本論文探討了利用機器學習技術提升量子態斷層成像的方法，特別是通過量子態的協 方差矩陣表示。隨著量子感測在引力波觀測站中的應用，光學態的研究，特別是壓縮 態，成為科學界的焦點。此外，通過操縱光學量子位來實現量子信息協議的潛力，為 量子技術的未來帶來了希望。因此，對於在受控實驗環境中創建的態進行快速且準確 的斷層成像方法有著迫切的需求。
我們提出了一種監督學習模型，該模型將正交序列作為量子態的實驗照片輸入，並 估計協方差矩陣及其壓縮角。我們的方法能夠從單模協方差矩陣中完全恢復純度、壓 縮和反壓縮水平以及維格納函數，並且相比其他方法具有較高的保真度。使用實際數 據的估計結果成功地與其他估計結果相符，並且具有輕量但強大的模型優勢。
此外，這些結果激發了對雙模態斷層成像的研究，這在重建正交序列數據中的糾纏 信息方面具有高度興趣。我們提出了一種雙模態斷層成像的方法，該方法在未來實現 糾纏態的實驗中顯示出前景。
協方差矩陣結合適當的機器學習模型，已被證明是量子態精確單掃描測量的合適表 示。這一進展使我們更接近於實時量子態斷層成像，使量子態的創建和操縱在實驗環 境中成為可能並可實現。

The present thesis explores the enhancement of quantum state tomography using Ma- chine Learning techniques, specifically through the covariance matrix representation of a quantum state. With the implementation of quantum sensing in gravitational wave observatories, the study of optical states, particularly squeezed states, has become a focus of the scientific community. Additionally, the potential for implementing quantum information protocols through the manipulation of optical qubits holds promise for the future of quantum technologies. Consequently, there is a pressing need for fast and accurate methods to perform the tomography of states created in controlled experimental environments.
We propose a supervised learning model that inputs the quadrature sequence as an experimental photograph of the quantum state and estimates the covariance matrix along with the squeezing angle. Our method fully recovers purity, squeezing and anti-squeezing levels, and Wigner functions from the single-mode covariance matrix with high fidelity compared to other methods. Estimations using real data fit successfully with other estimations, with the added advantage of being a lightweight yet powerful model.
Furthermore, these results inspired the study of two-mode state tomography, which is of high interest as it aims to reconstruct entanglement information from quadrature sequence data. We proposed a method for two-mode tomography that shows promise for future experimental realizations of entangled states.
The covariance matrix, combined with an appropriate Machine Learning model, has proven to be a suitable representation of quantum states for achieving precise single-scan measurements. This advancement brings us closer to real-time quantum state tomography, making the creation and manipulation of quantum states feasible and realizable in an experimental environments.

Acknowledgements v Abstract vi 摘要 vii
1 Introduction 1
1.1 Nonclassicallight ............................... 2
1.1.1 Squeezedstates............................ 3
1.1.2 Two-modesqueezedoperator..................... 4
1.1.3 Homodynedetection ......................... 5
1.2 Gaussianquantuminformation ....................... 7
1.2.1 Symplecticstructureofthephasespace. . . . . . . . . . . . . . . 8 1.2.2 Covariancematrix .......................... 9
1.3 Quantumstatetomography ......................... 12
1.4 Problemstatement .............................. 14
1.5 Thesisstructure................................ 15
2 Quantum State Reconstruction 16
2.1 Homodynemeasurement ........................... 16
2.2 Matrixrepresentationofopticalquantumstates . . . . . . . . . . . . . . 19
2.3 SimulationofGaussianopticalstates .................... 20
2.4 Physicalconstraintsintrainingprocess................... 23
2.4.1 Densitymatrixdescription...................... 23
2.4.2 Covariancematrixdescription.................... 24
2.5 MachineLearningtomography........................ 26 2.5.1 NeuralNetworks(NN) ........................ 26
2.5.2 ConvolutionalNeuralNetworks(CNN) . . . . . . . . . . . . . . . 28
2.5.3 ResidualNetworks(ResNet)...................... 31
3 Single-Mode Reconstruction with ML 33
3.1 Covariancematrixcurves........................... 33 3.2 Trainingprocess................................ 36 3.3 Squeezing/anti-squeezingestimation..................... 37 3.4 Purity ..................................... 38 3.5 Reconstructionresults ............................ 39 3.6 BenchmarkingofCM-QST........................... 41 3.7 Commentsonthemodel ........................... 43
4 Two-Mode Reconstruction with ML 44
4.1 Covariancematrixrepresentation ...................... 44 4.2 Simulatingentangledstates ......................... 46 4.3 Trainingprocess ............................... 48 4.4 Entanglementreconstruction......................... 48 4.5 Reconstructing the 4-dimensional Wigner space . . . . . . . . . . . . . . 49 4.6 Commentsonthemodel ........................... 50
5 Conclusions 52
A Appendix 54
A.1 ArchitectureoftheCM-QSTmodel...................... 54 A.2 ArchitectureoftheCM-DMmodel ..................... 55 A.3 Architectureofthe2CM-QSTmodel .................... 56
References ..............................59

[1] S. Slussarenko and G. J. Pryde, “Photonic quantum information processing: A concise review,” Applied Physics Reviews, vol. 6, Dec 2019.
[2] U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Physica Scripta, vol. 91, Apr 2016.
[3] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys., vol. 84, pp. 621–669, May 2012.
[4] P. Fritschel, M. Evans, and V. Frolov, “Balanced homodyne readout for quantum limited gravitational wave detectors,” Opt. Express, vol. 22, pp. 4224–4234, Feb 2014.
[5] F. Arute and et al., “Quantum supremacy using a programmable superconducting processor,” Nature, vol. 574, pp. 505–510, Oct 2019.
[6] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010.
[7] M. O. Scully and M. S. Zubairy, Quantum Optics. Cambridge University Press, 1997.
[8] G. Badurek, Z. Hradil, A. Lvovsky, G. Molina-Teriza, H. Rauch, J. Řeháček, A. Vaziri, and M. Zawisky, 10 Maximum-Likelihood Estimationin Experimental Quantum Physics, pp. 373–414. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004.
[9] J. L. Bromberg, “The Birth of the Laser,” Physics Today, vol. 41, pp. 26–33, Oct 1988.
[10] R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev., vol. 130, pp. 2529–2539, Jun 1963.
[11] K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A, vol. 55, pp. 3117–3123, Apr 1997.
[12] A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods. CRC
Press, 2017.
[13] M. M. Wilde, “Scribe Notes: Lecture 09 - Gaussian Quantum Information,” 2019.
[14] M. M. Wilde, “Scribe Notes: Lecture 06 - Gaussian Quantum Information,” 2019.
[15] A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information. Napoli Series on physics and Astrophysics, Bibliopolis, 2005.
[16] M. M. Wilde, “Scribe Notes: Lecture 10 - Gaussian Quantum Information,” 2019.
[17] G. Cariolaro and R. Corvaja, “Implementation of Two-Mode Gaussian States Whose
Covariance Matrix Has the Standard Form,” Symmetry, vol. 14, Jul 2022.
[18] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion for
Continuous Variable Systems,” Phys. Rev. Lett., vol. 84, pp. 2722–2725, Mar 2000.
[19] H.-Y. Hsieh, Y.-R. Chen, H.-C. Wu, H. L. Chen, J. Ning, Y.-C. Huang, C.-M. Wu, and R.-K. Lee, “Extract the Degradation Information in Squeezed States with Machine Learning,” Phys. Rev. Lett., vol. 128, Feb 2022.
[20] H.-Y. Hsieh, J. Ning, Y.-R. Chen, H.-C. Wu, H. L. Chen, C.-M. Wu, and R.-K. Lee, “Direct Parameter Estimations from Machine Learning-Enhanced Quantum State
Tomography,” Symmetry, vol. 14, Apr 2022.
[21] S. Nolan, A. Smerzi, and L. Pezzè, “A machine learning approach to Bayesian
parameter estimation,” npj Quantum Information, vol. 7, p. 169, Dec 2021.
[22] R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, “Neural Ordinary
Differential Equations,” 2019.
[23] L. Dinh, J. Sohl-Dickstein, and S. Bengio, “Density estimation using Real NVP,” in
International Conference on Learning Representations, 2017.
[24] G. Adesso, S. Ragy, and A. R. Lee, “Continuous Variable Quantum Information: Gaussian States and Beyond,” Open Systems & Information Dynamics, vol. 21, no. 01n02, 2014.
[25] S. Ross, Simulation. Statistical Modeling and Decision Science, Elsevier Science, 2006.
[26] S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms. Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014.
[27] K. Schütt, S. Chmiela, O. von Lilienfeld, A. Tkatchenko, K. Tsuda, and K. Müller, Machine Learning Meets Quantum Physics. Lecture Notes in Physics, Springer International Publishing, 2020.
[28] R. Bracewell, The Fourier Transform and Its Applications. Circuits and systems, McGraw Hill, 2000.
[29] K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” 2015.
[30] S. Abdoli, P. Cardinal, and A. Lameiras Koerich, “End-to-end environmental sound classification using a 1D convolutional neural network,” Expert Systems with Applica- tions, vol. 136, pp. 252–263, 2019.
[31] J. J. Rodriguez-Andina, M. J. Moure, and M. D. Valdes, “Features, Design Tools, and Application Domains of FPGAs,” IEEE Transactions on Industrial Electronics, vol. 54, no. 4, pp. 1810–1823, 2007.
[32] C. Macchiavello, A. Riccardi, and M. F. Sacchi, “Quantum thermodynamics of two bosonic systems,” Phys. Rev. A, vol. 101, Jun 2020.