量子領域和經典領域之間的劃分仍然是關鍵的研究領域，維格納函數是表徵量子態的關鍵工具。建構維格納函數的傳統方法，例如斷層掃描測量、各種準分佈表示的變換以及逐點相空間掃描，本質上是勞動密集且耗時的。在處理量子過程的非經典性時，這些方法變得更加複雜，因為數據可能包含負值，使得標準數學方法變得不可靠。
深度學習是機器學習的子集，在處理複雜資料模式和做出準確預測方面顯示出巨大的前景。具體來說，深度學習中的解碼器模型通常由反捲積層、恆等區塊和殘差結構組成，對於從部分輸入重建詳細資訊非常有效。這些模型可以學習資料中複雜的關係，使其適合需要精確重建的應用，例如影像生成和訊號處理。
在我們的研究中，我們提出了一種創新方法，利用基於深度學習的解碼器模型從三個邊緣分佈預測維格納函數。此模型的架構包含恆等區塊和委託區塊，每個區塊都包含以殘差結構配置的反捲積層，從而增強了模型的深度和效能。這種設計使模型能夠有效地捕捉資料中的複雜特徵和關係，從而促進更準確的預測。
我們的方法使得重建維格納函數並測量其非經典性變得更加容易和快速。透過使用這種深度學習模型，我們減少了實驗工作量和所需時間。此方法也可用於各種量子態，如諧波態、相干態和貓態，顯示其靈活性和有效性。
該方法為量子態分析的傳統技術提供了一種實用且有效的替代方法，顯著提高了效率並減少了維格納函數構造中涉及的實驗工作。透過整合先進的深度學習技術，我們的目標是提高量子態分析的精度和可靠性，為更廣泛地理解和應用量子力學做出貢獻。

The delineation between quantum and classical realms remains a pivotal research area, with the Wigner function serving as a critical tool for characterizing quantum states. Traditional approaches to constructing Wigner functions, such as tomographic measurements, transformations from various quasi-distribution representations, and point-by-point phase space scans, are inherently labor-intensive and time-consuming. These methods become even more complicated when dealing with the nonclassicality of quantum processes because the data can include negative values, making standard mathematical methods unreliable.
Deep learning is a subset of machine learning, has shown great promise in handling complex data patterns and making accurate predictions. Specifically, decoder models in deep learning, which typically consist of deconvolution layers, identity blocks, and residual structures, are highly effective for reconstructing detailed information from partial inputs. These models can learn intricate relationships within data, making them suitable for applications requiring precise reconstructions, such as image generation and signal processing.
In our study, we propose an innovative method leveraging a deep learning-based decoder model to predict the Wigner function from three marginal distributions. The architecture of the model incorporates identity blocks and devolution blocks, each containing deconvolution layers configured in a residual structure, thereby enhancing the model's depth and performance. This design allows the model to effectively capture complex features and relationships within the data, facilitating more accurate predictions.
Our approach makes it easier and faster to reconstruct the Wigner function and measure its nonclassicality. By using this deep learning model, we reduce the amount of experimental work and time needed. This method can also be used for various quantum states, such as harmonic states, coherent states, and cat states, showing its flexibility and effectiveness.
This methodology provides a practical and effective alternative to traditional techniques for the analysis of quantum states, offering substantial improvements in efficiency and reducing the experimental effort involved in Wigner function construction. By integrating advanced deep learning techniques, we aim to enhance the precision and reliability of quantum state analysis, contributing to the broader understanding and application of quantum mechanics.

[1] W. B. Case, Am. J. Phys. 76, 937 (2008).
[2] U. Leonhardt, Phys. Rev. A 53, 2998 (1996).
[3] D. S. Dean, P. Le Doussal, S. N. Majumdar, and G. Schehr, Phys. Rev. A 97, 063614 (2018).
[4] G. Nogues, A. Rauschenbeutel, S. Osnaghi, P. Bertet, M. Brune, J. Raimond, S. Haroche, L. Lutterbach, and L. Davidovich, Phys. Rev. A 62, 054101 (2000).
[5] M. Benedict and A. Czirj’ak, Phys. Rev. A 60, 4034 (1999).
[6] J. Weinbub and D. Ferry, Appl. Phys. Rev. 5 (2018).
[7] C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
[8] D. Leibfried, T. Pfau, and C. Monroe, Phys. Today 51, 22 (1998).
[9] W. P. Schleich, G. S"ussmann, J. Philpott, M. C. Teich, and B. E. Saleh, Phys. Today 44, 146 (1991).
[10] R. A. Campos, Am. J. Phys. 67, 641 (1999).
[11] C. Gerry and P. Knight, Am. J. Phys. 65, 964 (1997).
[12] A. Ekert and P. Knight, Am. J. Phys. 57, 692 (1989).
[13] M. Raymer, Contemp. Phys. 38, 343 (1997).
[14] A. Kenfack and K. .Zyczkowski, J. Opt. B: Quantum Semiclassical Opt. 6, 396 (2004).
[15] D. Smithey, M. Beck, J. Cooper, M. Raymer, and A. Faridani, Phys. Scr. 1993, 35 (1993).
[16] K. Banaszek, C. Radzewicz, K. W’odkiewicz, and J. Krasi’nski, Phys. Rev. A 60, 674 (1999).
[17] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J.-M. Raimond, and S. Haroche, Phys. Rev. Lett. 89, 200402 (2002).
[18] D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[19] F.-R. Winkelmann, C. A. Weidner, G. Ramola, W. Alt, D. Meschede, and A. Al- berti, J. Phys. B: At. Mol. Opt. Phys. 55, 194004 (2022).
[20] R. Salakhutdinov, Annu. Rev. Stat. Its Appl. 2, 361 (2015).
[21] E. Nalisnick, A. Matsukawa, Y. W. Teh, D. Gorur, and B. Lakshminarayanan, arXiv preprint arXiv:1810.09136 (2018).
[22] Z. Hu, Z. Yang, R. Salakhutdinov, and E. P. Xing, arXiv preprint arXiv:1706.00550 (2017).
[23] S. Ahmed, C. S. Munoz, F. Nori, and A. F. Kockum, Phys. Rev. Res. 3, 033278 (2021).
[24] S. Ahmed, C. S. Mu noz, F. Nori, and A. F. Kockum, Phys. Rev. Lett. 127, 140502 (2021).
[25] E. Y. Zhu, S. Johri, D. Bacon, M. Esencan, J. Kim, M. Muir, N. Murgai, J. Nguyen, N. Pisenti, A. Schouela, et al., Phys. Rev. Res. 4, 043092 (2022).
[26] S. Chen, O. Savchuk, S. Zheng, B. Chen, H. Stoecker, L. Wang, and K. Zhou, Phys. Rev. D 107, 056001 (2023).
[27] M. Bild, M. Fadel, Y. Yang, U. von L"upke, P. Martin, A. Bruno, and Y. Chu, Science 380, 274 (2023).
[28] M. Kim, F. De Oliveira, and P. Knight, Phys. Rev. A 40, 2494 (1989).
[29] J. Song and H.-y. Fan, Int. J. Theor. Phys. 51, 229 (2012).
[30] T. Kiesel, W. Vogel, B. Hage, and R. Schnabel, Phys. Rev. Lett. 107, 113604(2011).
[31] T. Kiesel, W. Vogel, B. Hage, J. DiGuglielmo, A. Samblowski, and R. Schnabel Phys. Rev. A 79, 022122 (2009).
[32] M. C. Teich and B. E. Saleh, Phys. Today 43, 26 (1990).
[33] Z. Zhang, L. Shao, W. Lu, and X. Wang, Phys. Rev. A 106, 043721 (2022).
[34] D. S. Schlegel, F. Minganti, and V. Savona, Phys. Rev. A 106, 022431 (2022).
[35] M. Xiang-Guo, W. Ji-Suo, L. Bao-Long, and L. Hong-Qi, Chin. Phys. B 17, 1791 (2008).
[36] J. Etesse, M. Bouillard, B. Kanseri, and R. Tualle-Brouri, Phys. Rev. Lett. 114, 193602 (2015).
[37] Z. Yang, C.-H. Yu, and M. J. Buehler, Sci. Adv. 7, eabd7416 (2021).
[38] V. Dumoulin and F. Visin, arXiv preprint arXiv:1603.07285 (2016).
[39] K. He, X. Zhang, S. Ren, and J. Sun, in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. (2016) pp. 770–778.
[40] H.-B. Chen, C. Gneiting, P.-Y. Lo, Y.-N. Chen, and F. Nori, Phys. Rev. Lett. 120, 030403 (2018).
[41] H.-B. Chen, P.-Y. Lo, C. Gneiting, J. Bae, Y.-N. Chen, and F. Nori, Nat. Commun. 10, 3794 (2019).
[42] H.-B. Chen and Y.-N. Chen, Sci. Rep. 11, 10046 (2021).
[43] C. M. Kropf, C. Gneiting, and A. Buchleitner, Phys. Rev. X 6, 031023 (2016).
[44] H.-B. Chen, Sci. Rep. 12, 2869 (2022).
[45] H.-B. Chen, C. Gneiting, P.-Y. Lo, Y.-N. Chen, and F. Nori, Phys. Rev. Lett. 120, 030403 (2018).
[46] H.-B. Chen, P.-Y. Lo, C. Gneiting, J. Bae, Y.-N. Chen, and F. Nori, Nat. Commun. 10, 3794 (2019).
[47] H.-B. Chen and Y.-N. Chen, Sci. Rep. 11, 10046 (2021).
[48] E. Wigner, Phys. Rev. 40, 749 (1932).
[49] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[50] Y. Kim and E. P. Wigner, Am. J. Phys. 58, 439 (1990).
[51] J. Snygg, Am. J. Phys. 45, 58 (1977).
[52] J. Snygg, Am. J. Phys. 48, 964 (1980).
[53] N. Mukunda, Am. J. Phys. 47, 182 (1979).
[54] S. Stenholm, Eur. J. Phys 1, 244 (1980).
[55] G. Mourgues, J. Andrieux, and M. Feix, Eur. J. Phys 5, 112 (1984).
[56] M. Casas, H. Krivine, and J. Martorell, Eur. J. Phys 12, 105 (1991).
[57] M. Belloni, M. Doncheski, and R. W. Robinett, Am. J. Phys. 72, 1183 (2004).
[58] H.-W. Lee, Am. J. Phys. 50, 438 (1982).
[59] W. H. Zurek, Phys. Today 44, 36 (1991).