ability to allow for the representation of quantum states in a phase-space formulation. It is a vital tool that provides us with invaluable insights into the peculiarities of the quantum world.The mathematical representations of these states through the Wigner function, while rigorous, can be laborious and time-consuming to experimentally validate. In the complex world of quantum mechanics, experiments require precise control and measurement, which often demands vast amounts of time, resources, and technical expertise. Furthermore, as the dimensions of the quantum system increase, the computational costs associated with the mathematical evaluation of the Wigner function grow exponentially.
This is where the potential for a decoder model comes into play. The idea here is to leverage the power of artificial intelligence and machine learning to create a model that can predict the Wigner function of a given quantum state. Such a model could help to streamline the process of calculating and understanding the Wigner function. By learning from numerous examples of quantum states and their corresponding Wigner functions, the model could be trained to accurately predict the Wigner function for new, unseen states. This would potentially save a significant amount of time and resources, thus increasing the efficiency of the research process.

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