在物理理論中，區分古典力學與量子力學的差異一直是廣泛探討的議題。在量子力學中，漢米頓系綜（Hamiltonian Ensemble）是一種用於描述量子狀態的工具，可用以計算量子狀態隨時間的演化行為。透過李代數（Lie Algebra）的數學結構，配合群傅立葉轉換（Fourier Transform on Group，簡稱 FToG）形式主義，可以將量子狀態轉換為聯合擬機率分佈（joint quasi-distribution）。此方法稱為「規範漢米頓系綜表徵」（Canonical Hamiltonian Ensemble Representation，簡稱 CHER），能將量子動力學從時域映射到頻域，更清楚地揭示其中的非古典性特徵，是量子理論研究中常用的數學工具之一。
然而，這種方法在計算聯合擬機率分佈時經常會遇到 NP-困難（NP-hard）問題，使得無法有效獲得真實標籤資料，產生所謂「真實值不足問題」（Ground Truth Deficiency）。為了解決此挑戰，本研究引入深度學習與人工智慧的技術，設計出一種深度生成模型（Deep Generative Model，簡稱 DGM）。本模型採用卷積神經網路（Convolutional Neural Network，CNN）架構，並由數個恆等區塊（identity block）與反卷積區塊（deconvolution block）所構成，其功能分別為圖像特徵的提取與圖像解析度的放大。透過訓練良好的深度生成模型，僅需輸入三個邊際分佈（marginals），即可預測完整的聯合擬機率分佈，進而大幅降低傳統方法所需的時間與實驗資源。
此外，模型所預測之圖像中的負值區域，亦可用來識別量子動力學中的非古典性。為驗證模型效能，我們將本技術應用於延伸自旋玻色子模型（extended spin-boson model）的模擬中，針對兩種不同的頻譜密度函數（spectral density），包括超歐姆模型（super-Ohmic model）與德魯德-洛倫茲模型（Drude--Lorentz model），探討在不同溫度與參數條件下，量子非古典性表現的變化情形。為了提高預測準確度，我們亦針對模型參數與訓練資料進行多組調整與測試，統整出目前模型在不同情境下的適用範圍。
另一方面，本研究亦擴展深度生成模型的應用至維格納函數（Wigner function）的預測，該函數為描述量子態在相空間中之表徵。相較於傳統計算雙變數聯合擬機率分佈的低效率，我們的方法僅需輸入三個邊際分佈，即可進行快速且有效的預測。此模型已可成功模擬諸如簡諧態（harmonic state）、相干態（coherent state）與貓態（cat state）等常見量子態。為貼近實際實驗情境，我們亦於資料中納入退相干效應（decoherence effect）與熱噪聲模型（thermal noise model）等相關參數，使預測結果能更廣泛應用於真實量子系統。
綜上所述，本研究展示了深度生成模型於量子動力學非古典性分析上的潛力，有效提升了分析效率與解析度，並為理論與實驗量子力學的發展提供了一個創新的數學工具。

In physics theory, the distinction between classical and quantum mechanics has been a subject of extensive discussion. In quantum mechanics, the Hamiltonian Ensemble (HE) provides a framework for describing the statistical properties and time evolution of quantum states, particularly in open quantum systems. By leveraging the structure of Lie algebra and applying the Fourier-transform-on-group (FToG) formalism, quantum dynamics can be represented through a joint quasi-probability distribution. This method, known as the Canonical Hamiltonian Ensemble Representation (CHER), facilitates the transformation of quantum dynamics from time domain to frequency domain. CHER has become a widely used tool in quantum mechanics research, which can represent the nonclassicality according to the negative regions of the distributions.
However, when computing the joint quasi-probability distribution often faces challenges related to NP-hard problems, particularly the issue of ground truth deficiency. To address this, we developed a deep generative model (DGM) powered by advancements in deep learning and artificial intelligence. The architecture of the model employs Convolutional Neural Networks (CNNs) and consists of several identity blocks and deconvolution blocks to perform sampling and layer expansion. With the assistance of this well-trained DGM, we can reconstruct the full joint quasi-probability distribution from three marginals, significantly reducing the experimental cost in terms of time and resources. The predicted negative regions of the reconstructed distribution can then be used to identify the nonclassicality in quantum dynamics.
In our work, we applied this technique to predict the dynamics of an extended spin-boson model with two spectral density models including super-Ohmic model and Drude--Lorentz model. For the purpose of analyzing the changes of nonclassicality under different conditions, including variations in temperature and parameters. To ensure accuracy, we generated multiple sets of distributions under various parameter configurations, systematically evaluating the predictive range of the model.
Additionally, the DGM can be utilized to predict the Wigner function, a phase-space representation of quantum states. Compared to traditional methods, which are computationally inefficient for obtaining bivariate joint quasi-probability distributions, this approach requires only three marginals to make accurate predictions. Common quantum states, such as the harmonic state, coherent state, and cat state, can all be effectively modeled. Taking practical experimental conditions into account, we incorporated parameters related to decoherence effects and thermal noise models to broaden the applicability of the results.
This work demonstrates the potential of DGM in efficiently analyzing quantum dynamics and its nonclassicality characteristics, offering significant advancements in both theoretical and experimental quantum mechanics research.

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